(*^ ::[paletteColors = 128; automaticGrouping; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = input, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12;] :[font = title; inactive; preserveAspect; startGroup; ] Schauder Basis for C[0,1] :[font = subtitle; inactive; preserveAspect; ] From Meyer's Wavelets - Algorithms & Applications SIAM, 1993, Pages 15-17 ;[s] 7:0,0;14,1;50,2;67,3;68,4;69,5;72,6;75,-1; 7:1,16,12,Times,1,18,0,0,0;1,16,12,Times,2,18,0,0,0;1,16,12,Times,1,18,0,0,0;1,16,12,Times,1,18,0,0,0;1,16,12,Times,1,18,0,0,0;1,16,12,Times,1,18,0,0,0;1,16,12,Times,1,18,0,0,0; :[font = subsubtitle; inactive; preserveAspect; ] Jack K. Cohen Colorado School of Mines, 1993 jkc@keller.mines.colorado.edu :[font = input; preserveAspect; ] <=1 by setting n = 2^j + k (0 <= k < 2^j) and using j and k to define a scaled and shifted triangle. Here is the second `octave': :[font = input; preserveAspect; startGroup; ] Plot[Evaluate@N@ {SchauderTriangle[2, x], SchauderTriangle[3, x]}, {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.588604 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(0.2)] 0.01131 0.13244 1 0 Msboxa [(0.4)] 0.01131 0.25016 1 0 Msboxa [(0.6)] 0.01131 0.36788 1 0 Msboxa [(0.8)] 0.01131 0.4856 1 0 Msboxa [(1)] 0.01131 0.60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.01472 moveto 0.17619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.01472 moveto 0.25238 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.01472 moveto 0.29048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.01472 moveto 0.32857 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.01472 moveto 0.36667 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.01472 moveto 0.44286 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.01472 moveto 0.48095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.01472 moveto 0.51905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.01472 moveto 0.55714 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.01472 moveto 0.63333 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.01472 moveto 0.67143 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.01472 moveto 0.70952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.13244 moveto 0.03006 0.13244 lineto stroke grestore [(0.2)] 0.01131 0.13244 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.25016 moveto 0.03006 0.25016 lineto stroke grestore [(0.4)] 0.01131 0.25016 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.36788 moveto 0.03006 0.36788 lineto stroke grestore [(0.6)] 0.01131 0.36788 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.4856 moveto 0.03006 0.4856 lineto stroke grestore [(0.8)] 0.01131 0.4856 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.60332 moveto 0.03006 0.60332 lineto stroke grestore [(1)] 0.01131 0.60332 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.03826 moveto 0.02756 0.03826 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0618 moveto 0.02756 0.0618 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.08535 moveto 0.02756 0.08535 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10889 moveto 0.02756 0.10889 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.15598 moveto 0.02756 0.15598 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.17952 moveto 0.02756 0.17952 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20307 moveto 0.02756 0.20307 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.22661 moveto 0.02756 0.22661 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.2737 moveto 0.02756 0.2737 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29724 moveto 0.02756 0.29724 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32079 moveto 0.02756 0.32079 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.34433 moveto 0.02756 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0.06349 0.01472 lineto 0.10317 0.01472 lineto 0.14286 0.01472 lineto 0.18254 0.01472 lineto 0.22222 0.01472 lineto 0.2619 0.01472 lineto 0.30159 0.01472 lineto 0.34127 0.01472 lineto 0.38095 0.01472 lineto 0.42063 0.01472 lineto 0.46032 0.01472 lineto 0.48016 0.01472 lineto 0.49008 0.01472 lineto 0.49504 0.01472 lineto 0.49752 0.01472 lineto 0.49876 0.01472 lineto 0.5 0.01472 lineto 0.50248 0.02085 lineto 0.50496 0.02698 lineto 0.50992 0.03924 lineto 0.51984 0.06377 lineto 0.53968 0.11282 lineto 0.57937 0.21092 lineto 0.61905 0.30902 lineto 0.65873 0.40712 lineto 0.69841 0.50522 lineto 0.71825 0.55427 lineto 0.72817 0.57879 lineto 0.73313 0.59106 lineto 0.73562 0.59719 lineto 0.73686 0.60025 lineto 0.7381 0.60332 lineto 0.74058 0.59719 lineto 0.74306 0.59106 lineto 0.74802 0.57879 lineto 0.75794 0.55427 lineto 0.77778 0.50522 lineto 0.81746 0.40712 lineto 0.85714 0.30902 lineto 0.89683 0.21092 lineto 0.93651 0.11282 lineto 0.97619 0.01472 lineto stroke grestore grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; preserveAspect; ] Here's a plotting module that generates the m^th octave: :[font = input; preserveAspect; ] PlotSchauderOctave[m_] := Module[{string, label, octave}, string = StringForm["Schauder Octave: m = `1`", m]; label = FontForm[string, {"Helvetica-Bold", 16}]; octave = Array[Hold@SchauderTriangle[#, x]&, 2^(m-1), 2^(m-1)]; Plot[Evaluate@N@Release@octave, {x, 0, 1}, DefaultFont -> {"Courier", 12}, PlotLabel -> label ] ] :[font = input; preserveAspect; startGroup; ] PlotSchauderOctave[3] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 12 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.588604 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa gsave /Helvetica-Bold findfont 16 scalefont setfont [(Schauder Octave: m = 3)] 0.5 0.61803 0 -2 Msboxa grestore [(0.2)] 0.01131 0.13244 1 0 Msboxa [(0.4)] 0.01131 0.25016 1 0 Msboxa [(0.6)] 0.01131 0.36788 1 0 Msboxa [(0.8)] 0.01131 0.4856 1 0 Msboxa [(1)] 0.01131 0.60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.01472 moveto 0.17619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.01472 moveto 0.25238 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.01472 moveto 0.29048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.01472 moveto 0.32857 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.01472 moveto 0.36667 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.01472 moveto 0.44286 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.01472 moveto 0.48095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.01472 moveto 0.51905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.01472 moveto 0.55714 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.01472 moveto 0.63333 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.01472 moveto 0.67143 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.01472 moveto 0.70952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave /Helvetica-Bold findfont 16 scalefont setfont [(Schauder Octave: m = 3)] 0.5 0.61803 0 -2 Mshowa grestore gsave 0.002 setlinewidth 0.02381 0.13244 moveto 0.03006 0.13244 lineto stroke grestore [(0.2)] 0.01131 0.13244 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.25016 moveto 0.03006 0.25016 lineto stroke grestore [(0.4)] 0.01131 0.25016 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.36788 moveto 0.03006 0.36788 lineto stroke grestore [(0.6)] 0.01131 0.36788 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.4856 moveto 0.03006 0.4856 lineto stroke grestore [(0.8)] 0.01131 0.4856 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.60332 moveto 0.03006 0.60332 lineto stroke grestore [(1)] 0.01131 0.60332 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.03826 moveto 0.02756 0.03826 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0618 moveto 0.02756 0.0618 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.08535 moveto 0.02756 0.08535 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10889 moveto 0.02756 0.10889 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.15598 moveto 0.02756 0.15598 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.17952 moveto 0.02756 0.17952 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20307 moveto 0.02756 0.20307 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.22661 moveto 0.02756 0.22661 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.2737 moveto 0.02756 0.2737 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29724 moveto 0.02756 0.29724 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32079 moveto 0.02756 0.32079 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.34433 moveto 0.02756 0.34433 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.39142 moveto 0.02756 0.39142 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.41497 moveto 0.02756 0.41497 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43851 moveto 0.02756 0.43851 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.46205 moveto 0.02756 0.46205 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.50914 moveto 0.02756 0.50914 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53269 moveto 0.02756 0.53269 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55623 moveto 0.02756 0.55623 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.57977 moveto 0.02756 0.57977 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.21092 lineto 0.10317 0.40712 lineto 0.12302 0.50522 lineto 0.13294 0.55427 lineto 0.1379 0.57879 lineto 0.14038 0.59106 lineto 0.14162 0.59719 lineto 0.14286 0.60332 lineto 0.14534 0.59106 lineto 0.14782 0.57879 lineto 0.15278 0.55427 lineto 0.1627 0.50522 lineto 0.18254 0.40712 lineto 0.22222 0.21092 lineto 0.24206 0.11282 lineto 0.25198 0.06377 lineto 0.25694 0.03924 lineto 0.25942 0.02698 lineto 0.26066 0.02085 lineto 0.2619 0.01472 lineto 0.26438 0.01472 lineto 0.26687 0.01472 lineto 0.27183 0.01472 lineto 0.28175 0.01472 lineto 0.30159 0.01472 lineto 0.34127 0.01472 lineto 0.38095 0.01472 lineto 0.42063 0.01472 lineto 0.46032 0.01472 lineto 0.5 0.01472 lineto 0.53968 0.01472 lineto 0.57937 0.01472 lineto 0.61905 0.01472 lineto 0.65873 0.01472 lineto 0.69841 0.01472 lineto 0.7381 0.01472 lineto 0.77778 0.01472 lineto 0.81746 0.01472 lineto 0.85714 0.01472 lineto 0.89683 0.01472 lineto 0.93651 0.01472 lineto 0.97619 0.01472 lineto stroke grestore grestore gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.01472 lineto 0.10317 0.01472 lineto 0.14286 0.01472 lineto 0.18254 0.01472 lineto 0.22222 0.01472 lineto 0.24206 0.01472 lineto 0.25198 0.01472 lineto 0.25694 0.01472 lineto 0.25942 0.01472 lineto 0.26066 0.01472 lineto 0.2619 0.01472 lineto 0.26438 0.02698 lineto 0.26687 0.03924 lineto 0.27183 0.06377 lineto 0.28175 0.11282 lineto 0.30159 0.21092 lineto 0.34127 0.40712 lineto 0.36111 0.50522 lineto 0.37103 0.55427 lineto 0.37599 0.57879 lineto 0.37847 0.59106 lineto 0.37971 0.59719 lineto 0.38095 0.60332 lineto 0.38343 0.59106 lineto 0.38591 0.57879 lineto 0.39087 0.55427 lineto 0.40079 0.50522 lineto 0.42063 0.40712 lineto 0.46032 0.21092 lineto 0.48016 0.11282 lineto 0.49008 0.06377 lineto 0.49504 0.03924 lineto 0.49752 0.02698 lineto 0.49876 0.02085 lineto 0.5 0.01472 lineto 0.50248 0.01472 lineto 0.50496 0.01472 lineto 0.50992 0.01472 lineto 0.51984 0.01472 lineto 0.53968 0.01472 lineto 0.57937 0.01472 lineto 0.61905 0.01472 lineto 0.65873 0.01472 lineto 0.69841 0.01472 lineto 0.7381 0.01472 lineto 0.77778 0.01472 lineto 0.81746 0.01472 lineto 0.85714 0.01472 lineto 0.89683 0.01472 lineto Mistroke 0.93651 0.01472 lineto 0.97619 0.01472 lineto Mfstroke grestore grestore gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.01472 lineto 0.10317 0.01472 lineto 0.14286 0.01472 lineto 0.18254 0.01472 lineto 0.22222 0.01472 lineto 0.2619 0.01472 lineto 0.30159 0.01472 lineto 0.34127 0.01472 lineto 0.38095 0.01472 lineto 0.42063 0.01472 lineto 0.46032 0.01472 lineto 0.48016 0.01472 lineto 0.49008 0.01472 lineto 0.49504 0.01472 lineto 0.49752 0.01472 lineto 0.49876 0.01472 lineto 0.5 0.01472 lineto 0.50248 0.02698 lineto 0.50496 0.03924 lineto 0.50992 0.06377 lineto 0.51984 0.11282 lineto 0.53968 0.21092 lineto 0.57937 0.40712 lineto 0.59921 0.50522 lineto 0.60913 0.55427 lineto 0.61409 0.57879 lineto 0.61657 0.59106 lineto 0.61781 0.59719 lineto 0.61905 0.60332 lineto 0.62153 0.59106 lineto 0.62401 0.57879 lineto 0.62897 0.55427 lineto 0.63889 0.50522 lineto 0.65873 0.40712 lineto 0.69841 0.21092 lineto 0.71825 0.11282 lineto 0.72817 0.06377 lineto 0.73313 0.03924 lineto 0.73562 0.02698 lineto 0.73686 0.02085 lineto 0.7381 0.01472 lineto 0.74058 0.01472 lineto 0.74306 0.01472 lineto 0.74802 0.01472 lineto 0.75794 0.01472 lineto 0.77778 0.01472 lineto 0.81746 0.01472 lineto 0.85714 0.01472 lineto 0.89683 0.01472 lineto Mistroke 0.93651 0.01472 lineto 0.97619 0.01472 lineto Mfstroke grestore grestore gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.01472 lineto 0.10317 0.01472 lineto 0.14286 0.01472 lineto 0.18254 0.01472 lineto 0.22222 0.01472 lineto 0.2619 0.01472 lineto 0.30159 0.01472 lineto 0.34127 0.01472 lineto 0.38095 0.01472 lineto 0.42063 0.01472 lineto 0.46032 0.01472 lineto 0.5 0.01472 lineto 0.53968 0.01472 lineto 0.57937 0.01472 lineto 0.61905 0.01472 lineto 0.65873 0.01472 lineto 0.69841 0.01472 lineto 0.71825 0.01472 lineto 0.72817 0.01472 lineto 0.73313 0.01472 lineto 0.73562 0.01472 lineto 0.73686 0.01472 lineto 0.7381 0.01472 lineto 0.74058 0.02698 lineto 0.74306 0.03924 lineto 0.74802 0.06377 lineto 0.75794 0.11282 lineto 0.77778 0.21092 lineto 0.81746 0.40712 lineto 0.8373 0.50522 lineto 0.84722 0.55427 lineto 0.85218 0.57879 lineto 0.85466 0.59106 lineto 0.8559 0.59719 lineto 0.85714 0.60332 lineto 0.85962 0.59106 lineto 0.8621 0.57879 lineto 0.86706 0.55427 lineto 0.87698 0.50522 lineto 0.89683 0.40712 lineto 0.93651 0.21092 lineto 0.97619 0.01472 lineto stroke grestore grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; preserveAspect; ] Here are the two exceptional basis functions: :[font = input; preserveAspect; startGroup; ] Plot[Evaluate@N@SchauderTriangle[-1, x], {x, 0, 1}, PlotRange -> {0, 1.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 1.94614e-17 0.561849 [ [(0)] 0.02381 0 0 2 Msboxa [(0.2)] 0.21429 0 0 2 Msboxa [(0.4)] 0.40476 0 0 2 Msboxa [(0.6)] 0.59524 0 0 2 Msboxa [(0.8)] 0.78571 0 0 2 Msboxa [(1)] 0.97619 0 0 2 Msboxa [(0.2)] 0.01131 0.11237 1 0 Msboxa [(0.4)] 0.01131 0.22474 1 0 Msboxa [(0.6)] 0.01131 0.33711 1 0 Msboxa [(0.8)] 0.01131 0.44948 1 0 Msboxa [(1)] 0.01131 0.56185 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.00625 lineto stroke grestore [(0)] 0.02381 0 0 2 Mshowa gsave 0.002 setlinewidth 0.21429 0 moveto 0.21429 0.00625 lineto stroke grestore [(0.2)] 0.21429 0 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0 moveto 0.40476 0.00625 lineto stroke grestore [(0.4)] 0.40476 0 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0 moveto 0.59524 0.00625 lineto stroke grestore [(0.6)] 0.59524 0 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0 moveto 0.78571 0.00625 lineto stroke grestore [(0.8)] 0.78571 0 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0 moveto 0.97619 0.00625 lineto stroke grestore [(1)] 0.97619 0 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0 moveto 0.0619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0 moveto 0.1 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0 moveto 0.1381 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0 moveto 0.17619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0 moveto 0.25238 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0 moveto 0.29048 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0 moveto 0.32857 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0 moveto 0.36667 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0 moveto 0.44286 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0 moveto 0.48095 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0 moveto 0.51905 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0 moveto 0.55714 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0 moveto 0.63333 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0 moveto 0.67143 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0 moveto 0.70952 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0 moveto 0.74762 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0 moveto 0.82381 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0 moveto 0.8619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0 moveto 0.9 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0 moveto 0.9381 0.00375 lineto stroke grestore gsave 0.002 setlinewidth 0 0 moveto 1 0 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.11237 moveto 0.03006 0.11237 lineto stroke grestore [(0.2)] 0.01131 0.11237 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.22474 moveto 0.03006 0.22474 lineto stroke grestore [(0.4)] 0.01131 0.22474 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.33711 moveto 0.03006 0.33711 lineto stroke grestore [(0.6)] 0.01131 0.33711 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.44948 moveto 0.03006 0.44948 lineto stroke grestore [(0.8)] 0.01131 0.44948 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.56185 moveto 0.03006 0.56185 lineto stroke grestore [(1)] 0.01131 0.56185 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.02247 moveto 0.02756 0.02247 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.04495 moveto 0.02756 0.04495 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.06742 moveto 0.02756 0.06742 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0899 moveto 0.02756 0.0899 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13484 moveto 0.02756 0.13484 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.15732 moveto 0.02756 0.15732 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.17979 moveto 0.02756 0.17979 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20227 moveto 0.02756 0.20227 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24721 moveto 0.02756 0.24721 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.26969 moveto 0.02756 0.26969 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29216 moveto 0.02756 0.29216 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.31464 moveto 0.02756 0.31464 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.35958 moveto 0.02756 0.35958 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.38206 moveto 0.02756 0.38206 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40453 moveto 0.02756 0.40453 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42701 moveto 0.02756 0.42701 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.47195 moveto 0.02756 0.47195 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49443 moveto 0.02756 0.49443 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.5169 moveto 0.02756 0.5169 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53938 moveto 0.02756 0.53938 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58432 moveto 0.02756 0.58432 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.6068 moveto 0.02756 0.6068 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave 0.004 setlinewidth 0.02381 0.56185 moveto 0.06349 0.56185 lineto 0.10317 0.56185 lineto 0.14286 0.56185 lineto 0.18254 0.56185 lineto 0.22222 0.56185 lineto 0.2619 0.56185 lineto 0.30159 0.56185 lineto 0.34127 0.56185 lineto 0.38095 0.56185 lineto 0.42063 0.56185 lineto 0.46032 0.56185 lineto 0.5 0.56185 lineto 0.53968 0.56185 lineto 0.57937 0.56185 lineto 0.61905 0.56185 lineto 0.65873 0.56185 lineto 0.69841 0.56185 lineto 0.7381 0.56185 lineto 0.77778 0.56185 lineto 0.81746 0.56185 lineto 0.85714 0.56185 lineto 0.89683 0.56185 lineto 0.93651 0.56185 lineto 0.97619 0.56185 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup; ] Plot[Evaluate@N@SchauderTriangle[0, x], {x, 0, 1}, PlotRange -> {0, 1.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 1.94614e-17 0.561849 [ [(0)] 0.02381 0 0 2 Msboxa [(0.2)] 0.21429 0 0 2 Msboxa [(0.4)] 0.40476 0 0 2 Msboxa [(0.6)] 0.59524 0 0 2 Msboxa [(0.8)] 0.78571 0 0 2 Msboxa [(1)] 0.97619 0 0 2 Msboxa [(0.2)] 0.01131 0.11237 1 0 Msboxa [(0.4)] 0.01131 0.22474 1 0 Msboxa [(0.6)] 0.01131 0.33711 1 0 Msboxa [(0.8)] 0.01131 0.44948 1 0 Msboxa [(1)] 0.01131 0.56185 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.00625 lineto stroke grestore [(0)] 0.02381 0 0 2 Mshowa gsave 0.002 setlinewidth 0.21429 0 moveto 0.21429 0.00625 lineto stroke grestore [(0.2)] 0.21429 0 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0 moveto 0.40476 0.00625 lineto stroke grestore [(0.4)] 0.40476 0 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0 moveto 0.59524 0.00625 lineto stroke grestore [(0.6)] 0.59524 0 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0 moveto 0.78571 0.00625 lineto stroke grestore [(0.8)] 0.78571 0 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0 moveto 0.97619 0.00625 lineto stroke grestore [(1)] 0.97619 0 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0 moveto 0.0619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0 moveto 0.1 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0 moveto 0.1381 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0 moveto 0.17619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0 moveto 0.25238 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0 moveto 0.29048 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0 moveto 0.32857 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0 moveto 0.36667 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0 moveto 0.44286 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0 moveto 0.48095 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0 moveto 0.51905 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0 moveto 0.55714 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0 moveto 0.63333 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0 moveto 0.67143 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0 moveto 0.70952 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0 moveto 0.74762 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0 moveto 0.82381 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0 moveto 0.8619 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0 moveto 0.9 0.00375 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0 moveto 0.9381 0.00375 lineto stroke grestore gsave 0.002 setlinewidth 0 0 moveto 1 0 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.11237 moveto 0.03006 0.11237 lineto stroke grestore [(0.2)] 0.01131 0.11237 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.22474 moveto 0.03006 0.22474 lineto stroke grestore [(0.4)] 0.01131 0.22474 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.33711 moveto 0.03006 0.33711 lineto stroke grestore [(0.6)] 0.01131 0.33711 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.44948 moveto 0.03006 0.44948 lineto stroke grestore [(0.8)] 0.01131 0.44948 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.56185 moveto 0.03006 0.56185 lineto stroke grestore [(1)] 0.01131 0.56185 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.02247 moveto 0.02756 0.02247 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.04495 moveto 0.02756 0.04495 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.06742 moveto 0.02756 0.06742 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0899 moveto 0.02756 0.0899 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13484 moveto 0.02756 0.13484 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.15732 moveto 0.02756 0.15732 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.17979 moveto 0.02756 0.17979 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20227 moveto 0.02756 0.20227 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24721 moveto 0.02756 0.24721 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.26969 moveto 0.02756 0.26969 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29216 moveto 0.02756 0.29216 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.31464 moveto 0.02756 0.31464 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.35958 moveto 0.02756 0.35958 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.38206 moveto 0.02756 0.38206 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40453 moveto 0.02756 0.40453 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42701 moveto 0.02756 0.42701 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.47195 moveto 0.02756 0.47195 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49443 moveto 0.02756 0.49443 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.5169 moveto 0.02756 0.5169 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.53938 moveto 0.02756 0.53938 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58432 moveto 0.02756 0.58432 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.6068 moveto 0.02756 0.6068 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave 0.004 setlinewidth 0.02381 0 moveto 0.06349 0.02341 lineto 0.10317 0.04682 lineto 0.14286 0.07023 lineto 0.18254 0.09364 lineto 0.22222 0.11705 lineto 0.2619 0.14046 lineto 0.30159 0.16387 lineto 0.34127 0.18728 lineto 0.38095 0.21069 lineto 0.42063 0.2341 lineto 0.46032 0.25751 lineto 0.5 0.28092 lineto 0.53968 0.30433 lineto 0.57937 0.32775 lineto 0.61905 0.35116 lineto 0.65873 0.37457 lineto 0.69841 0.39798 lineto 0.7381 0.42139 lineto 0.77778 0.4448 lineto 0.81746 0.46821 lineto 0.85714 0.49162 lineto 0.89683 0.51503 lineto 0.93651 0.53844 lineto 0.97619 0.56185 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] SchauderCoefficient Usage :[font = input; preserveAspect; startGroup; ] f[x_] := Exp[x]; Table[N@SchauderCoefficient[f[x], {x, n}], {n, -1, 7}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] {1., 1.718281828459045, -0.2104196435293943, -0.04033521866232271, -0.06650153296691208, -0.00886425527704438, -0.01138192907573376, -0.01461468622417961, -0.01876562856876296} ;[o] {1., 1.71828, -0.21042, -0.0403352, -0.0665015, -0.00886426, -0.0113819, -0.0146147, -0.0187656} :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Schauder Approximant Usage :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup; ] f[x] = Exp[x] :[font = input; preserveAspect; ] f[x_] := Exp[x] :[font = text; inactive; preserveAspect; ] You can print out the approximation function, but it's not very illuminating: :[font = input; preserveAspect; startGroup; ] SchauderApproximant[f[x], {x, 0}] :[font = output; output; inactive; preserveAspect; endGroup; ] Which[0 <= x <= 1, 1, True, 0] + (-1 + E)*Which[0 <= x <= 1, x, True, 0] ;[o] Which[0 <= x <= 1, 1, True, 0] + (-1 + E) Which[0 <= x <= 1, x, True, 0] :[font = text; inactive; preserveAspect; ] Hey, for some reason, inside a Plot command SchauderApproximant requires an Evaluate. I don't pretend to understand. It's that thing about ``Plot evaluates its arguments in a non-standard way.'' In any case, you want Evaluate anyway to speed up the Plot. :[font = input; preserveAspect; startGroup; ] nmax = 0; Plot[Evaluate@N@ {f[x], SchauderApproximant[f[x], {x, nmax}]}, {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 -0.327839 0.342554 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(1.25)] 0.01131 0.10035 1 0 Msboxa [(1.5)] 0.01131 0.18599 1 0 Msboxa [(1.75)] 0.01131 0.27163 1 0 Msboxa [(2)] 0.01131 0.35727 1 0 Msboxa [(2.25)] 0.01131 0.44291 1 0 Msboxa [(2.5)] 0.01131 0.52855 1 0 Msboxa [(2.75)] 0.01131 0.61418 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 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setlinewidth 0.74762 0.01472 moveto 0.74762 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.01472 moveto 0.82381 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.01472 moveto 0.8619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.01472 moveto 0.9 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.01472 moveto 0.9381 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.10035 moveto 0.03006 0.10035 lineto stroke grestore [(1.25)] 0.01131 0.10035 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.18599 moveto 0.03006 0.18599 lineto stroke grestore [(1.5)] 0.01131 0.18599 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.27163 moveto 0.03006 0.27163 lineto stroke grestore [(1.75)] 0.01131 0.27163 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.35727 moveto 0.03006 0.35727 lineto stroke grestore [(2)] 0.01131 0.35727 1 0 Mshowa gsave 0.002 setlinewidth 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup; ] nmax = 1; Plot[Evaluate@N@ {f[x], SchauderApproximant[f[x],{x, nmax}]}, {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 -0.327839 0.342554 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(1.25)] 0.01131 0.10035 1 0 Msboxa [(1.5)] 0.01131 0.18599 1 0 Msboxa [(1.75)] 0.01131 0.27163 1 0 Msboxa [(2)] 0.01131 0.35727 1 0 Msboxa [(2.25)] 0.01131 0.44291 1 0 Msboxa [(2.5)] 0.01131 0.52855 1 0 Msboxa [(2.75)] 0.01131 0.61418 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 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0.01131 0.35727 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.44291 moveto 0.03006 0.44291 lineto stroke grestore [(2.25)] 0.01131 0.44291 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.52855 moveto 0.03006 0.52855 lineto stroke grestore [(2.5)] 0.01131 0.52855 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.61418 moveto 0.03006 0.61418 lineto stroke grestore [(2.75)] 0.01131 0.61418 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.03184 moveto 0.02756 0.03184 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.04897 moveto 0.02756 0.04897 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0661 moveto 0.02756 0.0661 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.08323 moveto 0.02756 0.08323 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.11748 moveto 0.02756 0.11748 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13461 moveto 0.02756 0.13461 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.15174 moveto 0.02756 0.15174 lineto stroke 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newpath gsave gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.02929 lineto 0.10317 0.04448 lineto 0.14286 0.06033 lineto 0.18254 0.07684 lineto 0.22222 0.09406 lineto 0.2619 0.11201 lineto 0.30159 0.13072 lineto 0.34127 0.15023 lineto 0.38095 0.17057 lineto 0.42063 0.19178 lineto 0.46032 0.21389 lineto 0.5 0.23694 lineto 0.53968 0.26097 lineto 0.57937 0.28602 lineto 0.61905 0.31214 lineto 0.65873 0.33936 lineto 0.69841 0.36775 lineto 0.7381 0.39735 lineto 0.77778 0.4282 lineto 0.81746 0.46037 lineto 0.85714 0.49391 lineto 0.89683 0.52887 lineto 0.93651 0.56532 lineto 0.97619 0.60332 lineto stroke grestore grestore gsave gsave 0.004 setlinewidth 0.02381 0.01472 moveto 0.06349 0.03323 lineto 0.10317 0.05175 lineto 0.14286 0.07027 lineto 0.18254 0.08879 lineto 0.22222 0.10731 lineto 0.2619 0.12583 lineto 0.30159 0.14434 lineto 0.34127 0.16286 lineto 0.38095 0.18138 lineto 0.42063 0.1999 lineto 0.46032 0.21842 lineto 0.48016 0.22768 lineto 0.49008 0.23231 lineto 0.49504 0.23462 lineto 0.49752 0.23578 lineto 0.49876 0.23636 lineto 0.5 0.23694 lineto 0.50248 0.23885 lineto 0.50496 0.24075 lineto 0.50992 0.24457 lineto 0.51984 0.2522 lineto 0.53968 0.26747 lineto 0.57937 0.298 lineto 0.61905 0.32853 lineto 0.65873 0.35906 lineto 0.69841 0.3896 lineto 0.7381 0.42013 lineto 0.77778 0.45066 lineto 0.81746 0.48119 lineto 0.85714 0.51172 lineto 0.89683 0.54226 lineto 0.93651 0.57279 lineto 0.97619 0.60332 lineto stroke grestore grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup; ] nmax = 3; Plot[Evaluate@N@ {f[x], SchauderApproximant[f[x], {x, nmax}]}, {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 -0.327839 0.342554 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(1.25)] 0.01131 0.10035 1 0 Msboxa [(1.5)] 0.01131 0.18599 1 0 Msboxa [(1.75)] 0.01131 0.27163 1 0 Msboxa [(2)] 0.01131 0.35727 1 0 Msboxa [(2.25)] 0.01131 0.44291 1 0 Msboxa [(2.5)] 0.01131 0.52855 1 0 Msboxa [(2.75)] 0.01131 0.61418 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.01472 moveto 0.21429 0.02097 lineto stroke grestore [(0.2)] 0.21429 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.01472 moveto 0.40476 0.02097 lineto stroke grestore [(0.4)] 0.40476 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.01472 moveto 0.59524 0.02097 lineto stroke grestore [(0.6)] 0.59524 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.01472 moveto 0.78571 0.02097 lineto stroke grestore [(0.8)] 0.78571 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(1)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.01472 moveto 0.0619 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.01472 moveto 0.1 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.01472 moveto 0.1381 0.01847 lineto stroke grestore gsave 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0.85714 0.02809 lineto 0.85838 0.02813 lineto 0.85962 0.02824 lineto 0.86086 0.02843 lineto 0.8621 0.02869 lineto 0.86334 0.02903 lineto 0.86458 0.02945 lineto 0.86706 0.0305 lineto 0.86954 0.03184 lineto 0.87202 0.03349 lineto 0.87698 0.03766 lineto 0.8869 0.04948 lineto 0.89683 0.06573 lineto 0.91667 0.11037 lineto 0.93651 0.16855 lineto 0.97619 0.30902 lineto Mfstroke grestore grestore gsave 0.5 setgray gsave 0.004 setlinewidth 0.02381 0.30902 moveto 0.06349 0.4489 lineto 0.08333 0.50766 lineto 0.09325 0.53095 lineto 0.10317 0.55125 lineto 0.10813 0.5599 lineto 0.11062 0.56423 lineto 0.11186 0.56639 lineto 0.1131 0.56856 lineto 0.11558 0.57122 lineto 0.11806 0.57389 lineto 0.12302 0.57921 lineto 0.1255 0.58188 lineto 0.12674 0.58321 lineto 0.12798 0.58454 lineto 0.13046 0.58544 lineto 0.13294 0.58634 lineto 0.1379 0.58814 lineto 0.14038 0.58904 lineto 0.14162 0.58949 lineto 0.14286 0.58994 lineto 0.14534 0.58904 lineto 0.14782 0.58814 lineto 0.15278 0.58634 lineto 0.15526 0.58544 lineto 0.1565 0.58499 lineto 0.15774 0.58454 lineto 0.16022 0.58188 lineto 0.1627 0.57921 lineto 0.16766 0.57389 lineto 0.17014 0.57122 lineto 0.17138 0.56989 lineto 0.17262 0.56856 lineto 0.1751 0.56423 lineto 0.17758 0.5599 lineto 0.18254 0.55125 lineto 0.20238 0.50766 lineto 0.22222 0.4489 lineto 0.2619 0.30902 lineto 0.30159 0.16913 lineto 0.32143 0.11037 lineto 0.33135 0.08708 lineto 0.34127 0.06678 lineto 0.34623 0.05813 lineto 0.34871 0.0538 lineto 0.34995 0.05164 lineto 0.35119 0.04948 lineto 0.35367 0.04681 lineto 0.35615 0.04415 lineto 0.36111 0.03882 lineto Mistroke 0.36359 0.03615 lineto 0.36483 0.03482 lineto 0.36607 0.03349 lineto 0.36855 0.03259 lineto 0.37103 0.03169 lineto 0.37599 0.02989 lineto 0.37847 0.02899 lineto 0.37971 0.02854 lineto 0.38095 0.02809 lineto 0.38343 0.02899 lineto 0.38591 0.02989 lineto 0.39087 0.03169 lineto 0.39335 0.03259 lineto 0.39459 0.03304 lineto 0.39583 0.03349 lineto 0.39831 0.03615 lineto 0.40079 0.03882 lineto 0.40575 0.04415 lineto 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MathPictureEnd :[font = section; inactive; preserveAspect; startGroup; ] The Haar Basis for C[0,1] :[font = text; inactive; preserveAspect; ] Meyer defines the Haar basis so that EVERY basis function vanishes at x = 1. Thus the approximation can NOT be uniform at x = 1. I am not sure if this is intrinsic or whether there is some other definition of the basis that would cure this ``defect.'' I tried many other things, but either they were neutral or created ``glitiches'' in the interior of [0, 10]. On the other hand, if we merely adopt the convention that the value of the Approximant at x = 1 should be the LIMIT from the interior, then the approximation becomes uniform (as you will see from the following graphical output). You can tell I don't understand---right? jkc 11/09/93 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Haar Usage :[font = text; inactive; preserveAspect; ] This function generates the functions in the Haar Basis for C[0, 1]. Here is the generating function: :[font = input; preserveAspect; startGroup; ] Plot[Haar[x], {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.309017 0.294302 [ [(0.2)] 0.21429 0.30902 0 2 Msboxa [(0.4)] 0.40476 0.30902 0 2 Msboxa [(0.6)] 0.59524 0.30902 0 2 Msboxa [(0.8)] 0.78571 0.30902 0 2 Msboxa [(1)] 0.97619 0.30902 0 2 Msboxa [(-1)] 0.01131 0.01472 1 0 Msboxa [(-0.5)] 0.01131 0.16187 1 0 Msboxa [(0.5)] 0.01131 0.45617 1 0 Msboxa [(1)] 0.01131 0.60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.30902 moveto 0.21429 0.31527 lineto stroke grestore [(0.2)] 0.21429 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.30902 moveto 0.40476 0.31527 lineto stroke grestore [(0.4)] 0.40476 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.30902 moveto 0.59524 0.31527 lineto stroke grestore [(0.6)] 0.59524 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.30902 moveto 0.78571 0.31527 lineto stroke grestore [(0.8)] 0.78571 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.30902 moveto 0.97619 0.31527 lineto stroke grestore [(1)] 0.97619 0.30902 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.30902 moveto 0.0619 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.30902 moveto 0.1 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.30902 moveto 0.1381 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.30902 moveto 0.17619 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.30902 moveto 0.25238 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.29048 0.30902 moveto 0.29048 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.32857 0.30902 moveto 0.32857 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.36667 0.30902 moveto 0.36667 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.44286 0.30902 moveto 0.44286 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.48095 0.30902 moveto 0.48095 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.51905 0.30902 moveto 0.51905 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.55714 0.30902 moveto 0.55714 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.63333 0.30902 moveto 0.63333 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.67143 0.30902 moveto 0.67143 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.70952 0.30902 moveto 0.70952 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.74762 0.30902 moveto 0.74762 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.82381 0.30902 moveto 0.82381 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.8619 0.30902 moveto 0.8619 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.9 0.30902 moveto 0.9 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.9381 0.30902 moveto 0.9381 0.31277 lineto stroke grestore gsave 0.002 setlinewidth 0 0.30902 moveto 1 0.30902 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.01472 moveto 0.03006 0.01472 lineto stroke grestore [(-1)] 0.01131 0.01472 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.16187 moveto 0.03006 0.16187 lineto stroke grestore [(-0.5)] 0.01131 0.16187 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.45617 moveto 0.03006 0.45617 lineto stroke grestore [(0.5)] 0.01131 0.45617 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.60332 moveto 0.03006 0.60332 lineto stroke grestore [(1)] 0.01131 0.60332 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.04415 moveto 0.02756 0.04415 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.07358 moveto 0.02756 0.07358 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10301 moveto 0.02756 0.10301 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13244 moveto 0.02756 0.13244 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.1913 moveto 0.02756 0.1913 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.22073 moveto 0.02756 0.22073 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.25016 moveto 0.02756 0.25016 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27959 moveto 0.02756 0.27959 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.33845 moveto 0.02756 0.33845 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.36788 moveto 0.02756 0.36788 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.39731 moveto 0.02756 0.39731 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42674 moveto 0.02756 0.42674 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.4856 moveto 0.02756 0.4856 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.51503 moveto 0.02756 0.51503 lineto stroke grestore gsave 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lineto 0.65873 0.01472 lineto 0.69841 0.01472 lineto 0.7381 0.01472 lineto 0.77778 0.01472 lineto 0.81746 0.01472 lineto 0.85714 0.01472 lineto 0.89683 0.01472 lineto 0.93651 0.01472 lineto 0.94643 0.01472 lineto 0.95635 0.01472 lineto 0.96627 0.01472 lineto 0.96875 0.01472 lineto 0.97123 0.01472 lineto 0.97371 0.01472 lineto 0.97495 0.01472 lineto 0.97619 0.30902 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup; ] Plot[Evaluate@N@{Haar[2, x], Haar[3, x]}, {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.309017 0.208103 [ [(0.2)] 0.21429 0.30902 0 2 Msboxa [(0.4)] 0.40476 0.30902 0 2 Msboxa [(0.6)] 0.59524 0.30902 0 2 Msboxa [(0.8)] 0.78571 0.30902 0 2 Msboxa [(1)] 0.97619 0.30902 0 2 Msboxa [(-1)] 0.01131 0.10091 1 0 Msboxa [(-0.5)] 0.01131 0.20497 1 0 Msboxa [(0.5)] 0.01131 0.41307 1 0 Msboxa [(1)] 0.01131 0.51712 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.21429 0.30902 moveto 0.21429 0.31527 lineto stroke grestore [(0.2)] 0.21429 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.40476 0.30902 moveto 0.40476 0.31527 lineto stroke grestore [(0.4)] 0.40476 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.59524 0.30902 moveto 0.59524 0.31527 lineto stroke grestore [(0.6)] 0.59524 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.78571 0.30902 moveto 0.78571 0.31527 lineto stroke grestore [(0.8)] 0.78571 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.30902 moveto 0.97619 0.31527 lineto stroke grestore [(1)] 0.97619 0.30902 0 2 Mshowa gsave 0.001 setlinewidth 0.0619 0.30902 moveto 0.0619 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.1 0.30902 moveto 0.1 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.1381 0.30902 moveto 0.1381 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.17619 0.30902 moveto 0.17619 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.25238 0.30902 moveto 0.25238 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; preserveAspect; ] Here's a plotting module that generates the m^th octave: :[font = input; preserveAspect; ] PlotHaarOctave[m_] := Module[{string, label, octave}, string = StringForm["Haar Octave: m = `1`", m]; label = FontForm[string, {"Helvetica-Bold", 16}]; octave = Array[Hold@Haar[#, x]&, 2^(m-1), 2^(m-1)]; Plot[Evaluate@N@Release@octave, {x, 0, 1}, DefaultFont -> {"Courier", 12}, PlotLabel -> label ] ] :[font = input; preserveAspect; startGroup; ] PlotHaarOctave[3] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 12 scalefont setfont % Scaling calculations 0.02381 0.952381 0.309017 0.147151 [ [(0.2)] 0.21429 0.30902 0 2 Msboxa [(0.4)] 0.40476 0.30902 0 2 Msboxa [(0.6)] 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] HaarCoefficient Usage :[font = input; preserveAspect; startGroup; ] J = 3; f[x_] := 1; Table[HaarCoefficient[f[x], {x, n}], {n, 0, 2^J - 1}] :[font = output; output; inactive; preserveAspect; endGroup; ] {1, 0, 0, 0, 0, 0, 0, 0} ;[o] {1, 0, 0, 0, 0, 0, 0, 0} :[font = input; preserveAspect; startGroup; ] J = 3; f[x_] := Exp[x]; Table[N@HaarCoefficient[f[x], {x, n}], {n, 0, 2^J - 1}] :[font = output; output; inactive; preserveAspect; endGroup; ] {1.718281828459045, -0.4208392870587887, -0.1140852265470823, -0.1880947396808171, -0.0354570211081775, -0.04552771630293507, -0.05845874489671843, -0.07506251427505184} ;[o] {1.71828, -0.420839, -0.114085, -0.188095, -0.035457, -0.0455277, -0.0584587, -0.0750625} :[font = text; inactive; preserveAspect; ] Here's a surprise: :[font = input; preserveAspect; startGroup; ] J = 2; f[x_] := Sin[4Pi (x - 1/16)] Table[HaarCoefficient[f[x], {x, n}], {n, 0, 2^J - 1}] :[font = output; output; inactive; preserveAspect; endGroup; ] {0, 0, ((-1/2 + I/2)*(-1)^(5/4)*2^(1/2))/Pi, ((-1/2 + I/2)*(-1)^(5/4)*2^(1/2))/Pi} ;[o] 1 I 5/4 (-(-) + -) (-1) Sqrt[2] 2 2 {0, 0, --------------------------, Pi 1 I 5/4 (-(-) + -) (-1) Sqrt[2] 2 2 --------------------------} Pi :[font = text; inactive; preserveAspect; ] Complex Numbers? Well, they are due to the Integrates I have inside the HaarCoefficient module (and Plot will not like them). Here's the scoop: :[font = input; preserveAspect; startGroup; ] f[x_] := Sin[4Pi (x - 1/16)] n = 3; j = Floor@N@Log[2, n]; power = 2^j; k = n - power; temp = Integrate[f[x], {x, k/power, (k+1/2)/power}] - Integrate[f[x], {x, (k+1/2)/power, (k+1)/power}] :[font = output; output; inactive; preserveAspect; endGroup; ] ((-1/2 + I/2)*(-1)^(5/4))/Pi ;[o] 1 I 5/4 (-(-) + -) (-1) 2 2 ------------------ Pi :[font = text; inactive; preserveAspect; ] OK, so take N and cure the problem? Well, not quite: :[font = input; preserveAspect; startGroup; ] N[%] :[font = output; output; inactive; preserveAspect; endGroup; ] 0.2250790790392765 - 1.766974823035287*10^-17*I ;[o] -17 0.225079 - 1.76697 10 I :[font = text; inactive; preserveAspect; ] Yes, indeed, now Mathematica is hung up on the round-off error: ;[s] 3:0,0;17,1;28,2;63,-1; 3:1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0; :[font = input; preserveAspect; startGroup; ] Chop[%] :[font = output; output; inactive; preserveAspect; endGroup; ] 0.2250790790392765 ;[o] 0.225079 :[font = text; inactive; preserveAspect; ] So you will have to put Chop@N@ on some functions. However, don't despair because the GOOD way to get Haar expansions doesn't use this module! Stay tuned. :[font = input; preserveAspect; startGroup; ] J = 3; f[x_] := Sin[4Pi (x - 1/16)] Table[Chop@N@HaarCoefficient[f[x], {x, n}], {n, 0, 2^J - 1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] {0, 0, 0.3183098861837907, 0.3183098861837907, -0.2250790790392765, 0.2250790790392765, -0.2250790790392765, 0.2250790790392765} ;[o] {0, 0, 0.31831, 0.31831, -0.225079, 0.225079, -0.225079, 0.225079} :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] DirectHaarApproximant Usage :[font = text; inactive; preserveAspect; ] For everyday use, I prefer the routine in the next subsection (HaarApproximant). :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup; ] f[x] = Exp[x] :[font = input; preserveAspect; ] f[x_] := Exp[x] :[font = input; preserveAspect; startGroup; ] nmax = 1; Plot[Evaluate@N@ {f[x], DirectHaarApproximant[f[x], {x, nmax}]}, {x, 0, 1} ] :[font = postscript; PostScript; formatAsPostScript; output; inactive; 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] HaarApproximant Usage :[font = text; inactive; preserveAspect; ] This routine integrates f up to F, finds the Schauder coefficients of F (by function evaluations, not integrations) and then uses a scaled version of these coefficients to obtain the Haar Expansion. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup; ] f[x] = Exp[x] :[font = input; preserveAspect; ] f[x_] := Exp[x] :[font = input; preserveAspect; startGroup; ] nmax = 3; Plot[Evaluate@N@ {f[x], HaarApproximant[f[x], {x, nmax}]}, {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 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grestore [(0.5)] 0.01131 0.12298 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.23125 moveto 0.03006 0.23125 lineto stroke grestore [(1)] 0.01131 0.23125 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.33952 moveto 0.03006 0.33952 lineto stroke grestore [(1.5)] 0.01131 0.33952 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.44779 moveto 0.03006 0.44779 lineto stroke grestore [(2)] 0.01131 0.44779 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.55605 moveto 0.03006 0.55605 lineto stroke grestore [(2.5)] 0.01131 0.55605 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.03637 moveto 0.02756 0.03637 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.05802 moveto 0.02756 0.05802 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.07968 moveto 0.02756 0.07968 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10133 moveto 0.02756 0.10133 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.14464 moveto 0.02756 0.14464 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 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0.41926 lineto 0.65873 0.43647 lineto 0.69841 0.45441 lineto 0.7381 0.47312 lineto 0.77778 0.49262 lineto 0.81746 0.51296 lineto 0.85714 0.53416 lineto 0.89683 0.55626 lineto 0.93651 0.5793 lineto 0.97619 0.60332 lineto stroke grestore grestore gsave gsave 0.004 setlinewidth 0.02381 0.26072 moveto 0.06349 0.26072 lineto 0.10317 0.26072 lineto 0.14286 0.26072 lineto 0.18254 0.26072 lineto 0.22222 0.26072 lineto 0.23214 0.26072 lineto 0.24206 0.26072 lineto 0.25198 0.26072 lineto 0.25446 0.26072 lineto 0.25694 0.26072 lineto 0.25942 0.26072 lineto 0.26066 0.26072 lineto 0.2619 0.33059 lineto 0.30159 0.33059 lineto 0.34127 0.33059 lineto 0.38095 0.33059 lineto 0.42063 0.33059 lineto 0.44048 0.33059 lineto 0.46032 0.33059 lineto 0.48016 0.33059 lineto 0.48512 0.33059 lineto 0.49008 0.33059 lineto 0.49504 0.33059 lineto 0.49628 0.33059 lineto 0.49752 0.33059 lineto 0.49876 0.33059 lineto 0.5 0.42031 lineto 0.50248 0.42031 lineto 0.50496 0.42031 lineto 0.50992 0.42031 lineto 0.51984 0.42031 lineto 0.53968 0.42031 lineto 0.57937 0.42031 lineto 0.61905 0.42031 lineto 0.65873 0.42031 lineto 0.69841 0.42031 lineto 0.70833 0.42031 lineto 0.71825 0.42031 lineto 0.72817 0.42031 lineto 0.73065 0.42031 lineto 0.73313 0.42031 lineto 0.73562 0.42031 lineto 0.73686 0.42031 lineto 0.7381 0.53551 lineto 0.77778 0.53551 lineto 0.81746 0.53551 lineto 0.85714 0.53551 lineto 0.89683 0.53551 lineto 0.93651 0.53551 lineto Mistroke 0.94643 0.53551 lineto 0.95635 0.53551 lineto 0.96627 0.53551 lineto 0.96875 0.53551 lineto 0.97123 0.53551 lineto 0.97371 0.53551 lineto 0.97495 0.53551 lineto 0.97619 0.01472 lineto Mfstroke grestore grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup; ] nmax = 7; Plot[Evaluate@N@ {f[x], HaarApproximant[f[x], {x, nmax}]}, {x, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.216535 [ [(0.2)] 0.21429 0.01472 0 2 Msboxa [(0.4)] 0.40476 0.01472 0 2 Msboxa [(0.6)] 0.59524 0.01472 0 2 Msboxa [(0.8)] 0.78571 0.01472 0 2 Msboxa [(1)] 0.97619 0.01472 0 2 Msboxa [(0.5)] 0.01131 0.12298 1 0 Msboxa [(1)] 0.01131 0.23125 1 0 Msboxa [(1.5)] 0.01131 0.33952 1 0 Msboxa [(2)] 0.01131 0.44779 1 0 Msboxa [(2.5)] 0.01131 0.55605 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin 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Observe the glitches at the inflection points for n = 31. They occur for DirectHaarApproximant too, so that's not the problem. I _hope_ it's just a Plot artifact, but can't establish that. 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