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Meyer Wavelets
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Jack K. Cohen
Colorado School of Mines, 1993
jkc@keller.mines.colorado.edu 
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Initialization
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<<Meyer.m
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pi = N[Pi]; sqrt2 = N[Sqrt[2]];
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The Taper Function
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The Meyer wavelets depend on a tapering function (Ten Lectures, p. 117).  This tapering function is implemented as a polynomial by the function NuTaper.  I define the `order', n, of the taper as the degree of vanishing at the left endpoint.  For example, for n=2, we have
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nu[x_] = NuTaper[2, x]
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(3 - 2*x)*x^2
;[o]
           2
(3 - 2 x) x
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Note that this taper is symmetric in the sense that it also joins up with x = 1 to the same order:
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Factor@(nu[x] - 1)
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-((-1 + x)^2*(1 + 2*x))
;[o]
          2
-((-1 + x)  (1 + 2 x))
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Plot[nu[x], {x, 0, 1}]
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Notice that the degree of the order n taper is 2n-1.  Ten Lectures has a persistent type in the degree 7 taper (e.g, figure caption, p. 119).  The correct polynomial is
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NuTaper[4, x]
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x^4*(35 - 84*x + 70*x^2 - 20*x^3)
;[o]
 4                  2       3
x  (35 - 84 x + 70 x  - 20 x )
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This typo isn't terribly important to the world at large, but observing that the stated polynomial wasn't equal to 1 at the right endpoint started me off on making Mathematica notebooks (constructing the 7th degree tapering polynomial by hand did not appeal to me).
;[s]
3:0,0;164,1;175,2;266,-1;
3:1,14,11,Times,0,16,0,0,0;1,14,11,Times,2,16,0,0,0;1,14,11,Times,0,16,0,0,0;
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Meyer Low and High Pass Filters
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The subroutines in Meyer.m honor the Fourier convention that puts a 2Pi in the exponent.  This eliminates 2PI's popping up all over the place.  The effect in plotting M0 and M1 is to replace the half-period point Pi by the half-period point 1/2.  Note that the exponential phase factor in M1 is defined variously in Ten Lectures.  Here are definitions of M0 and M1:
PhiHat[xi] = M0[xi/2] PhiHat[xi/2]  (5.1.17)
PsiHat[xi] = M1[xi/2] PhiHat[xi/2]  (5.1.29)  (but phase choice of p. 193)
;[s]
1:0,0;485,-1;
1:1,13,10,Times,0,14,0,0,0;
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Basic usage
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n = 3;
v[xi_] = NuTaper[n, xi]
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xi^3*(10 - 15*xi + 6*xi^2)
;[o]
  3                   2
xi  (10 - 15 xi + 6 xi )
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Check that the orthogonality condition is verified:
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Clear[norm];
norm[xi_] = N@Abs@MeyerM0[v, xi]^2 +
				N@Abs@MeyerM1[v, xi]^2;
Table[{xi, norm[xi]}, {xi, 0.0, 1.0, 0.1}]//
TableForm
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   {0.8, 1.}, {0.9, 1.}, {1., 1.}}]
;[o]
0.    1.

0.1   1.

0.2   1.

0.3   1.

0.4   1.

0.5   1.

0.6   1.

0.7   1.

0.8   1.

0.9   1.

1.    1.
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Plotting the filters
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PlotFilters[n_] :=
Module[{string, label, v},
  v[xi_] = NuTaper[n, xi];
  string = StringForm["Meyer M0, M1: n = `1`", n];
  label = FontForm[string, {"Helvetica-Bold", 16}];
  Plot[Evaluate@Abs@{MeyerM0[v, xi], MeyerM1[v, xi]},
   	{xi, 0, 1/2},
   	PlotRange->{0, 1.1},
   	PlotStyle -> {GrayLevel[0], GrayLevel[0.5]},
 	DefaultFont -> {"Courier", 12},
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PlotFilters[2]
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        {0.2552083333333333, 0.842014639093515}, 
        {0.2578125, 0.891740694750052}, 
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        {0.2734375, 0.996463803330757}, 
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        {0.2779947916666666, 0.999163431882002}, 
        {0.2786458333333333, 0.999332285891332}, 
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        {0.3072916666666666, 0.99999999998462}, 
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        {0.30859375, 0.999999999996156}, 
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        {0.3138020833333333, 0.999999999999995}, 
        {0.314453125, 0.999999999999998}, 
        {0.3151041666666666, 0.999999999999999}, 
        {0.3157552083333333, 1.}, {0.31640625, 1.}, 
        {0.3170572916666666, 1.}, 
        {0.3177083333333333, 1.}, 
        {0.3190104166666666, 1.}, {0.3203125, 1.}, 
        {0.3229166666666666, 1.}, {0.328125, 1.}, 
        {0.3333333333333333, 1.}, 
        {0.3541666666666666, 1.}, {0.375, 1.}, 
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        {0.4166666666666666, 1.}, {0.4375, 1.}, 
        {0.4583333333333333, 1.}, 
        {0.4791666666666666, 1.}, {0.5, 1.}}]}}}, 
   {PlotRange -> {0, 1.1}, 
    AspectRatio -> GoldenRatio^(-1), 
    DisplayFunction :> $DisplayFunction, 
    ColorOutput -> Automatic, Axes -> Automatic, 
    AxesOrigin -> Automatic, 
    PlotLabel -> 
     FontForm[StringForm["Meyer M0, M1: n = `1`", 
       16], {"Helvetica-Bold", 16}], 
    AxesLabel -> None, Ticks -> Automatic, 
    GridLines -> None, Prolog -> {}, Epilog -> {}, 
    AxesStyle -> Automatic, Background -> Automatic, 
    DefaultColor -> Automatic, 
    DefaultFont :> {"Courier", 12}, 
    RotateLabel -> True, Frame -> False, 
    FrameStyle -> Automatic, FrameTicks -> Automatic, 
    FrameLabel -> None, PlotRegion -> Automatic}]}
;[o]
{-Graphics-, -Graphics-, -Graphics-}
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Meyer PhiHat and PsiHat
:[font = text; inactive; preserveAspect; ]
These functions have compact support (sort of dual to the Daubechies wavelets which are compact support in spatial domain).
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ]
Basic usage
:[font = input; preserveAspect; startGroup; ]
Clear[phihat];
n = 3;
v[xi_] = NuTaper[n, xi];
phihat[xi_] = N@MeyerPhiHat[v, xi];
Table[{xi, phihat[xi]}, {xi, 0, 1.0, 0.1}]//TableForm
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TableForm[{{0, 1.}, {0.1, 1.}, {0.2, 1.}, {0.3, 1.}, 
   {0.4, 0.995864126251826}, 
   {0.5, 0.7071067811865476}, 
   {0.6, 0.0908550606443414}, {0.7, 0}, {0.8, 0}, 
   {0.9, 0}, {1., 0}}]
;[o]
0     1.

0.1   1.

0.2   1.

0.3   1.

0.4   0.995864

0.5   0.707107

0.6   0.0908551

0.7   0

0.8   0

0.9   0

1.    0
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Plotting the transforms
:[font = input; preserveAspect; ]
PlotPhiHat[n_] :=
Module[{string, label, v, phihat},
  v[xi_] = NuTaper[n, xi];
  phihat[xi_] = N@MeyerPhiHat[v, xi];
  string = StringForm["Meyer Phi^: n = `1`", n];
  label = FontForm[string, {"Helvetica-Bold", 16}];
  Plot[Evaluate@phihat[xi], {xi, -1, 1},
	   	PlotRange->{0, 1.1},
	 	DefaultFont -> {"Courier", 12},
	    PlotLabel -> label
  ]
]

PlotPsiHat[n_] :=
Module[{string, label, v, abspsihat},
  v[xi_] = NuTaper[n, xi];
  abspsihat[xi_] = N@Abs@MeyerPsiHat[v, xi];
  string = StringForm["Meyer |Psi^|: n = `1`", n];
  label = FontForm[string, {"Helvetica-Bold", 16}];
  Plot[Evaluate@abspsihat[xi], {xi, -1.5, 1.5},
	   	PlotRange->{0, 1.1},
	 	DefaultFont -> {"Courier", 12},
	    PlotLabel -> label
  ]
]
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n = 2
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PlotPhiHat[2]
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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-
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PlotPsiHat[2]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ]
%!
%%Creator: Mathematica
%%AspectRatio: 0.61803 
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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-
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Meyer Phi and Psi
:[font = text; inactive; preserveAspect; ]
These functions have infinite support, but die off rapidly.  Hence they have `effective' compact support.  The parameter L lets one picks the plot range to include this effective support.  It would be nice if someone figured out the function L(n, accuracy) .
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ]
Basic usage
:[font = input; preserveAspect; startGroup; ]
Clear[phi];
n = 3;
v[xi_] = NuTaper[n, xi];
L = 10;
phi[x_] = N@MeyerPhi[v, x, L];
Table[{x, phi[x]}, {x, 0.0, 1.0, 0.1}]//TableForm
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TableForm[{{0., 1.058924543168759}, 
   {0.1, 1.038954945441438}, 
   {0.2, 0.980446278122835}, 
   {0.3, 0.887478431671814}, 
   {0.4, 0.7664612085460743}, 
   {0.5, 0.6255887251035412}, 
   {0.6, 0.4741502124426491}, 
   {0.7, 0.3217625007015882}, 
   {0.8, 0.1775959179580986}, 
   {0.9, 0.04966450642936748}, 
   {1., -0.0557564429225268}}]
;[o]
0.    1.05892

0.1   1.03895

0.2   0.980446

0.3   0.887478

0.4   0.766461

0.5   0.625589

0.6   0.47415

0.7   0.321763

0.8   0.177596

0.9   0.0496645

1.    -0.0557564
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Plotting the father and mother wavelet
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PlotPhi[n_, L_] :=
Module[{string, label, v, phi},
  v[xi_] = NuTaper[n, xi];
  phi[x_] = N@MeyerPhi[v, x, L];
  string = StringForm["Meyer Phi: n = `1`", n];
  label = FontForm[string, {"Helvetica-Bold", 16}];
  Plot[Evaluate@phi[x], {x, -L, L},
		PlotRange -> {-0.25, 1.2},
	 	DefaultFont -> {"Courier", 12},
	    PlotLabel -> label
  ]
]

PlotPsi[n_, L_] :=
Module[{string, label, v, psi},
  v[xi_] = NuTaper[n, xi];
  psi[x_] = N@MeyerPsi[v, x, L];
  string = StringForm["Meyer Psi: n = `1`", n];
  label = FontForm[string, {"Helvetica-Bold", 16}];
  Plot[Evaluate@psi[x], {x, -L, L},
		PlotRange -> {-1.0, 1.5},
		DefaultFont -> {"Courier", 12},
		PlotLabel -> label
  ]
]
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n = 2
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n = 2; L = 10;
PlotPhi[n, L]
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Mfstroke
grestore
grestore
% End of Graphics
MathPictureEnd
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; 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 = section; inactive; Cclosed; preserveAspect; startGroup; ]
Computation of  wavelet coefficients
:[font = text; inactive; preserveAspect; ]
This is old stuff, maybe it can be done better.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ]
v[x] = x
:[font = text; inactive; preserveAspect; ]
This case can be done analytically, but we would anticipate slow fall off of the h_n coefficients.
:[font = input; preserveAspect; startGroup; ]
int1[n_] = Integrate[Cos[2 Pi n x],
	{x, 0, 1/6}] //Together //Simplify
:[font = output; output; inactive; preserveAspect; endGroup; ]
Sin[(n*Pi)/3]/(2*n*Pi)
;[o]
    n Pi
Sin[----]
     3
---------
 2 n Pi
:[font = input; preserveAspect; startGroup; ]
int2[n_] = Integrate[Cos[2 Pi n x] Cos[Pi(6x - 1)/2],
	{x, 1/6, 1/3}] //Together //Simplify
:[font = output; output; inactive; preserveAspect; endGroup; ]
(3*Cos[(2*n*Pi)/3] + 2*n*Sin[(n*Pi)/3])/
  ((9 - 4*n^2)*Pi)
;[o]
      2 n Pi            n Pi
3 Cos[------] + 2 n Sin[----]
        3                3
-----------------------------
                2
        (9 - 4 n ) Pi
:[font = input; preserveAspect; startGroup; ]
h[n_] = 2 Sqrt[2] (int1[n] + int2[n]) //Simplify
:[font = output; output; inactive; preserveAspect; endGroup; ]
(3*2^(3/2)*(2*n*Cos[(2*n*Pi)/3] + 3*Sin[(n*Pi)/3]))/
  (18*n*Pi - 8*n^3*Pi)
;[o]
   3/2          2 n Pi          n Pi
3 2    (2 n Cos[------] + 3 Sin[----])
                  3              3
--------------------------------------
                       3
          18 n Pi - 8 n  Pi
:[font = input; preserveAspect; startGroup; ]
h[0] = Limit[h[n], n -> 0]
:[font = output; output; inactive; preserveAspect; endGroup; ]
(2^(1/2)*(2 + Pi))/(3*Pi)
;[o]
Sqrt[2] (2 + Pi)
----------------
      3 Pi
:[font = input; preserveAspect; startGroup; ]
Table[{n, h[n]}, {n, 0, 96}] //N//TableForm
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; ]
TableForm[{{0, 0.7715099595100671}, 
   {1., 0.4316322262631766}, 
   {2., -0.05769190407715884}, 
   {3., -0.1000351462396784}, 
   {4., 0.04050242501133717}, 
   {5., 0.02255166588292273}, 
   {6., -0.0200070292479357}, 
   {7., 0.004541394739570721}, 
   {8., 0.003691882686403709}, 
   {9., -0.00857444110625816}, 
   {10., 0.004351248170820495}, 
   {11., 0.003514613363089722}, 
   {12., -0.004763578392365642}, 
   {13., 0.001620059111926194}, 
   {14., 0.001419171155184118}, 
   {15., -0.003031368067869044}, 
   {16., 0.001546565713874513}, 
   {17., 0.001357336359175227}, 
   {18., -0.002098639431601646}, 
   {19., 0.000812410761229635}, 
   {20., 0.0007385560615701333}, 
   {21., -0.001539002249841207}, 
   {22., 0.0007835796112359306}, 
   {23., 0.0007133478160370548}, 
   {24., -0.001176884073407982}, 
   {25., 0.0004858005018119622}, 
   {26., 0.0004510304085139945}, 
   {27., -0.000929119005322091}, 
   {28., 0.0004719484773647723}, 
   {29., 0.0004385877522831149}, 
   {30., -0.0007521439566893119}, 
   {31., 0.000322631728946073}, 
   {32., 0.0003036040155522813}, 
   {33., -0.0006213363120476924}, 
   {34., 0.0003149880041397231}, 
   {35., 0.0002966103467231335}, 
   {36., -0.0005219225021200616}, 
   {37., 0.0002296774254429172}, 
   {38., 0.0002181622955262861}, 
   {39., -0.0004446006499541266}, 
   {40., 0.0002250336980714203}, 
   {41., 0.0002138571783201554}, 
   {42.00000000000001, -0.000383276422374247}, 
   {43., 0.0001717718014569335}, 
   {44.00000000000001, 0.0001642834849292397}, 
   {45., -0.0003338214001324086}, 
   {46., 0.0001687462047679322}, 
   {47.00000000000001, 0.0001614508095196979}, 
   {48., -0.0002933582001163592}, 
   {49.00000000000001, 0.0001332850094798434}, 
   {50., 0.0001281455071689149}, 
   {51., -0.0002598315486744895}, 
   {52.00000000000001, 0.0001312064620968706}, 
   {53., 0.000126184657169192}, 
   {54.00000000000001, -0.0002317416515204906}, 
   {55., 0.0001064164336770546}, 
   {56., 0.0001027379550852546}, 
   {57.00000000000001, -0.0002079732770055685}, 
   {58., 0.0001049281022958842}, 
   {59.00000000000001, 0.0001013253952087498}, 
   {60., -0.0001876832012001472}, 
   {61., 0.0000869214022017619}, 
   {62.00000000000001, 0.0000841988402012055}, 
   {63., -0.0001702242987629242}, 
   {64.00000000000001, 0.0000858196583555665}, 
   {65., 0.0000831480364425929}, 
   {66., -0.0001550932499839976}, 
   {67.00000000000001, 0.0000723300149679842}, 
   {68., 0.00007025893286098759}, 
   {69.00000000000001, -0.0001418938244534446}, 
   {70., 0.00007149190138478142}, 
   {71., 0.00006945630747361704}, 
   {72.00000000000001, -0.0001303106551103063}, 
   {73., 0.00006112596671468288}, 
   {74.00000000000001, 0.00005951406620534568}, 
   {75., -0.000120090211572243}, 
   {76., 0.0000604737273090904}, 
   {77.00000000000001, 0.00005888729786188271}, 
   {78., -0.0001110267993781115}, 
   {79.00000000000001, 0.00005233666060126738}, 
   {80., 0.00005105765812849301}, 
   {81., -0.0001029521230597034}, 
   {82., 0.00005181919564576752}, 
   {83., 0.00005055894262078755}, 
   {84., -0.0000957274126695488}, 
   {85., 0.00004531502572731973}, 
   {86., 0.00004428322521721094}, 
   {87., -0.0000892374185902573}, 
   {88., 0.0000448976437300886}, 
   {89., 0.00004387994778873397}, 
   {90., -0.0000833857845843388}, 
   {91., 0.00003961703273470687}, 
   {92., 0.0000387726277276289}, 
   {93., -0.00007809144905517444}, 
   {94., 0.00003927551676618129}, 
   {95., 0.00003844192404227885}, 
   {96., -0.00007328582142100987}}]
;[o]
0     0.77151

1.    0.431632

2.    -0.0576919

3.    -0.100035

4.    0.0405024

5.    0.0225517

6.    -0.020007

7.    0.00454139

8.    0.00369188

9.    -0.00857444

10.   0.00435125

11.   0.00351461

12.   -0.00476358

13.   0.00162006

14.   0.00141917

15.   -0.00303137

16.   0.00154657

17.   0.00135734

18.   -0.00209864

19.   0.000812411

20.   0.000738556

21.   -0.001539

22.   0.00078358

23.   0.000713348

24.   -0.00117688

25.   0.000485801

26.   0.00045103

27.   -0.000929119

28.   0.000471948

29.   0.000438588

30.   -0.000752144

31.   0.000322632

32.   0.000303604

33.   -0.000621336

34.   0.000314988

35.   0.00029661

36.   -0.000521923

37.   0.000229677

38.   0.000218162

39.   -0.000444601

40.   0.000225034

41.   0.000213857

42.   -0.000383276

43.   0.000171772

44.   0.000164283

45.   -0.000333821

46.   0.000168746

47.   0.000161451

48.   -0.000293358

49.   0.000133285

50.   0.000128146

51.   -0.000259832

52.   0.000131206

53.   0.000126185

54.   -0.000231742

55.   0.000106416

56.   0.000102738

57.   -0.000207973

58.   0.000104928

59.   0.000101325

60.   -0.000187683

61.   0.0000869214

62.   0.0000841988

63.   -0.000170224

64.   0.0000858197

65.   0.000083148

66.   -0.000155093

67.   0.00007233

68.   0.0000702589

69.   -0.000141894

70.   0.0000714919

71.   0.0000694563

72.   -0.000130311

73.   0.000061126

74.   0.0000595141

75.   -0.00012009

76.   0.0000604737

77.   0.0000588873

78.   -0.000111027

79.   0.0000523367

80.   0.0000510577

81.   -0.000102952

82.   0.0000518192

83.   0.0000505589

84.   -0.0000957274

85.   0.000045315

86.   0.0000442832

87.   -0.0000892374

88.   0.0000448976

89.   0.0000438799

90.   -0.0000833858

91.   0.000039617

92.   0.0000387726

93.   -0.0000780914

94.   0.0000392755

95.   0.0000384419

96.   -0.0000732858
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ]
v[x] =  sine square taper
:[font = input; preserveAspect; ]
v[xi_] = N@Sin[Pi/2 xi];
phihat[xi_] = N@MeyerPhiHat[v, xi];
:[font = input; preserveAspect; ]
Clear[h];
h[n_] := h[n] = 2 sqrt2 *
	NIntegrate[Cos[2 pi n x] phihat[2x], {x, 0, 1/3}]
:[font = input; preserveAspect; startGroup; ]
h[0]
:[font = output; output; inactive; preserveAspect; endGroup; ]
0.6939080277230163
;[o]
0.693908
:[font = input; preserveAspect; startGroup; ]
Table[{n, h[n]}, {n, 0, 32}] //TableForm
:[font = message; inactive; preserveAspect; ]
NIntegrate::ploss: 
   Numerical integration stopping due to loss of
     precision. Achieved neither the requested
     PrecisionGoal nor AccuracyGoal; suspect one of
     the following: highly oscillatory integrand or
     the true value of the integral is 0.
:[font = message; inactive; preserveAspect; ]
NIntegrate::ploss: 
   Numerical integration stopping due to loss of
     precision. Achieved neither the requested
     PrecisionGoal nor AccuracyGoal; suspect one of
     the following: highly oscillatory integrand or
     the true value of the integral is 0.
:[font = message; inactive; preserveAspect; ]
NIntegrate::ploss: 
   Numerical integration stopping due to loss of
     precision. Achieved neither the requested
     PrecisionGoal nor AccuracyGoal; suspect one of
     the following: highly oscillatory integrand or
     the true value of the integral is 0.
:[font = message; inactive; preserveAspect; ]
General::stop: 
   Further output of NIntegrate::ploss
     will be suppressed during this calculation.
:[font = output; output; inactive; preserveAspect; endGroup; ]
TableForm[{{0, 0.6939080277230163}, 
   {1, 0.4385271066703539}, 
   {2, 0.01227943936371829}, 
   {3, -0.1177104768232206}, 
   {4, -0.00982287097184306}, 
   {5, 0.0436710055018155}, 
   {6, 0.006597796439027268}, 
   {7, -0.01234764488002501}, 
   {8, -0.0035064669244163}, 
   {9, -0.000140642406593802}, 
   {10, 0.00122596428161226}, 
   {11, 0.002930159132090298}, 
   {12, 1.580241025458046*10^-6}, 
   {13, -0.001717760646004666}, 
   {14, -0.0003517982708427788}, 
   {15, -(1.09677674426084*10^-8)}, 
   {16, 0.000223633660387509}, 
   {17, 0.000746893491646411}, 
   {18, 5.170024459328841*10^-11}, 
   {19, -0.0005310771672959831}, 
   {20, -0.0001080745374066267}, 
   {21, -(1.76298690367621*10^-13)}, 
   {22, 0.00007981981179236322}, 
   {23, 0.000296672497109758}, 
   {24, 4.931071628053731*10^-16}, 
   {25, -0.0002303333044328527}, 
   {26, -0.00004727269929651999}, 
   {27, -(1.180777150314324*10^-16)}, 
   {28, 0.00003755690978250394}, 
   {29, 0.0001469543184906331}, 
   {30, -(2.167310244481949*10^-16)}, 
   {31, -0.0001201283036014624}, 
   {32, -0.00002487845594682621}}]
;[o]
0    0.693908

1    0.438527

2    0.0122794

3    -0.11771

4    -0.00982287

5    0.043671

6    0.0065978

7    -0.0123476

8    -0.00350647

9    -0.000140642

10   0.00122596

11   0.00293016

               -6
12   1.58024 10

13   -0.00171776

14   -0.000351798

                -8
15   -1.09678 10

16   0.000223634

17   0.000746893

               -11
18   5.17002 10

19   -0.000531077

20   -0.000108075

                -13
21   -1.76299 10

22   0.0000798198

23   0.000296672

               -16
24   4.93107 10

25   -0.000230333

26   -0.0000472727

                -16
27   -1.18078 10

28   0.0000375569

29   0.000146954

                -16
30   -2.16731 10

31   -0.000120128

32   -0.0000248785
:[font = input; preserveAspect; ]
For[n = 1, n < 33, ++n,
	h[-n] = h[n]
]
:[font = input; preserveAspect; startGroup; ]
For[n = -32, n < 33, ++n,
	g[n] = (-1)^n h[1 - n]
]
:[font = message; inactive; preserveAspect; endGroup; ]
NIntegrate::ploss: 
   Numerical integration stopping due to loss of
     precision. Achieved neither the requested
     PrecisionGoal nor AccuracyGoal; suspect one of
     the following: highly oscillatory integrand or
     the true value of the integral is 0.
:[font = input; preserveAspect; startGroup; ]
Table[{n, g[n]}, {n, -32, 32}] //TableForm
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; ]
TableForm[{{-32, -(2.134727759377427*10^-16)}, 
   {-31, 0.00002487845594682621}, 
   {-30, -0.0001201283036014624}, 
   {-29, 2.167310244481949*10^-16}, 
   {-28, 0.0001469543184906331}, 
   {-27, -0.00003755690978250394}, 
   {-26, -(1.180777150314324*10^-16)}, 
   {-25, 0.00004727269929651999}, 
   {-24, -0.0002303333044328527}, 
   {-23, -(4.931071628053731*10^-16)}, 
   {-22, 0.000296672497109758}, 
   {-21, -0.00007981981179236322}, 
   {-20, -(1.76298690367621*10^-13)}, 
   {-19, 0.0001080745374066267}, 
   {-18, -0.0005310771672959831}, 
   {-17, -(5.170024459328841*10^-11)}, 
   {-16, 0.000746893491646411}, 
   {-15, -0.000223633660387509}, 
   {-14, -(1.09677674426084*10^-8)}, 
   {-13, 0.0003517982708427788}, 
   {-12, -0.001717760646004666}, 
   {-11, -(1.580241025458046*10^-6)}, 
   {-10, 0.002930159132090298}, 
   {-9, -0.00122596428161226}, 
   {-8, -0.000140642406593802}, 
   {-7, 0.0035064669244163}, 
   {-6, -0.01234764488002501}, 
   {-5, -0.006597796439027268}, 
   {-4, 0.0436710055018155}, 
   {-3, 0.00982287097184306}, 
   {-2, -0.1177104768232206}, 
   {-1, -0.01227943936371829}, 
   {0, 0.4385271066703539}, 
   {1, -0.6939080277230163}, 
   {2, 0.4385271066703539}, 
   {3, -0.01227943936371829}, 
   {4, -0.1177104768232206}, 
   {5, 0.00982287097184306}, 
   {6, 0.0436710055018155}, 
   {7, -0.006597796439027268}, 
   {8, -0.01234764488002501}, 
   {9, 0.0035064669244163}, 
   {10, -0.000140642406593802}, 
   {11, -0.00122596428161226}, 
   {12, 0.002930159132090298}, 
   {13, -(1.580241025458046*10^-6)}, 
   {14, -0.001717760646004666}, 
   {15, 0.0003517982708427788}, 
   {16, -(1.09677674426084*10^-8)}, 
   {17, -0.000223633660387509}, 
   {18, 0.000746893491646411}, 
   {19, -(5.170024459328841*10^-11)}, 
   {20, -0.0005310771672959831}, 
   {21, 0.0001080745374066267}, 
   {22, -(1.76298690367621*10^-13)}, 
   {23, -0.00007981981179236322}, 
   {24, 0.000296672497109758}, 
   {25, -(4.931071628053731*10^-16)}, 
   {26, -0.0002303333044328527}, 
   {27, 0.00004727269929651999}, 
   {28, -(1.180777150314324*10^-16)}, 
   {29, -0.00003755690978250394}, 
   {30, 0.0001469543184906331}, 
   {31, 2.167310244481949*10^-16}, 
   {32, -0.0001201283036014624}}]
;[o]
                 -16
-32   -2.13473 10

-31   0.0000248785

-30   -0.000120128

                -16
-29   2.16731 10

-28   0.000146954

-27   -0.0000375569

                 -16
-26   -1.18078 10

-25   0.0000472727

-24   -0.000230333

                 -16
-23   -4.93107 10

-22   0.000296672

-21   -0.0000798198

                 -13
-20   -1.76299 10

-19   0.000108075

-18   -0.000531077

                 -11
-17   -5.17002 10

-16   0.000746893

-15   -0.000223634

                 -8
-14   -1.09678 10

-13   0.000351798

-12   -0.00171776

                 -6
-11   -1.58024 10

-10   0.00293016

-9    -0.00122596

-8    -0.000140642

-7    0.00350647

-6    -0.0123476

-5    -0.0065978

-4    0.043671

-3    0.00982287

-2    -0.11771

-1    -0.0122794

0     0.438527

1     -0.693908

2     0.438527

3     -0.0122794

4     -0.11771

5     0.00982287

6     0.043671

7     -0.0065978

8     -0.0123476

9     0.00350647

10    -0.000140642

11    -0.00122596

12    0.00293016

                 -6
13    -1.58024 10

14    -0.00171776

15    0.000351798

                 -8
16    -1.09678 10

17    -0.000223634

18    0.000746893

                 -11
19    -5.17002 10

20    -0.000531077

21    0.000108075

                 -13
22    -1.76299 10

23    -0.0000798198

24    0.000296672

                 -16
25    -4.93107 10

26    -0.000230333

27    0.0000472727

                 -16
28    -1.18078 10

29    -0.0000375569

30    0.000146954

                -16
31    2.16731 10

32    -0.000120128
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ]
v[x] =  5th order polynomial
:[font = input; preserveAspect; ]
v[xi_] = NuTaper[3, xi];
phihat[xi_] = N@MeyerPhiHat[v, xi];
:[font = input; preserveAspect; ]
Clear[h];
h[n_] := h[n] = 2 sqrt2 *
	NIntegrate[Cos[2 pi n x] phihat[2x], {x, 0, 1/3}]
:[font = input; preserveAspect; startGroup; ]
h[0]
:[font = output; output; inactive; preserveAspect; endGroup; ]
0.7487909596455448
;[o]
0.748791
:[font = input; preserveAspect; startGroup; ]
Table[{n, h[n]}, {n, 0, 32}] //TableForm
:[font = output; output; inactive; preserveAspect; endGroup; ]
TableForm[{{0, 0.7487909596455448}, 
   {1, 0.4423468545272427}, 
   {2, -0.0394312714106252}, 
   {3, -0.1279283652837368}, 
   {4, 0.03327754102997376}, 
   {5, 0.05712030029141813}, 
   {6, -0.02480743961682835}, 
   {7, -0.02530995829614579}, 
   {8, 0.0160002566077106}, 
   {9, 0.00953839991965583}, 
   {10, -0.00855580716089063}, 
   {11, -0.002450712958354055}, 
   {12, 0.003415723335711926}, 
   {13, 0.00005813993271627958}, 
   {14, -0.0006472379343091984}, 
   {15, 0.0002249912346414393}, 
   {16, -0.0003286744488452197}, 
   {17, 0.00006082096623831612}, 
   {18, 0.0003328576326912566}, 
   {19, -0.0002311377294304242}, 
   {20, -0.00005896300412449642}, 
   {21, 0.0001740687527579166}, 
   {22, -0.0001153955027954965}, 
   {23, -0.00002743812722073463}, 
   {24, 0.0001146254505817485}, 
   {25, -0.00006656963359713712}, 
   {26, -0.00002770199641750342}, 
   {27, 0.00006633025204554301}, 
   {28, -0.00003976825913463582}, 
   {29, -0.00001491368093388662}, 
   {30, 0.00004596032778803772}, 
   {31, -0.00002668800811296295}, 
   {32, -0.00001281218531023024}}]
;[o]
0    0.748791

1    0.442347

2    -0.0394313

3    -0.127928

4    0.0332775

5    0.0571203

6    -0.0248074

7    -0.02531

8    0.0160003

9    0.0095384

10   -0.00855581

11   -0.00245071

12   0.00341572

13   0.0000581399

14   -0.000647238

15   0.000224991

16   -0.000328674

17   0.000060821

18   0.000332858

19   -0.000231138

20   -0.000058963

21   0.000174069

22   -0.000115396

23   -0.0000274381

24   0.000114625

25   -0.0000665696

26   -0.000027702

27   0.0000663303

28   -0.0000397683

29   -0.0000149137

30   0.0000459603

31   -0.000026688

32   -0.0000128122
:[font = input; preserveAspect; ]
For[n = 1, n < 33, ++n,
	h[-n] = h[n]
]
:[font = input; preserveAspect; ]
For[n = -32, n < 33, ++n,
	g[n] = (-1)^n h[1 - n]
]
:[font = input; preserveAspect; startGroup; ]
Table[{n, g[n]}, {n, -32, 32}] //TableForm
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; ]
TableForm[{{-32, 0.00003043007620528805}, 
   {-31, 0.00001281218531023024}, 
   {-30, -0.00002668800811296295}, 
   {-29, -0.00004596032778803772}, 
   {-28, -0.00001491368093388662}, 
   {-27, 0.00003976825913463582}, 
   {-26, 0.00006633025204554301}, 
   {-25, 0.00002770199641750342}, 
   {-24, -0.00006656963359713712}, 
   {-23, -0.0001146254505817485}, 
   {-22, -0.00002743812722073463}, 
   {-21, 0.0001153955027954965}, 
   {-20, 0.0001740687527579166}, 
   {-19, 0.00005896300412449642}, 
   {-18, -0.0002311377294304242}, 
   {-17, -0.0003328576326912566}, 
   {-16, 0.00006082096623831612}, 
   {-15, 0.0003286744488452197}, 
   {-14, 0.0002249912346414393}, 
   {-13, 0.0006472379343091984}, 
   {-12, 0.00005813993271627958}, 
   {-11, -0.003415723335711926}, 
   {-10, -0.002450712958354055}, 
   {-9, 0.00855580716089063}, 
   {-8, 0.00953839991965583}, 
   {-7, -0.0160002566077106}, 
   {-6, -0.02530995829614579}, 
   {-5, 0.02480743961682835}, 
   {-4, 0.05712030029141813}, 
   {-3, -0.03327754102997376}, 
   {-2, -0.1279283652837368}, 
   {-1, 0.0394312714106252}, 
   {0, 0.4423468545272427}, 
   {1, -0.7487909596455448}, 
   {2, 0.4423468545272427}, {3, 0.0394312714106252}, 
   {4, -0.1279283652837368}, 
   {5, -0.03327754102997376}, 
   {6, 0.05712030029141813}, 
   {7, 0.02480743961682835}, 
   {8, -0.02530995829614579}, 
   {9, -0.0160002566077106}, 
   {10, 0.00953839991965583}, 
   {11, 0.00855580716089063}, 
   {12, -0.002450712958354055}, 
   {13, -0.003415723335711926}, 
   {14, 0.00005813993271627958}, 
   {15, 0.0006472379343091984}, 
   {16, 0.0002249912346414393}, 
   {17, 0.0003286744488452197}, 
   {18, 0.00006082096623831612}, 
   {19, -0.0003328576326912566}, 
   {20, -0.0002311377294304242}, 
   {21, 0.00005896300412449642}, 
   {22, 0.0001740687527579166}, 
   {23, 0.0001153955027954965}, 
   {24, -0.00002743812722073463}, 
   {25, -0.0001146254505817485}, 
   {26, -0.00006656963359713712}, 
   {27, 0.00002770199641750342}, 
   {28, 0.00006633025204554301}, 
   {29, 0.00003976825913463582}, 
   {30, -0.00001491368093388662}, 
   {31, -0.00004596032778803772}, 
   {32, -0.00002668800811296295}}]
;[o]
-32   0.0000304301

-31   0.0000128122

-30   -0.000026688

-29   -0.0000459603

-28   -0.0000149137

-27   0.0000397683

-26   0.0000663303

-25   0.000027702

-24   -0.0000665696

-23   -0.000114625

-22   -0.0000274381

-21   0.000115396

-20   0.000174069

-19   0.000058963

-18   -0.000231138

-17   -0.000332858

-16   0.000060821

-15   0.000328674

-14   0.000224991

-13   0.000647238

-12   0.0000581399

-11   -0.00341572

-10   -0.00245071

-9    0.00855581

-8    0.0095384

-7    -0.0160003

-6    -0.02531

-5    0.0248074

-4    0.0571203

-3    -0.0332775

-2    -0.127928

-1    0.0394313

0     0.442347

1     -0.748791

2     0.442347

3     0.0394313

4     -0.127928

5     -0.0332775

6     0.0571203

7     0.0248074

8     -0.02531

9     -0.0160003

10    0.0095384

11    0.00855581

12    -0.00245071

13    -0.00341572

14    0.0000581399

15    0.000647238

16    0.000224991

17    0.000328674

18    0.000060821

19    -0.000332858

20    -0.000231138

21    0.000058963

22    0.000174069

23    0.000115396

24    -0.0000274381

25    -0.000114625

26    -0.0000665696

27    0.000027702

28    0.0000663303

29    0.0000397683

30    -0.0000149137

31    -0.0000459603

32    -0.000026688
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ]
v[x] =  7th order polynomial
:[font = input; preserveAspect; ]
v[xi_] = NuTaper[4, xi];
phihat[xi_] = N@MeyerPhiHat[v, xi];
:[font = input; preserveAspect; ]
Clear[h];
h[n_] := h[n] = 2 sqrt2 *
	NIntegrate[Cos[2 pi n x] phihat[2x], {x, 0, 1/3}]
:[font = input; preserveAspect; startGroup; ]
h[0]
:[font = output; output; inactive; preserveAspect; endGroup; ]
0.7437504000617336
;[o]
0.74375
:[font = input; preserveAspect; startGroup; ]
Table[{n, h[n]}, {n, 0, 32}] //TableForm
:[font = output; output; inactive; preserveAspect; endGroup; ]
TableForm[{{0, 0.7437504000617336}, 
   {1, 0.4440946696487148}, 
   {2, -0.03504825887364074}, 
   {3, -0.1326965074832326}, 
   {4, 0.03062121905768919}, 
   {5, 0.06366724912749276}, 
   {6, -0.02432176421295161}, 
   {7, -0.03209403724122456}, 
   {8, 0.01740387407471206}, 
   {9, 0.01525090072481833}, 
   {10, -0.01104463290478378}, 
   {11, -0.006386723407097954}, 
   {12, 0.006045505648088331}, 
   {13, 0.002194773565737437}, 
   {14, -0.002702166518011627}, 
   {15, -0.0005781853577488331}, 
   {16, 0.000859499528586013}, 
   {17, 0.0001611897079741688}, 
   {18, -0.000093512851016903}, 
   {19, -0.000139913020951992}, 
   {20, -0.00007559278860226008}, 
   {21, 0.0001497133598240797}, 
   {22, 0.00002446195021200165}, 
   {23, -0.000102778900388185}, 
   {24, 0.00003740960075704535}, 
   {25, 0.00003234129866603939}, 
   {26, -0.00004642870438504407}, 
   {27, 0.00001499347595743307}, 
   {28, 0.00002010638421302857}, 
   {29, -0.00002443796270328764}, 
   {30, 6.187538827319182*10^-6}, 
   {31, 0.00001050391845351309}, 
   {32, -0.0000142068925198614}}]
;[o]
0    0.74375

1    0.444095

2    -0.0350483

3    -0.132697

4    0.0306212

5    0.0636672

6    -0.0243218

7    -0.032094

8    0.0174039

9    0.0152509

10   -0.0110446

11   -0.00638672

12   0.00604551

13   0.00219477

14   -0.00270217

15   -0.000578185

16   0.0008595

17   0.00016119

18   -0.0000935129

19   -0.000139913

20   -0.0000755928

21   0.000149713

22   0.000024462

23   -0.000102779

24   0.0000374096

25   0.0000323413

26   -0.0000464287

27   0.0000149935

28   0.0000201064

29   -0.000024438

               -6
30   6.18754 10

31   0.0000105039

32   -0.0000142069
:[font = input; preserveAspect; ]
For[n = 1, n < 33, ++n,
	h[-n] = h[n]
]
:[font = input; preserveAspect; ]
For[n = -32, n < 33, ++n,
	g[n] = (-1)^n h[1 - n]
]
:[font = input; preserveAspect; startGroup; ]
Table[{n, g[n]}, {n, -32, 32}] //TableForm
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; ]
TableForm[{{-32, 4.849036798898432*10^-6}, 
   {-31, 0.0000142068925198614}, 
   {-30, 0.00001050391845351309}, 
   {-29, -(6.187538827319182*10^-6)}, 
   {-28, -0.00002443796270328764}, 
   {-27, -0.00002010638421302857}, 
   {-26, 0.00001499347595743307}, 
   {-25, 0.00004642870438504407}, 
   {-24, 0.00003234129866603939}, 
   {-23, -0.00003740960075704535}, 
   {-22, -0.000102778900388185}, 
   {-21, -0.00002446195021200165}, 
   {-20, 0.0001497133598240797}, 
   {-19, 0.00007559278860226008}, 
   {-18, -0.000139913020951992}, 
   {-17, 0.000093512851016903}, 
   {-16, 0.0001611897079741688}, 
   {-15, -0.000859499528586013}, 
   {-14, -0.0005781853577488331}, 
   {-13, 0.002702166518011627}, 
   {-12, 0.002194773565737437}, 
   {-11, -0.006045505648088331}, 
   {-10, -0.006386723407097954}, 
   {-9, 0.01104463290478378}, 
   {-8, 0.01525090072481833}, 
   {-7, -0.01740387407471206}, 
   {-6, -0.03209403724122456}, 
   {-5, 0.02432176421295161}, 
   {-4, 0.06366724912749276}, 
   {-3, -0.03062121905768919}, 
   {-2, -0.1326965074832326}, 
   {-1, 0.03504825887364074}, 
   {0, 0.4440946696487148}, 
   {1, -0.7437504000617336}, 
   {2, 0.4440946696487148}, 
   {3, 0.03504825887364074}, 
   {4, -0.1326965074832326}, 
   {5, -0.03062121905768919}, 
   {6, 0.06366724912749276}, 
   {7, 0.02432176421295161}, 
   {8, -0.03209403724122456}, 
   {9, -0.01740387407471206}, 
   {10, 0.01525090072481833}, 
   {11, 0.01104463290478378}, 
   {12, -0.006386723407097954}, 
   {13, -0.006045505648088331}, 
   {14, 0.002194773565737437}, 
   {15, 0.002702166518011627}, 
   {16, -0.0005781853577488331}, 
   {17, -0.000859499528586013}, 
   {18, 0.0001611897079741688}, 
   {19, 0.000093512851016903}, 
   {20, -0.000139913020951992}, 
   {21, 0.00007559278860226008}, 
   {22, 0.0001497133598240797}, 
   {23, -0.00002446195021200165}, 
   {24, -0.000102778900388185}, 
   {25, -0.00003740960075704535}, 
   {26, 0.00003234129866603939}, 
   {27, 0.00004642870438504407}, 
   {28, 0.00001499347595743307}, 
   {29, -0.00002010638421302857}, 
   {30, -0.00002443796270328764}, 
   {31, -(6.187538827319182*10^-6)}, 
   {32, 0.00001050391845351309}}]
;[o]
                -6
-32   4.84904 10

-31   0.0000142069

-30   0.0000105039

                 -6
-29   -6.18754 10

-28   -0.000024438

-27   -0.0000201064

-26   0.0000149935

-25   0.0000464287

-24   0.0000323413

-23   -0.0000374096

-22   -0.000102779

-21   -0.000024462

-20   0.000149713

-19   0.0000755928

-18   -0.000139913

-17   0.0000935129

-16   0.00016119

-15   -0.0008595

-14   -0.000578185

-13   0.00270217

-12   0.00219477

-11   -0.00604551

-10   -0.00638672

-9    0.0110446

-8    0.0152509

-7    -0.0174039

-6    -0.032094

-5    0.0243218

-4    0.0636672

-3    -0.0306212

-2    -0.132697

-1    0.0350483

0     0.444095

1     -0.74375

2     0.444095

3     0.0350483

4     -0.132697

5     -0.0306212

6     0.0636672

7     0.0243218

8     -0.032094

9     -0.0174039

10    0.0152509

11    0.0110446

12    -0.00638672

13    -0.00604551

14    0.00219477

15    0.00270217

16    -0.000578185

17    -0.0008595

18    0.00016119

19    0.0000935129

20    -0.000139913

21    0.0000755928

22    0.000149713

23    -0.000024462

24    -0.000102779

25    -0.0000374096

26    0.0000323413

27    0.0000464287

28    0.0000149935

29    -0.0000201064

30    -0.000024438

                 -6
31    -6.18754 10

32    0.0000105039
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ]
v[x] =  19th order polynomial
:[font = input; preserveAspect; startGroup; ]
Clear[v];
epsilon = 5 * 0.0001;
order = 10;
v[x_] = NuTaper[order, x];
v[x]
:[font = output; output; inactive; preserveAspect; endGroup; ]
x^10*(92378 - 755820*x + 2771340*x^2 - 5969040*x^3 + 
    8314020*x^4 - 7759752*x^5 + 4849845*x^6 - 
    1956240*x^7 + 461890*x^8 - 48620*x^9)
;[o]
 10                              2            3
x   (92378 - 755820 x + 2771340 x  - 5969040 x  + 
 
             4            5            6            7
    8314020 x  - 7759752 x  + 4849845 x  - 1956240 x  + 
 
            8          9
    461890 x  - 48620 x )
:[font = text; inactive; preserveAspect; ]
Note that for Meyer, m0 has support in [-1/3, 1/3], so we can shrink the integration interval.
:[font = input; preserveAspect; startGroup; ]

Clear[c];
c[n_] := c[n] = 4 *
	NIntegrate[Cos[2 Pi n x] MeyerM0[v, x], {x, 0, 1/3}]
c[0]
:[font = output; output; inactive; preserveAspect; endGroup; ]
1.033676101512571
;[o]
1.03368
:[font = input; preserveAspect; startGroup; ]
Clear[sum];
k = 0; sum[0] = c[0]/2;
While[Abs[c[k]] > epsilon/10, (* must remain small *)
	++k;
	c[k];
	sum[k] = sum[k-1] + c[k]
]

Table[c[n]/2, {n, 0, k}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup; endGroup; ]
{0.5168380507562857, 0.3164781455384325, 
  -0.01649966087407518, -0.100729786538015, 
  0.01552319831450735, 0.05509498413488373, 
  -0.01401782193930871, -0.03424007455777174, 
  0.01214390289121548, 0.0221083104743232, 
  -0.01008599397405697, -0.01432556477339673, 
  0.00802405124120529, 0.00916062224090211, 
  -0.006108867168218027, -0.005732295835107676, 
  0.004446084594891164, 0.003499573188474467, 
  -0.003090648607010155, -0.002088572913421177, 
  0.002050918800946534, 0.00122774266824506, 
  -0.001299645172757643, -0.0007203190016253333, 
  0.0007880578180533944, 0.0004287822151138096, 
  -0.0004594960824917243, -0.0002623683736274173, 
  0.000260025546131652, 0.0001652924939436777, 
  -0.0001448886071958024, -0.0001059005166251523, 
  0.0000809309024611647, 0.00006759851658899325, 
  -0.00004601316881936313, -0.00004214093435491416, 
  0.00002674507517734536, 0.00002531353758986662, 
  -0.00001574531517873311}
;[o]
{0.516838, 0.316478, -0.0164997, -0.10073, 0.0155232, 
 
  0.055095, -0.0140178, -0.0342401, 0.0121439, 
 
  0.0221083, -0.010086, -0.0143256, 0.00802405, 
 
  0.00916062, -0.00610887, -0.0057323, 0.00444608, 
 
  0.00349957, -0.00309065, -0.00208857, 0.00205092, 
 
  0.00122774, -0.00129965, -0.000720319, 0.000788058, 
 
  0.000428782, -0.000459496, -0.000262368, 0.000260026, 
 
  0.000165292, -0.000144889, -0.000105901, 
 
  0.0000809309, 0.0000675985, -0.0000460132, 
 
  -0.0000421409, 0.0000267451, 0.0000253135, 
 
  -0.0000157453}
^*)