(*^ ::[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, 16, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 11, "Ohlfs"; ; 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; ] Battle-Lemarie Wavelets :[font = subsubtitle; inactive; preserveAspect; ] Jack K. 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Battle-Lemarie PhiHat and PsiHat :[font = text; inactive; preserveAspect; ] Again, recall that following Ten Lectures, the scaling function with the phrase ``sharp'' in its name is the orthonormalized one. These functions have infinite support, but die off rapidly. Hence they have `effective' compact support. The parameter L lets one pick 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 = text; inactive; preserveAspect; ] Here's the non-orthogonalized one: :[font = input; preserveAspect; startGroup; ] Clear[phihat]; n = 3; phihat[xi_] = N@SplinePhiHat[n, xi]; Table[{xi, phihat[xi]}, {xi, 0, 1.0, 0.1}]//TableForm :[font = output; output; inactive; preserveAspect; endGroup; ] TableForm[{{0, 1.}, {0.1, 0.936116640921296}, {0.2, 0.7658703698019802}, {0.3, 0.5429327867076874}, {0.4, 0.3280846004720078}, {0.5, 0.1642557160749494}, {0.6, 0.06480683466113739}, {0.7, 0.0183163497389932}, {0.8, 0.002991681132038987}, {0.9, 0.0001426789576164157}, {1., 1.057676877775121*10^-63}}] ;[o] 0 1. 0.1 0.936117 0.2 0.76587 0.3 0.542933 0.4 0.328085 0.5 0.164256 0.6 0.0648068 0.7 0.0183163 0.8 0.00299168 0.9 0.000142679 -63 1. 1.05768 10 :[font = text; inactive; preserveAspect; ] And after normalization: :[font = input; preserveAspect; startGroup; ] Clear[phihat]; n = 3; phihat[xi_] = N@SplinePhiHatSharp[n, xi]; Table[{xi, phihat[xi]}, {xi, 0, 1.0, 0.1}]//TableForm :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] TableForm[{{0, 1.}, {0.1, 0.99999998600774}, {0.2, 0.999992058558031}, {0.3, 0.999426887002791}, {0.4, 0.98101511264068}, {0.5, 0.7070519235004946}, {0.6, 0.1937807629907519}, {0.7, 0.03371660884932365}, {0.8, 0.003906218978742313}, {0.9, 0.0001524157881432325}, {1., 1.057676877775121*10^-63}}] ;[o] 0 1. 0.1 1. 0.2 0.999992 0.3 0.999427 0.4 0.981015 0.5 0.707052 0.6 0.193781 0.7 0.0337166 0.8 0.00390622 0.9 0.000152416 -63 1. 1.05768 10 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Plotting the transforms :[font = input; preserveAspect; ] PlotPhiHatSharp[n_, L_] := Module[{string, label, phihat}, phihat[xi_] = N@SplinePhiHatSharp[n, xi]; string = StringForm["B-L: phi^#(xi); n = `1`", n]; label = FontForm[string, {"Helvetica-Bold", 16}]; 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Battle-Lemarie Phi and Psi :[font = text; inactive; preserveAspect; ] SplinePhiSharp and SplinePsi are computed by using a Fourier Cosine Series. The series is constructed for the interval [-L, L], using the argument L (default 10) to define the `effective' support of the scaling function. SplinePhiSharp is continued periodically outside [-L, L]. Now use A to control the plot range (sorry about that). These functions are slow; maybe someone can do better. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Basic usage :[font = text; inactive; preserveAspect; ] The non-normalized father wavelet (scaling function): :[font = input; preserveAspect; startGroup; ] Clear[phi]; n = 1; phi[xi_] = N@SplinePhi[n, xi]; Table[{xi, phi[xi]}, {xi, 0, 5.0, 0.5}]//TableForm :[font = output; output; inactive; preserveAspect; endGroup; ] TableForm[{{0, 0.995798922444134}, {0.5, 0.5000154668801171}, {1., 0.002116240424202598}, {1.5, -(5.411666548083005*10^-6)}, {2., -0.00001358209816126887}, {2.5, -0.00001425943351799352}, {3., -(8.4871619802529*10^-6)}, {3.5, -(6.798945908492727*10^-8)}, {4., 6.354788610596352*10^-6}, {4.5, 7.787532497993688*10^-6}, {5., 4.31342879125829*10^-6}}] ;[o] 0 0.995799 0.5 0.500015 1. 0.00211624 -6 1.5 -5.41167 10 2. -0.0000135821 2.5 -0.0000142594 -6 3. -8.48716 10 -8 3.5 -6.79895 10 -6 4. 6.35479 10 -6 4.5 7.78753 10 -6 5. 4.31343 10 :[font = text; inactive; preserveAspect; ] Normalization changes the values: :[font = input; preserveAspect; startGroup; ] Clear[phi]; n = 1; phi[xi_] = N@SplinePhiSharp[n, xi]; Table[{xi, phi[xi]}, {xi, 0, 5.0, 0.5}]//TableForm :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] TableForm[{{0, 1.286723417215949}, {0.5, 0.5585057075594229}, {1., -0.1718328952016476}, {1.5, -0.06972611502451691}, {2., 0.0347830569863749}, {2.5, 0.0136678059149019}, {3., -0.007785257189852614}, {3.5, -0.003013224422318406}, {4., 0.001827659534438453}, {4.5, 0.0007010108753664924}, {5., -0.0004411100072016061}}] ;[o] 0 1.28672 0.5 0.558506 1. -0.171833 1.5 -0.0697261 2. 0.0347831 2.5 0.0136678 3. -0.00778526 3.5 -0.00301322 4. 0.00182766 4.5 0.000701011 5. -0.00044111 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Plotting the father and mother wavelets :[font = input; preserveAspect; ] PlotPhiSharp[n_, L_, A_] := Module[{string, label, phihat}, phi[x_] = N@SplinePhiSharp[n, 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