Hinleitung zur Fouriertransformation

Buch:Höhere Mathematik sehen und verstehen, Haftendorn, Riebesehl, Dammer, 
Springer Spektrum, Feb. 2021
Datei FourierSeriesToTransformForTriangle.nb zu Abschnitt .9.6 Seite 121, Abb. 1.81

f[x_]:=Piecewise[{{1,-(π/2)≤x<(π/2)},{0,True}}]

f[x_]:=(E^(-(x^2)))

f[x_]:=(1/(1+(x^2)))

f[x_]:=Piecewise[{{(((x+π))/π),-π≤x<0},{(((π-x))/π),0≤x<π},{0,True}}]

Plot[f[x],{x,-4π,4π},PlotRange->All]

Plot[f[Mod[x+π,2π]-π],{x,-4π,4π},PlotRange->All]

Plot[f[Mod[x+2π,4π]-2π],{x,-6π,6π},PlotRange->All]

s1=Table[{n,(1/π)Integrate[f[x]Cos[n x],{x,-π,π}]},{n,0,5}]

Table[{(n/2),(1/π)Integrate[f[x]Cos[((n x)/2)],{x,-2π,2π}]},{n,0,10}]

p1=ListPlot[s1,PlotStyle->{PointSize[0.055`],RGBColor[1,0.7,0]},AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},PlotRange->All,AxesOrigin->{0,0},    BaseStyle->{FontSize->18}]

p1s=ListPlot[{#[[1]],1#[[2]]}&/@s1,PlotStyle->{PointSize[0.025`],RGBColor[1,0.7,0]},PlotRange->All,BaseStyle->{FontSize->18}]

p1b=BarChart[Last/@s1,PlotRange->All,AxesOrigin->{0,0},BaseStyle->{FontSize->18},BarSpacing->0,ChartStyle->Opacity[1]]

p1b=BarChart[{Last/@s1,Last/@s1}//Transpose//Flatten,PlotRange->All,AxesOrigin->{0,0},BaseStyle->{FontSize->18},BarSpacing->0,ChartStyle->Opacity[0.5]]

s2=Table[{(n/2),(1./(2π))NIntegrate[f[x]Cos[((n x)/2)],{x,-2π,2π}]},{n,0,10}];

p2=ListPlot[s2,PlotStyle->{PointSize[0.045`],RGBColor[0,1,0]},AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},PlotRange->All,BaseStyle->{FontSize->18}]

p2s=ListPlot[{#[[1]],2#[[2]]}&/@s2,PlotStyle->{PointSize[0.025`],RGBColor[0,1,0]},PlotRange->All,BaseStyle->{FontSize->18}]

p2b=BarChart[2Last/@s2,PlotRange->All,AxesOrigin->{0,0},BaseStyle->{FontSize->18},ChartStyle->Hue[0.3],BarSpacing->0]

p21b=Show[p2b,p1b]

s3=Table[{(n/4),(1./(4π))NIntegrate[f[x]Cos[((n x)/4)],{x,-4π,4π},MaxRecursion->10]},{n,0,20}];

p3=ListPlot[s3,PlotStyle->{PointSize[0.035`],RGBColor[0,0,1]},AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},PlotRange->All,BaseStyle->{FontSize->18}]

p3s=ListPlot[{#[[1]],4#[[2]]}&/@s3,PlotStyle->{PointSize[0.025`],Hue[0.7`]},PlotRange->All,BaseStyle->{FontSize->18}]

p3b=BarChart[4Last/@s3,PlotRange->All,AxesOrigin->{0,0},BaseStyle->{FontSize->18},ChartStyle->Hue[0.7],BarSpacing->0]

p3c=BarChart[{4Last/@s3,4Last/@s3}//Transpose//Flatten,AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},PlotRange->All,AxesOrigin->{0,0},BaseStyle->{FontSize->18},BarSpacing->0,ChartStyle->{Hue[0.7]}]

s4=Table[{(n/8),(1./(8π))NIntegrate[f[x]Cos[((n x)/8)],{x,-8π,8π},MaxRecursion->10]},{n,0,40}];

p4=ListPlot[s4,PlotStyle->{PointSize[0.025`],RGBColor[1,0,0]},AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},PlotRange->All,BaseStyle->{FontSize->18}]

p4s=ListPlot[{#[[1]],8#[[2]]}&/@s4,PlotStyle->{PointSize[0.025`],Hue[0]},PlotRange->All,BaseStyle->{FontSize->18}]

p4b=BarChart[8Last/@s4,PlotRange->All,AxesOrigin->{0,0},BaseStyle->{FontSize->18},ChartStyle->Hue[0],BarSpacing->0,ChartStyle->Opacity[0.5]]

p34=Show[p3c,p4b]

F[t_]:=Simplify[(1/π)(∫_(-∞)^∞)f[x]Cos[t x]\[DifferentialD]x,t∈Reals]//Evaluate

F[t]//TeXForm

F[1]

Fps=Plot[F[t/8-1.5/8],{t,0,40},PlotStyle->{Thickness[0.008],RGBColor[0,0,0.7]},AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},BaseStyle->{FontSize->18}]

Fp=Plot[F[t],{t,0,5},PlotStyle->{Thickness[0.008],RGBColor[0,0,0.7], Opacity[0.6]},AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},BaseStyle->{FontSize->18}]

7

Fp4b=Show[Fps,p4b]

(∫_0^∞)(E^(-ω))Cos[ω x]\[DifferentialD]ω

pall=Show[p4s,p3s,p2s,p1s,Fp,AxesStyle->{{Thickness[0.006],Bold},{Thickness[0.006],Bold}},PlotRange->All]

FourierCosTransform[f[x],x,t]

FourierTransform[f[x],x,t]

f1[x_]:=f[Mod[x+π,2π]-π]

pf1=Plot[f1[x],{x,-π,15π},AspectRatio->0.2,Ticks->{Table[2i π,{i,0,7}],{0.5,1}},BaseStyle->FontSize->20]

f2[x_]:=f[Mod[x+2π,4π]-2π]

pf2=Plot[f2[x],{x,-π,15π},AspectRatio->0.2,Ticks->{Table[2i π,{i,0,7}],{0.5,1}},BaseStyle->FontSize->20]

f3[x_]:=f[Mod[x+4π,8π]-4π]

pf3=Plot[f3[x],{x,-π,15π},AspectRatio->0.2,Ticks->{Table[2i π,{i,0,7}],{0.5,1}},BaseStyle->FontSize->20]

f4[x_]:=f[Mod[x+8π,16π]-8π]

pf4=Plot[f4[x],{x,-π,15π},AspectRatio->0.2,Ticks->{Table[2i π,{i,0,7}],{0.5,1}},BaseStyle->FontSize->20]