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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::J:<6:\"\{\}-%%ROOTG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YGF'-%-BOUNDS_WIDTHG6#$"%+!)F*-%.BOUNDS_HEIGHTG6#$"$!pF*-%)CHILDRENG6" Approximations to Piecewise Continuous Functions Univ.-Prof. Dr.-Ing. habil. Josef BETTEN RWTH Aachen University Templergraben 55 D-52056 A a c h e n , Germany betten@mmw.rwth-aachen.de Abstract This worksheet is concerned with approximations to piecewise continuous functions. It has been illustrated that Maplesoft furnishes powerful tools in finding best approximations to several functions. Some examples are discussed in more detail. Keywords: step functions; HEAVISIDE functions; FOURIER approximation; L-two norm FOURIER Approximation restart: FOURIER_series(x):= a[0]/2+sum(a[k]*cos(k*x)+b[k]*sin(k*x),k=1..infinity);

NiM+LSUvRk9VUklFUl9zZXJpZXNHNiMlInhHLCYqJiMiIiIiIiMiIiImJSJhRzYjIiIhIiIiIiIiLSUkc3VtRzYkLCYqJiYlImFHNiMlImtHIiIiLSUkY29zRzYjKiYlImtHIiIiJSJ4RyIiIiIiIiIiIiomJiUiYkc2IyUia0ciIiItJSRzaW5HNiMqJiUia0ciIiIlInhHIiIiIiIiIiIiLyUia0c7IiIiJSlpbmZpbml0eUciIiI=

a[k]:=(1/Pi)*Int(f(x)*cos(k*x),x=0..2*Pi);

NiM+JiUiYUc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7IiIhLCQqJiIiIyIiIiUjUGlHIiIiIiIiIiIiIiIi

a[0]:=simplify(subs(k=0,%));

NiM+JiUiYUc2IyIiISwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkLSUiZkc2IyUieEcvJSJ4RzsiIiEsJComIiIjIiIiJSNQaUciIiIiIiIiIiIiIiI=

b[k]:=(1/Pi)*Int(f(x)*sin(k*x),x=0..2*Pi);

NiM+JiUiYkc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRzaW5HNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7IiIhLCQqJiIiIyIiIiUjUGlHIiIiIiIiIiIiIiIi

Unit Step Function or HEAVISIDE Unit Function F(x):=piecewise(x>0 and x<Pi,1,x>Pi and x<2*Pi,-1,x>2*Pi,1);

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

plot(%,x=0..2.1*Pi,labels=[x,F],color=black);

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Alternatively, the piecewise continuous function F(x) can be represented as a HEAVISIDE function: alias(H=Heaviside): HEAVISIDE[F]:=convert(F(x),H);

NiM+JiUqSEVBVklTSURFRzYjJSJGRywoLSUiSEc2IyUieEciIiIqJiIiIyIiIi0lIkhHNiMsJiUieEciIiIlI1BpRyEiIiIiIiEiIiomIiIjIiIiLSUiSEc2IywmKiYiIiMiIiIlI1BpRyIiIiEiIiUieEciIiIiIiIiIiI=

plot(%,x=0..2.1*Pi,color=black, title="HEAVISIDE unit step function");

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

A[0]:=value(subs(f(x)=F(x),a[0]));

NiM+JiUiQUc2IyIiISIiIQ==

A[k]:=simplify(value(subs(f(x)=F(x),a[k])));

NiM+JiUiQUc2IyUia0csJCosIiIjIiIiLSUkc2luRzYjKiYlImtHIiIiJSNQaUciIiIiIiIsJiIiIiEiIi0lJGNvc0c2IyomJSJrRyIiIiUjUGlHIiIiIiIiIiIiJSJrRyEiIiUjUGlHISIiISIi

A[k]:=subs(sin(k*Pi)=0,%);

NiM+JiUiQUc2IyUia0ciIiE=

B[k]:=simplify(value(subs(f(x)=F(x),b[k])));

NiM+JiUiQkc2IyUia0csJCosIiIjIiIiLSUkY29zRzYjKiYlImtHIiIiJSNQaUciIiIiIiIsJiIiIiEiIi0lJGNvc0c2IyomJSJrRyIiIiUjUGlHIiIiIiIiIiIiJSJrRyEiIiUjUGlHISIiIiIi

B[k]:=simplify(subs(cos(k*Pi)=(-1)^k,%));

NiM+JiUiQkc2IyUia0csJCoqIiIjIiIiLCYpISIiLCYiIiIiIiIlImtHIiIiIiIiKSEiIiwkKiYiIiMiIiIlImtHIiIiIiIiIiIiIiIiJSJrRyEiIiUjUGlHISIiIiIi

for i in [1,3,5] do y(x,n=i):=sum(B[k]*sin(k*x),k=1..i) od;

NiM+LSUieUc2JCUieEcvJSJuRyIiIiwkKigiIiUiIiIlI1BpRyEiIi0lJHNpbkc2IyUieEciIiIiIiI= NiM+LSUieUc2JCUieEcvJSJuRyIiJCwmKigiIiUiIiIlI1BpRyEiIi0lJHNpbkc2IyUieEciIiIiIiIqJiMiIiUiIiQiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIiIiIiIiI= NiM+LSUieUc2JCUieEcvJSJuRyIiJiwoKigiIiUiIiIlI1BpRyEiIi0lJHNpbkc2IyUieEciIiIiIiIqJiMiIiUiIiQiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIiIiIiIiIqJiMiIiUiIiYiIiIqJiUjUGlHISIiLSUkc2luRzYjLCQqJiIiJiIiIiUieEciIiIiIiIiIiIiIiIiIiI=

plot({y(x,n=1),y(x,n=3),y(x,n=5)},x=0..3.5*Pi, labels=[x,y],color=black, title="Approximations with n = [1, 3, 5]");

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for i in [1,3,99] do y(x,n=i):=sum(B[k]*sin(k*x),k=1..i) od: plot({y(x,n=1),y(x,n=3),y(x,n=99)},x=0..3.5*Pi, labels=[x,y],color=black, title="Approximations with n = [1, 3, 99]");

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L_zwei[n]:=sqrt((1/Pi)*Int((F(x_)-y(x,n))^2,x=0..Pi));

NiM+JiUnTF96d2VpRzYjJSJuRyokKSwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkKiQpLCYtJSJGRzYjJSN4X0ciIiItJSJ5RzYkJSJ4RyUibkchIiIiIiMiIiIvJSJ4RzsiIiElI1BpRyIiIiIiIiMiIiIiIiMiIiI=

for i in [1,3,5,99] do L_zwei[n=i]:= evalf(sqrt((1/Pi)*int((F(x)-y(x,n=i))^2,x=0.05..0.95*Pi)),4) od;

NiM+JiUnTF96d2VpRzYjLyUibkciIiIkIiVlTyEiJQ== NiM+JiUnTF96d2VpRzYjLyUibkciIiQkIiUkSCMhIiU= NiM+JiUnTF96d2VpRzYjLyUibkciIiYkIiVlOyEiJQ== NiM+JiUnTF96d2VpRzYjLyUibkciIyoqJCIlKkciISIm

Example: h(x) h(x):=piecewise(x>0 and x<Pi,Pi,x>Pi and x<2*Pi,x-Pi, x>2*Pi and x<3*Pi,Pi);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUYjNictRiM2Jy1JI21pR0YkNiVRImhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GN1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYwNiVRInhGJ0YzRjYvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRj1GPUZlbkY9LUY6Ni1RIzo9RidGPUY/RkJGREZGRkhGSkZML0ZPUSwwLjI3Nzc3NzhlbUYnL0ZSRlxvLUYjNiYtRjo2LVEifGZyRidGPS9GQEY1RkIvRkVGNUZGRkhGSkZML0ZPUSwwLjE2NjY2NjdlbUYnL0ZSRmZvLUknbXRhYmxlR0YkNjctSSRtdHJHRiQ2Jy1JJG10ZEdGJDYoLUYwNiVRJyYjOTYwO0YnL0Y0RkFGPS8lKXJvd2FsaWduR1EhRicvJSxjb2x1bW5hbGlnbkdGZ3AvJStncm91cGFsaWduR0ZncC8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZecS1GX3A2KC1GIzYpLUYjNictSSNtbkdGJDYkUSIwRidGPS1GOjYtUSI8RidGPUY/RkJGREZGRkhGSkZMRltvRl1vRlhGZW5GPS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC41ZW1GJy8lJmRlcHRoR0Zjci8lKmxpbmVicmVha0dRJWF1dG9GJy1GOjYvUSRhbmRGJy8lJWJvbGRHRjUvRjdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRmJzRj9GQkZERkZGSEZKRkxGTkZRRl5yLUYjNidGWEZbckZhcEZlbkY9RmVuRj1GZXBGaHBGanBGXHFGX3FGZXBGaHBGanAtRlxwNictRl9wNigtRiM2JkZYLUY6Ni1RKCZtaW51cztGJ0Y9Rj9GQkZERkZGSEZKRkwvRk9RLDAuMjIyMjIyMmVtRicvRlJGYXRGYXBGPUZlcEZocEZqcEZccUZfcS1GX3A2KC1GIzYpLUYjNidGYXBGW3JGWEZlbkY9Rl5yRlxzRl5yLUYjNidGWEZbci1GIzYmLUZocTYkUSIyRidGPS1GOjYtUTEmSW52aXNpYmxlVGltZXM7RidGPUY/RkJGREZGRkhGSkZMRk5GUUZhcEY9RmVuRj1GZW5GPUZlcEZocEZqcEZccUZfcUZlcEZocEZqcC1GXHA2J0ZecC1GX3A2KC1GIzYpLUYjNidGW3VGW3JGWEZlbkY9Rl5yRlxzRl5yLUYjNidGWEZbci1GIzYmLUZocTYkUSIzRidGPUZgdUZhcEY9RmVuRj1GZW5GPUZlcEZocEZqcEZccUZfcUZlcEZocEZqcC8lJmFsaWduR1ElYXhpc0YnL0ZmcFEpYmFzZWxpbmVGJy9GaXBRJ2NlbnRlckYnL0ZbcVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjUvJSxjb2x1bW53aWR0aEdGW3MvRmVyRltzLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EkMmVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zody8lJmZyYW1lR0Zody8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRkEvJS1lcXVhbGNvbHVtbnNHRkEvJS1kaXNwbGF5c3R5bGVHRkEvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dRJjAuOGVtRidGZW5GPUZlbkY9RmVuRj0=

plot(%,x=0..3*Pi,color=black,labels=[x,h]);

NictJSdDVVJWRVNHNiM3ZG83JCQiIiEhIiIkIiIhISIiNyQkIjEoKSpRZSE9ej5rISM9JCIxJHoqZWBFZlRKISM6NyQkIjJ1em42T2VSRyIhIz0kIjEkeiplYEVmVEohIzo3JCQiMSdwXjxhUGYjPiEjPCQiMSR6KmVgRWZUSiEjOjckJCIyW2ZOQnM7emMjISM9JCIxJHoqZWBFZlRKISM6NyQkIjEjUi5OM3Y9JlEhIzwkIjEkeiplYEVmVEohIzo3JCQiMiYqPXJZV0xlOCYhIz0kIjEkeiplYEVmVEohIzo3JCQiMSV5MXE7XVBxKCEjPCQiMSR6KmVgRWZUSiEjOjckJCIyekJNKilvbXItIiEjPCQiMSR6KmVgRWZUSiEjOjckJCIycE4sTS5dMmEiISM8JCIxJHoqZWBFZlRKISM6NyQkIjJlWm95UExWMCMhIzwkIjEkeiplYEVmVEohIzo3JCQiMTooejtpbCFbSCEjOyQiMSR6KmVgRWZUSiEjOjckJCIxYjRcbHl6VFEhIzskIjEkeiplYEVmVEohIzo3JCQiMTN0Km9pKSk+JmUhIzskIjEkeiplYEVmVEohIzo3JCQiMUtGYFBVXXZ5ISM7JCIxJHoqZWBFZlRKISM6NyQkIjEkMyNwSkNTKikpKiEjOyQiMSR6KmVgRWZUSiEjOjckJCIxJzRmYVFhYzwiISM6JCIxJHoqZWBFZlRKISM6NyQkIjJgMiYqWzEnKSpvOCEjOyQiMSR6KmVgRWZUSiEjOjckJCIyMWZKVicqSCpvOiEjOyQiMSR6KmVgRWZUSiEjOjckJCIyXyVvOVRFQm88ISM7JCIxJHoqZWBFZlRKISM6NyQkIjEwVm1UKlJLKD4hIzokIjEkeiplYEVmVEohIzo3JCQiMk4peWEkWzVROiMhIzskIjEkeiplYEVmVEohIzo3JCQiMm1FI3BYLDRkQiEjOyQiMSR6KmVgRWZUSiEjOjckJCIyRTdnT1wvN2MjISM7JCIxJHoqZWBFZlRKISM6NyQkIjIlKjQnUipbMXp2IyEjOyQiMSR6KmVgRWZUSiEjOjckJCIyWkBhd2g9cyVHISM7JCIxJHoqZWBFZlRKISM6NyQkIjBMN2Z1SWwkSCEjOSQiMSR6KmVgRWZUSiEjOjckJCIxdygpUUQ2aiopSCEjOiQiMSR6KmVgRWZUSiEjOjckJCIyOkFsW11KRi8kISM7JCIxJHoqZWBFZlRKISM6NyQkIjJVVy5ZcCJHcEkhIzskIjEkeiplYEVmVEohIzo3JCQiMW47TSUpPSRlNCQhIzokIjEkeiplYEVmVEohIzo3JCQiMiV5MkB6cDU0SiEjOyQiMSR6KmVgRWZUSiEjOjckJCIyKSopKXpTMiNRQUohIzskIjEkeiplYEVmVEohIzo3JCQiMmRXOTppPiFISiEjOyQiMSR6KmVgRWZUSiEjOjckJCIxLCFcKm9ybE5KISM6JCIxJHoqZWBFZlRKISM6NyQkIjJtYiRRO1pIVUohIzskIjFGXWpQemk/cSEjPjckJCIyRDY9UUVLKltKISM7JCIxOiU+JEctaFJ0ISM9NyQkIjEocEZxQikqKVFLISM6JCIxdS56ViRlMHQqISM8NyQkIjI7R1AtQGspR0whIzskIjEmKVtabWJycz0hIzs3JCQiMXBrKCpReTtRTiEjOiQiMWRuJ1EmPXZsUiEjOzckJCIxLmRhOnFWQlAhIzokIjEqNGYmPk9XPWUhIzs3JCQiMig9cSNlbTFuI1IhIzskIjFiQVBBLDleeSEjOzckJCIxOCl6PlduLTclISM6JCIxKT4rUil5dSd5KiEjOzckJCIxRWsuMXlBQVYhIzokIjJHallDOk4xPSIhIzs3JCQiMUZHV2wvcDJYISM6JCIyTy5gPSJ5NG04ISM7NyQkIjFNd3UpKnB0MlohIzokIjIwJXk6WFY5bTohIzs3JCQiMSRSKHpDIkhiIlwhIzokIi93P3JrJFJ4IiEjODckJCIxWmhPInk3azQmISM6JCIyUE53eDc/WyY+ISM7NyQkIjEvLCRmM3M8SCYhIzokIjIySlNCVnosOiMhIzs3JCQiMXlqWWp0ZiRcJiEjOiQiMSZldyk0WitfQiEjOjckJCIxPiYpb0FTLyJwJiEjOiQiMUUoKTRwOFhcRCEjOjckJCIxLVMrPVEzIyllISM6JCIxNFVUazZcU0YhIzo3JCQiMSZSMyRSND8lNCchIzokIjJBZz1kRzNFJkghIzs3JCQiMVQlKj50IypcKj0nISM6JCIyJFsnNCc+bSF6LyQhIzs3JCQiMSlbITQyd3olRychIzokIjEkeiplYEVmVEohIzo3JCQiMW03YT4qW2xRJyEjOiQiMSR6KmVgRWZUSiEjOjckJCIxVj8qPkIrJClbJyEjOiQiMSR6KmVgRWZUSiEjOjckJCIxVytmY1Fxc20hIzokIjEkeiplYEVmVEohIzo3JCQiMSNwYmNzMFYobyEjOiQiMSR6KmVgRWZUSiEjOjckJCIxQ2FDdHQqUjEoISM6JCIxJHoqZWBFZlRKISM6NyQkIjEtMDktKXpBRSghIzokIjEkeiplYEVmVEohIzo3JCQiMWhQNj5yOGN1ISM6JCIxJHoqZWBFZlRKISM6NyQkIjJsb0JVdygzZnchIzskIjEkeiplYEVmVEohIzo3JCQiMSUpM3NjTGJheSEjOiQiMSR6KmVgRWZUSiEjOjckJCIxTi54Y15XYSEpISM6JCIxJHoqZWBFZlRKISM6NyQkIjEkPSN6XjxvXyMpISM6JCIxJHoqZWBFZlRKISM6NyQkIjAmNGMlZlNbVikhIzkkIjEkeiplYEVmVEohIzo3JCQiMTkpKSlcPTpPaykhIzokIjEkeiplYEVmVEohIzo3JCQiMWZ1W25jTUkpKSEjOiQiMSR6KmVgRWZUSiEjOjckJCIxWyU9JzMkUiVIISohIzokIjEkeiplYEVmVEohIzo3JCQiMSkqcCZwSygqKj4jKiEjOiQiMSR6KmVgRWZUSiEjOjckJCItJD4lenhDJSohIzYkIjEkeiplYEVmVEohIzotJSVWSUVXRzYkOyQiIiEhIiIkIitpenhDJSohIiolKERFRkFVTFRHLSUrQVhFU0xBQkVMU0c2Jy1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRIng2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRImg2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii0lJUZPTlRHNiUlKERFRkFVTFRHJShERUZBVUxURyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRJJCEiIi0lKUJPVU5EU19ZRzYjJCIjISohIiItJS1CT1VORFNfV0lEVEhHNiMkIiVJQiEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiRTJiEiIi0lKUNISUxEUkVORzYi

alias(H=Heaviside): HEAVISIDE[h]:=convert(h(x),H);

NiM+JiUqSEVBVklTSURFRzYjJSJoRywuKiYlI1BpRyIiIi0lIkhHNiMlInhHIiIiIiIiKigiIiMiIiIlI1BpRyIiIi0lIkhHNiMsJiUieEciIiIlI1BpRyEiIiIiIiEiIiomJSJ4RyIiIi0lIkhHNiMsJiUieEciIiIlI1BpRyEiIiIiIiIiIiomJSJ4RyIiIi0lIkhHNiMsJiomIiIjIiIiJSNQaUciIiIhIiIlInhHIiIiIiIiISIiKigiIiMiIiIlI1BpRyIiIi0lIkhHNiMsJiomIiIjIiIiJSNQaUciIiIhIiIlInhHIiIiIiIiIiIiKiYlI1BpRyIiIi0lIkhHNiMsJiomIiIkIiIiJSNQaUciIiIhIiIlInhHIiIiIiIiISIi

plot(%,x=0..3*Pi,color=black, title="HEAVISIDE Function for h(x)");

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

A[0]:=value(subs(f(x)=h(x),a[0]));

NiM+JiUiQUc2IyIiISwkKigiIiQiIiIiIiMhIiIlI1BpRyIiIiIiIg==

A[k]:=simplify(value(subs(f(x)=h(x),a[k]))): A[k]:=subs({sin(k*Pi)=0,cos(k*Pi)=(-1)^k,(cos(k*Pi))^2=1},%);

NiM+JiUiQUc2IyUia0cqKCwmIiIiIiIiKSEiIiUia0chIiIiIiIlI1BpRyEiIiklImtHIiIjISIi

B[k]:=simplify(value(subs(f(x)=h(x),b[k]))): B[k]:=subs({sin(k*Pi)=0,cos(k*Pi)=(-1)^k,(cos(k*Pi))^2=1},%);

NiM+JiUiQkc2IyUia0csJComJSJrRyEiIikhIiIlImtHIiIiISIi

y(x,n):=A[0]/2+sum(A[k]*cos(k*x)+B[k]*sin(k*x),k=1..n);

NiM+LSUieUc2JCUieEclIm5HLCYqKCIiJCIiIiIiJSEiIiUjUGlHIiIiIiIiLSUkc3VtRzYkLCYqKiwmIiIiIiIiKSEiIiUia0chIiIiIiIlI1BpRyEiIiklImtHIiIjISIiLSUkY29zRzYjKiYlImtHIiIiJSJ4RyIiIiIiIiIiIiooJSJrRyEiIikhIiIlImtHIiIiLSUkc2luRzYjKiYlImtHIiIiJSJ4RyIiIiIiIiEiIi8lImtHOyIiIiUibkciIiI=

for i in [1,2,3] do y(x,n=i):=A[0]/2+sum(A[k]*cos(k*x)+B[k]*sin(k*x),k=1..i) od;

NiM+LSUieUc2JCUieEcvJSJuRyIiIiwoKigiIiQiIiIiIiUhIiIlI1BpRyIiIiIiIiooIiIjIiIiJSNQaUchIiItJSRjb3NHNiMlInhHIiIiIiIiLSUkc2luRzYjJSJ4RyIiIg== NiM+LSUieUc2JCUieEcvJSJuRyIiIywqKigiIiQiIiIiIiUhIiIlI1BpRyIiIiIiIiooIiIjIiIiJSNQaUchIiItJSRjb3NHNiMlInhHIiIiIiIiLSUkc2luRzYjJSJ4RyIiIiomIyIiIiIiIyIiIi0lJHNpbkc2IywkKiYiIiMiIiIlInhHIiIiIiIiIiIiISIi NiM+LSUieUc2JCUieEcvJSJuRyIiJCwuKigiIiQiIiIiIiUhIiIlI1BpRyIiIiIiIiooIiIjIiIiJSNQaUchIiItJSRjb3NHNiMlInhHIiIiIiIiLSUkc2luRzYjJSJ4RyIiIiomIyIiIiIiIyIiIi0lJHNpbkc2IywkKiYiIiMiIiIlInhHIiIiIiIiIiIiISIiKiYjIiIjIiIqIiIiKiYlI1BpRyEiIi0lJGNvc0c2IywkKiYiIiQiIiIlInhHIiIiIiIiIiIiIiIiIiIiKiYjIiIiIiIkIiIiLSUkc2luRzYjLCQqJiIiJCIiIiUieEciIiIiIiIiIiIiIiI=

alias(H=Heaviside,co=color): p[1]:=plot({y(x,n=1),y(x,n=2),y(x,n=3)},x=0..3*Pi,co=black): p[2]:=plot(1.72*H(x-3*Pi),x=0..3.01*Pi,co=black): p[3]:=plot(h(x),x=0..3*Pi, title="Approximations with n = [1, 2, 3]" ): plots[display]({seq(p[k],k=1..3)});

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for i in [1,2,50] do y(x,n=i):=A[0]/2+sum(A[k]*cos(k*x)+B[k]*sin(k*x),k=1..i) od: p[4]:=plot({y(x,n=1),y(x,n=2),y(x,n=50)},x=0..3*Pi,co=black): p[5]:=plot(1.72*H(x-3*Pi),x=0..3.01*Pi,co=black, title="Approximations with n = [1, 2, 50]"): plots[display]({p[4],p[5]});

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Smoothing of the FOURIER-Series y(x, n=3): g(k,N):=sin(Pi*k/N)/(Pi*k/N); # smoothing factor

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G(x,n,N):=A[0]/2+ sum(g(kappa,Nu)*(A[kappa]*cos(kappa*x)+B[kappa]*sin(kappa*x)), kappa=1..n); # smoothing function

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g(k,4):=subs(N=4,g(k,N)); # N := n + 1

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G(x,n=3,N=4):=subs({n=3,Nu=4,kappa=k,g(kappa,Nu)=g(k,4), A[kappa]=A[k],B[kappa]=B[k]},G(x,n,N));

NiM+LSUiR0c2JSUieEcvJSJuRyIiJC8lIk5HIiIlLCYqKCIiJCIiIiIiJSEiIiUjUGlHIiIiIiIiLSUkc3VtRzYkLCQqLCIiJSIiIi0lJHNpbkc2IywkKigiIiUhIiIlImtHIiIiJSNQaUciIiIiIiIiIiIlI1BpRyEiIiUia0chIiIsJioqLCYiIiIiIiIpISIiJSJrRyEiIiIiIiUjUGlHISIiKSUia0ciIiMhIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiIiIiKiglImtHISIiKSEiIiUia0ciIiItJSRzaW5HNiMqJiUia0ciIiIlInhHIiIiIiIiISIiIiIiIiIiLyUia0c7IiIiIiIkIiIi

alias(H=Heaviside,co=color): p[6]:=plot({y(x,n=3),G(x,n=3,N=4)},x=0..3*Pi,co=black): p[7]:=plot(1.72*H(x-3*Pi),x=0..3.01*Pi,co=black): p[8]:=plot(h(x),x=0..3*Pi, title="Approximation with n = 3 and its Smoothing"): plots[display](seq(p[k],k=6..8));

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Smoothing of the FOURIER-Series y(x, n = 2): g(k,3):=subs(N=3,g(k,N));

NiM+LSUiZ0c2JCUia0ciIiQsJCoqIiIkIiIiLSUkc2luRzYjLCQqKCIiJCEiIiUia0ciIiIlI1BpRyIiIiIiIiIiIiUjUGlHISIiJSJrRyEiIiIiIg==

G(x,n=2,N=3):=evalf(A[0]/2+ sum(g(k,3)*(A[k]*cos(k*x)+B[k]*sin(k*x)),k=1..2),4);

NiM+LSUiR0c2JSUieEcvJSJuRyIiIy8lIk5HIiIkLCokIiVjQiEiJCIiIiomJCIla18hIiUiIiItJSRjb3NHNiMlInhHIiIiIiIiKiYkIiVvIykhIiUiIiItJSRzaW5HNiMlInhHIiIiIiIiKiYkIiVuPyEiJSIiIi0lJHNpbkc2IywkKiYkIiIjIiIhIiIiJSJ4RyIiIiIiIiIiIiEiIg==

alias(H=Heaviside,co=color): p[9]:=plot({h(x),y(x,n=2),G(x,n=2,N=3)},x=0..3*Pi,co=black, title="Approximation with n = 2 and its Smoothing"): plots[display]({p[7],p[9]});

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"#5%+HORIZONTALG%+HORIZONTALG-%&TITLEG6$-%)_TYPESETG6#QKApproximation~with~n~=~2~and~its~Smoothing6"-%-TRANSPARENCYG6#$""!!""-%%ROOTG6'-%)BOUNDS_XG6#$"$!G!""-%)BOUNDS_YG6#$"$?#!""-%-BOUNDS_WIDTHG6#$"%5P!""-%.BOUNDS_HEIGHTG6#$"$!*)!""-%)CHILDRENG6"

The above Figures illustrate the good approximations to the given function h(x). Example: K(x) K(x):=piecewise(x>-Pi and x<0,3,x>0 and x<1/2,0, x>1/2 and x<Pi,1/x);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUYjNictRiM2Jy1JI21pR0YkNiVRIktGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GN1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYwNiVRInhGJ0YzRjYvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRj1GPUZlbkY9LUY6Ni1RIzo9RidGPUY/RkJGREZGRkhGSkZML0ZPUSwwLjI3Nzc3NzhlbUYnL0ZSRlxvLUYjNiYtRjo2LVEifGZyRidGPS9GQEY1RkIvRkVGNUZGRkhGSkZML0ZPUSwwLjE2NjY2NjdlbUYnL0ZSRmZvLUknbXRhYmxlR0YkNjctSSRtdHJHRiQ2Jy1JJG10ZEdGJDYoLUkjbW5HRiQ2JFEiM0YnRj0vJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmdwLyUrZ3JvdXBhbGlnbkdGZ3AvJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGXnEtRl9wNigtRiM2KS1GIzYnLUYjNiYtRjo2LVEqJnVtaW51czA7RidGPUY/RkJGREZGRkhGSkZML0ZPUSwwLjIyMjIyMjJlbUYnL0ZSRl1yLUYwNiVRJyYjOTYwO0YnL0Y0RkFGPUZlbkY9LUY6Ni1RIjxGJ0Y9Rj9GQkZERkZGSEZKRkxGW29GXW9GWEZlbkY9LUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjVlbUYnLyUmZGVwdGhHRltzLyUqbGluZWJyZWFrR1ElYXV0b0YnLUY6Ni9RJGFuZEYnLyUlYm9sZEdGNS9GN1ElYm9sZEYnLyUrZm9udHdlaWdodEdGanNGP0ZCRkRGRkZIRkpGTEZORlFGZnItRiM2J0ZYRmNyLUZicDYkUSIwRidGPUZlbkY9RmVuRj1GZXBGaHBGanBGXHFGX3FGZXBGaHBGanAtRlxwNictRl9wNihGX3RGZXBGaHBGanBGXHFGX3EtRl9wNigtRiM2KS1GIzYnRl90RmNyRlhGZW5GPUZmckZkc0Zmci1GIzYnRlhGY3ItSSZtZnJhY0dGJDYoLUZicDYkUSIxRidGPS1GYnA2JFEiMkYnRj0vJS5saW5ldGhpY2tuZXNzR0ZecS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zbdi8lKWJldmVsbGVkR0ZBRmVuRj1GZW5GPUZlcEZocEZqcEZccUZfcUZlcEZocEZqcC1GXHA2Jy1GX3A2KC1GX3U2KC1GIzYlRmF1RmVuRj1GVkZndUZpdUZcdkZedkZlcEZocEZqcEZccUZfcS1GX3A2KC1GIzYpLUYjNidGXnVGY3JGWEZlbkY9RmZyRmRzRmZyLUYjNidGWEZjckZfckZlbkY9RmVuRj1GZXBGaHBGanBGXHFGX3FGZXBGaHBGanAvJSZhbGlnbkdRJWF4aXNGJy9GZnBRKWJhc2VsaW5lRicvRmlwRlt2L0ZbcVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjUvJSxjb2x1bW53aWR0aEdGY3MvRl1zRmNzLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EkMmVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZleC8lJmZyYW1lR0ZleC8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRkEvJS1lcXVhbGNvbHVtbnNHRkEvJS1kaXNwbGF5c3R5bGVHRkEvJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dRJjAuOGVtRidGZW5GPUZlbkY9RmVuRj0=

plot(%,x=-Pi..Pi,color=black,labels=[x,K]);

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

alpha[0]:=(1/Pi)*Int(f(x),x=-Pi..Pi);

NiM+JiUmYWxwaGFHNiMiIiEsJComKiQlI1BpRyEiIiIiIi0lJEludEc2JC0lImZHNiMlInhHLyUieEc7LCQlI1BpRyEiIiUjUGlHIiIiIiIi

alpha[k]:=(1/Pi)*Int(f(x)*cos(k*x),x=-Pi..Pi);

NiM+JiUmYWxwaGFHNiMlImtHLCQqJiokJSNQaUchIiIiIiItJSRJbnRHNiQqJi0lImZHNiMlInhHIiIiLSUkY29zRzYjKiYlImtHIiIiJSJ4RyIiIiIiIi8lInhHOywkJSNQaUchIiIlI1BpRyIiIiIiIg==

beta[k]:=(1/Pi)*Int(f(x)*sin(k*x),x=-Pi..Pi);

NiM+JiUlYmV0YUc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRzaW5HNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7LCQlI1BpRyEiIiUjUGlHIiIiIiIi

Alpha[0]:=value(subs(f(x)=K(x),alpha[0]));

NiM+JiUmQWxwaGFHNiMiIiEqJiUjUGlHISIiLCgqJiIiJCIiIiUjUGlHIiIiIiIiLSUjbG5HNiMiIiMiIiItJSNsbkc2IyUjUGlHIiIiIiIi

Alpha[k]:=simplify(value(subs(f(x)=K(x),alpha[k])));

NiM+JiUmQWxwaGFHNiMlImtHLCQqKCwoKiYiIiQiIiItJSRzaW5HNiMqJiUia0ciIiIlI1BpRyIiIiIiIiEiIiomLSUjQ2lHNiMsJComIiIjISIiJSJrRyIiIiIiIiIiIiUia0ciIiIiIiIqJi0lI0NpRzYjKiYlImtHIiIiJSNQaUciIiIiIiIlImtHIiIiISIiIiIiJSJrRyEiIiUjUGlHISIiISIi

A[k]:=subs(sin(k*Pi)=0,%);

NiM+JiUiQUc2IyUia0csJCooLCYqJi0lI0NpRzYjLCQqJiIiIyEiIiUia0ciIiIiIiIiIiIlImtHIiIiIiIiKiYtJSNDaUc2IyomJSJrRyIiIiUjUGlHIiIiIiIiJSJrRyIiIiEiIiIiIiUia0chIiIlI1BpRyEiIiEiIg==

BETA[k]:=simplify(value(subs(f(x)=K(x),beta[k])));

NiM+JiUlQkVUQUc2IyUia0cqKCwqKiYiIiQiIiItJSRjb3NHNiMqJiUia0ciIiIlI1BpRyIiIiIiIiIiIiIiJCEiIiomLSUjU2lHNiMsJComIiIjISIiJSJrRyIiIiIiIiIiIiUia0ciIiIhIiIqJi0lI1NpRzYjKiYlImtHIiIiJSNQaUciIiIiIiIlImtHIiIiIiIiIiIiJSJrRyEiIiUjUGlHISIi

BETA[k]:=subs(cos(k*Pi)=(-1)^k,%);

NiM+JiUlQkVUQUc2IyUia0cqKCwqKiYiIiQiIiIpISIiJSJrRyIiIiIiIiIiJCEiIiomLSUjU2lHNiMsJComIiIjISIiJSJrRyIiIiIiIiIiIiUia0ciIiIhIiIqJi0lI1NpRzYjKiYlImtHIiIiJSNQaUciIiIiIiIlImtHIiIiIiIiIiIiJSJrRyEiIiUjUGlHISIi

y(x,n):=Alpha[0]/2+ sum(Alpha[k]*cos(k*x)+BETA[k]*sin(k*x),k=1..n): y(x,n):=subs(sin(k*Pi)=0,%);

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

y(x,1):=subs(n=1,y(x,n)): Y(x,1):=evalf(%,4);

NiM+LSUiWUc2JCUieEciIiIsKCQiJSN6IiEiJCIiIiomJCIlKSp6ISImIiIiLSUkY29zRzYjJSJ4RyIiIiIiIiomJCIleDkhIiQiIiItJSRzaW5HNiMlInhHIiIiISIi

y(x,3):=subs(n=3,y(x,n)): Y(x,3):=evalf(%,4);

NiM+LSUiWUc2JCUieEciIiQsMCQiJSN6IiEiJCIiIiomJCIlKSp6ISImIiIiLSUkY29zRzYjJSJ4RyIiIiIiIiomJCIleDkhIiQiIiItJSRzaW5HNiMlInhHIiIiISIiKiYkIiVZNiEiJSIiIi0lJGNvc0c2IywkKiYkIiIjIiIhIiIiJSJ4RyIiIiIiIiIiIiEiIiomJCIlLTohIiUiIiItJSRzaW5HNiMsJComJCIiIyIiISIiIiUieEciIiIiIiIiIiIiIiIqJiQiJWs5ISIlIiIiLSUkY29zRzYjLCQqJiQiIiQiIiEiIiIlInhHIiIiIiIiIiIiISIiKiYkIiVcXyEiJSIiIi0lJHNpbkc2IywkKiYkIiIkIiIhIiIiJSJ4RyIiIiIiIiIiIiEiIg==

y(x,149):=subs(n=149,y(x,n)): Y(x,149):=evalf(%): plot({Y(x,1),Y(x,3),Y(x,149)},x=-Pi..Pi,co=black, title="Approximations with n = [1, 3, 149]");

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L_two[n]:=sqrt((1/2/Pi)*Int((K(x_)-Y(x,n))^2,x=-Pi..Pi));

NiM+JiUmTF90d29HNiMlIm5HLCQqJiMiIiIiIiMiIiIqJikiIiMjIiIiIiIjIiIiKSwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkKiQpLCYtJSJLRzYjJSN4X0ciIiItJSJZRzYkJSJ4RyUibkchIiIiIiMiIiIvJSJ4RzssJCUjUGlHISIiJSNQaUciIiIiIiIjIiIiIiIjIiIiIiIiIiIi

for i in [1,3,149] do L_two[n=i]:=evalf(sqrt((1/2/Pi)*int((K(x)-Y(x,i))^2,x=-3..3)),4) od;

NiM+JiUmTF90d29HNiMvJSJuRyIiIiQiJWRpISIl NiM+JiUmTF90d29HNiMvJSJuRyIiJCQiJXdbISIl NiM+JiUmTF90d29HNiMvJSJuRyIkXCIkIiVObSEiJg==

Example: M(x) restart: M(x):=piecewise(-Pi<x and x<0,1, x>0 and x<1/2,2, x>1/2 and x<Pi,1/x);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUYjNictRiM2Jy1JI21pR0YkNiVRIk1GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GN1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUC1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYwNiVRInhGJ0YzRjYvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRj1GPUZlbkY9LUY6Ni1RIzo9RidGPUY/RkJGREZGRkhGSkZML0ZPUSwwLjI3Nzc3NzhlbUYnL0ZSRlxvLUYjNiYtRjo2LVEifGZyRidGPS9GQEY1RkIvRkVGNUZGRkhGSkZML0ZPUSwwLjE2NjY2NjdlbUYnL0ZSRmZvLUknbXRhYmxlR0YkNjctSSRtdHJHRiQ2Jy1JJG10ZEdGJDYoLUkjbW5HRiQ2JFEiMUYnRj0vJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmdwLyUrZ3JvdXBhbGlnbkdGZ3AvJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGXnEtRl9wNigtRiM2KS1GIzYnLUYjNiYtRjo2LVEqJnVtaW51czA7RidGPUY/RkJGREZGRkhGSkZML0ZPUSwwLjIyMjIyMjJlbUYnL0ZSRl1yLUYwNiVRJyYjOTYwO0YnL0Y0RkFGPUZlbkY9LUY6Ni1RIjxGJ0Y9Rj9GQkZERkZGSEZKRkxGW29GXW9GWEZlbkY9LUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjVlbUYnLyUmZGVwdGhHRltzLyUqbGluZWJyZWFrR1ElYXV0b0YnLUY6Ni9RJGFuZEYnLyUlYm9sZEdGNS9GN1ElYm9sZEYnLyUrZm9udHdlaWdodEdGanNGP0ZCRkRGRkZIRkpGTEZORlFGZnItRiM2J0ZYRmNyLUZicDYkUSIwRidGPUZlbkY9RmVuRj1GZXBGaHBGanBGXHFGX3FGZXBGaHBGanAtRlxwNictRl9wNigtRmJwNiRRIjJGJ0Y9RmVwRmhwRmpwRlxxRl9xLUZfcDYoLUYjNiktRiM2J0ZfdEZjckZYRmVuRj1GZnJGZHNGZnItRiM2J0ZYRmNyLUkmbWZyYWNHRiQ2KEZhcEZmdC8lLmxpbmV0aGlja25lc3NHRl5xLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmh1LyUpYmV2ZWxsZWRHRkFGZW5GPUZlbkY9RmVwRmhwRmpwRlxxRl9xRmVwRmhwRmpwLUZccDYnLUZfcDYoLUZidTYoLUYjNiVGYXBGZW5GPUZWRmR1RmZ1Rml1Rlt2RmVwRmhwRmpwRlxxRl9xLUZfcDYoLUYjNiktRiM2J0ZhdUZjckZYRmVuRj1GZnJGZHNGZnItRiM2J0ZYRmNyRl9yRmVuRj1GZW5GPUZlcEZocEZqcEZccUZfcUZlcEZocEZqcC8lJmFsaWduR1ElYXhpc0YnL0ZmcFEpYmFzZWxpbmVGJy9GaXBGaHUvRltxUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGNS8lLGNvbHVtbndpZHRoR0Zjcy9GXXNGY3MvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSQyZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmJ4LyUmZnJhbWVHRmJ4LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGQS8lLWVxdWFsY29sdW1uc0dGQS8lLWRpc3BsYXlzdHlsZUdGQS8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR1EmMC44ZW1GJ0ZlbkY9RmVuRj1GZW5GPQ==

plot(%,x=-Pi..Pi,color=black,labels=[x,M]);

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

a[0]:=(1/Pi)*Int(f(x),x=-Pi..Pi);

NiM+JiUiYUc2IyIiISwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkLSUiZkc2IyUieEcvJSJ4RzssJCUjUGlHISIiJSNQaUciIiIiIiI=

a[k]:=(1/Pi)*Int(f(x)*cos(k*x),x=-Pi..Pi);

NiM+JiUiYUc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7LCQlI1BpRyEiIiUjUGlHIiIiIiIi

b[k]:=(1/Pi)*Int(f(x)*sin(k*x),x=-Pi..Pi);

NiM+JiUiYkc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRzaW5HNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7LCQlI1BpRyEiIiUjUGlHIiIiIiIi

A[0]:=value(subs(f(x)=M(x),a[0]));

NiM+JiUiQUc2IyIiISomJSNQaUchIiIsKiIiIiIiIiUjUGlHIiIiLSUjbG5HNiMiIiMiIiItJSNsbkc2IyUjUGlHIiIiIiIi

A[k]:=simplify(value(subs(f(x)=M(x),a[k])));

NiM+JiUiQUc2IyUia0csJCooLCotJSRzaW5HNiMqJiUjUGlHIiIiJSJrRyIiIiEiIiomIiIjIiIiLSUkc2luRzYjLCQqJiIiIyEiIiUia0ciIiIiIiIiIiIhIiIqJi0lI0NpRzYjLCQqJiIiIyEiIiUia0ciIiIiIiIiIiIlImtHIiIiIiIiKiYtJSNDaUc2IyomJSNQaUciIiIlImtHIiIiIiIiJSJrRyIiIiEiIiIiIiUjUGlHISIiJSJrRyEiIiEiIg==

A[k]:=subs(sin(Pi*k)=0,%);

NiM+JiUiQUc2IyUia0csJCooLCgqJiIiIyIiIi0lJHNpbkc2IywkKiYiIiMhIiIlImtHIiIiIiIiIiIiISIiKiYtJSNDaUc2IywkKiYiIiMhIiIlImtHIiIiIiIiIiIiJSJrRyIiIiIiIiomLSUjQ2lHNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiUia0ciIiIhIiIiIiIlI1BpRyEiIiUia0chIiIhIiI=

B[k]:=simplify(value(subs(f(x)=M(x),b[k])));

NiM+JiUiQkc2IyUia0cqKCwsLSUkY29zRzYjKiYlI1BpRyIiIiUia0ciIiIiIiIiIiIiIiIqJiIiIyIiIi0lJGNvc0c2IywkKiYiIiMhIiIlImtHIiIiIiIiIiIiISIiKiYtJSNTaUc2IywkKiYiIiMhIiIlImtHIiIiIiIiIiIiJSJrRyIiIiEiIiomLSUjU2lHNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiUia0ciIiIiIiIiIiIlI1BpRyEiIiUia0chIiI=

B[k]:=subs(cos(Pi*k)=(-1)^k,%);

NiM+JiUiQkc2IyUia0cqKCwsKSEiIiUia0ciIiIiIiIiIiIqJiIiIyIiIi0lJGNvc0c2IywkKiYiIiMhIiIlImtHIiIiIiIiIiIiISIiKiYtJSNTaUc2IywkKiYiIiMhIiIlImtHIiIiIiIiIiIiJSJrRyIiIiEiIiomLSUjU2lHNiMqJiUjUGlHIiIiJSJrRyIiIiIiIiUia0ciIiIiIiIiIiIlI1BpRyEiIiUia0chIiI=

for i in [1,2,3,10] do y(x,n=i):=evalf(A[0]/2+ sum(A[k]*cos(k*x)+B[k]*sin(k*x),k=1..i),4) od;

NiM+LSUieUc2JCUieEcvJSJuRyIiIiwoJCIlOiYqISIlIiIiKiYkIiVhUSEiJSIiIi0lJGNvc0c2IyUieEciIiIiIiIqJiQiJWc3ISIlIiIiLSUkc2luRzYjJSJ4RyIiIiEiIg== NiM+LSUieUc2JCUieEcvJSJuRyIiIywsJCIlOiYqISIlIiIiKiYkIiVhUSEiJSIiIi0lJGNvc0c2IyUieEciIiIiIiIqJiQiJWc3ISIlIiIiLSUkc2luRzYjJSJ4RyIiIiEiIiomJCIlSzohIiUiIiItJSRjb3NHNiMsJComJCIiIyIiISIiIiUieEciIiIiIiIiIiIiIiIqJiQiJWtIISIlIiIiLSUkc2luRzYjLCQqJiQiIiMiIiEiIiIlInhHIiIiIiIiIiIiIiIi NiM+LSUieUc2JCUieEcvJSJuRyIiJCwwJCIlOiYqISIlIiIiKiYkIiVhUSEiJSIiIi0lJGNvc0c2IyUieEciIiIiIiIqJiQiJWc3ISIlIiIiLSUkc2luRzYjJSJ4RyIiIiEiIiomJCIlSzohIiUiIiItJSRjb3NHNiMsJComJCIiIyIiISIiIiUieEciIiIiIiIiIiIiIiIqJiQiJWtIISIlIiIiLSUkc2luRzYjLCQqJiQiIiMiIiEiIiIlInhHIiIiIiIiIiIiIiIiKiYkIiVIbCEiJiIiIi0lJGNvc0c2IywkKiYkIiIkIiIhIiIiJSJ4RyIiIiIiIiIiIiIiIiomJCIlVScqISImIiIiLSUkc2luRzYjLCQqJiQiIiQiIiEiIiIlInhHIiIiIiIiIiIiIiIi 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

alias(H=Heaviside,th=thickness,co=color): p[1]:=plot({seq(y(x,n=i),i=1..3)},x=-Pi..Pi,co=black): p[2]:=plot(M(x),x=-Pi..Pi,th=2,co=black): p[3]:=plot(0.7191*H(x+Pi),x=-1.001*Pi..-0.999*Pi,co=black): p[4]:=plot(0.7196*H(x-Pi),x=0.999*Pi..1.001*Pi,co=black, title="Approximations with n = [1, 2, 3]"): plots[display]({seq(p[k],k=1..4)});

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y(x,n=99):=evalf(A[0]/2+ sum(A[k]*cos(k*x)+B[k]*sin(k*x),k=1..99),4): plot({M(x),y(x,n=99)},x=-Pi..Pi,color=black, title="M(x) and y(x, n = 99)");

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L_two[n]:=sqrt((1/2/Pi)*Int((M(x_)-y(x,n))^2,x=-Pi..Pi));

NiM+JiUmTF90d29HNiMlIm5HLCQqJiMiIiIiIiMiIiIqJikiIiMjIiIiIiIjIiIiKSwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkKiQpLCYtJSJNRzYjJSN4X0ciIiItJSJ5RzYkJSJ4RyUibkchIiIiIiMiIiIvJSJ4RzssJCUjUGlHISIiJSNQaUciIiIiIiIjIiIiIiIjIiIiIiIiIiIi

for i in [1,2,3,99] do L_two[n=i]:=evalf(sqrt((1/2/Pi)*int((M(x)-y(x,n=i))^2, x=-3..3)),4) od;

NiM+JiUmTF90d29HNiMvJSJuRyIiIiQiJVtJISIl NiM+JiUmTF90d29HNiMvJSJuRyIiIyQiJWo+ISIl NiM+JiUmTF90d29HNiMvJSJuRyIiJCQiJXE8ISIl NiM+JiUmTF90d29HNiMvJSJuRyIjKiokIiV5QSEiJg==

Smoothing of the FOURIER-Series y(x, n = 3): g(k,N):=N*sin(k*Pi/N)/k/Pi;

NiM+LSUiZ0c2JCUia0clIk5HKiolIk5HIiIiLSUkc2luRzYjKiglImtHIiIiJSNQaUciIiIlIk5HISIiIiIiJSJrRyEiIiUjUGlHISIi

g(k,4):=subs(N=4,%);

NiM+LSUiZ0c2JCUia0ciIiUsJCoqIiIlIiIiLSUkc2luRzYjLCQqKCIiJSEiIiUjUGlHIiIiJSJrRyIiIiIiIiIiIiUia0chIiIlI1BpRyEiIiIiIg==

G(x,n=3,N=4):=evalf(A[0]/2+ sum(g(k,4)*(A[k]*cos(k*x)+B[k]*sin(k*x)),k=1..3),4);

NiM+LSUiR0c2JSUieEcvJSJuRyIiJC8lIk5HIiIlLDAkIiU6JiohIiUiIiIqJiQiJW9NISIlIiIiLSUkY29zRzYjJSJ4RyIiIiIiIiomJCIlTTYhIiUiIiItJSRzaW5HNiMlInhHIiIiISIiKiYkIiVfKCohIiYiIiItJSRjb3NHNiMsJComJCIiIyIiISIiIiUieEciIiIiIiIiIiIiIiIqJiQiJSgpPSEiJSIiIi0lJHNpbkc2IywkKiYkIiIjIiIhIiIiJSJ4RyIiIiIiIiIiIiIiIiomJCIlZj4hIiYiIiItJSRjb3NHNiMsJComJCIiJCIiISIiIiUieEciIiIiIiIiIiIiIiIqJiQiJSQqRyEiJiIiIi0lJHNpbkc2IywkKiYkIiIkIiIhIiIiJSJ4RyIiIiIiIiIiIiIiIg==

plot({M(x),y(x,n=3),G(x,n=3,N=4)},x=-Pi..Pi,color=black, title="Smoothing of the FOURIER-Series y(x, n = 3)");

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g(k,11):=subs(N=11,g(k,N));

NiM+LSUiZ0c2JCUia0ciIzYsJCoqIiM2IiIiLSUkc2luRzYjLCQqKCIjNiEiIiUjUGlHIiIiJSJrRyIiIiIiIiIiIiUia0chIiIlI1BpRyEiIiIiIg==

G(x,n=10,N=11):=evalf(A[0]/2+ sum(g(k,11)*(A[k]*cos(k*x)+B[k]*sin(k*x)),k=1..10),4);

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

plot({M(x),y(x,n=10),G(x,n=10,N=11)},x=-Pi..Pi,color=black, title="FOURIER-Series y(x, n = 10) and its Smoothing");

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R\303\251sum\303\251 In this worksheet it has been illustrated that Maplesoft furnishes powerful tools in finding best approximations to several piecewise continuous functions.