Approximations to Piecewise Continuous Functions

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1 Approximations to Piecewise Continuous Functions Univ.-Prof. Dr.-Ing. habil. Josef BETTEN RWTH Aachen University Templergraben 55 D A a c h e n, Germany betten@mmw.rwth-aachen.de Abstract This worsheet 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[]*cos(*x+b[]*sin(*x,=..infinity; FOURIER_series( x + 2 a 0 ( a cos( x + b sin( x = a[](/pi*int(f(x*cos(*x,x=0..2*pi; a a[0]simplify(subs(=0,%; 0 2 a 0 f( x cos( x 0 2 f( x b[](/pi*int(f(x*sin(*x,x=0..2*pi; 2 b f( x sin( x 0

2 Unit Step Function or HEAVISIDE Unit Function F(xpiecewise(x0 and x<pi,,xpi and x<2*pi,-,x2*pi,; 0 < x and x < F( x - < x and x < 2 2 < x plot(%,x=0..2.*pi,labels=[x,f],color=blac; Alternatively, the piecewise continuous function F(x can be represented as a HEAVISIDE function: alias(h=heaviside: HEAVISIDE[F]convert(F(x,H; HEAVISIDE F H( x 2 H ( x + 2 H( 2 + x plot(%,x=0..2.*pi,color=blac, title="heaviside unit step function"; A[0]value(subs(f(x=F(x,a[0]; A 0 0 A[]simplify(value(subs(f(x=F(x,a[]; A 2 sin( ( + cos( A[]subs(sin(*Pi=0,%; A 0 B[]simplify(value(subs(f(x=F(x,b[]; 2 cos( ( + cos( B B[]simplify(subs(cos(*Pi=(-^,%; 2( (- ( + + (- ( 2 B for i in [,3,5] do y(x,n=isum(b[]*sin(*x,=..i od; 2

3 4 sin( x y ( x, n = 4 sin( x 4 sin( 3 x y ( x, n = sin( x 4 sin( 3 x 4 sin( 5 x y ( x, n = plot({y(x,n=,y(x,n=3,y(x,n=5},x=0..3.5*pi, labels=[x,y],color=blac, title="approximations with n = [, 3, 5]"; for i in [,3,99] do y(x,n=isum(b[]*sin(*x,=..i od: plot({y(x,n=,y(x,n=3,y(x,n=99},x=0..3.5*pi, labels=[x,y],color=blac, title="approximations with n = [, 3, 99]"; L_zwei[n]sqrt((/Pi*Int((F(x_-y(x,n^2,x=0..Pi; L_zwei n ( F( x_ y ( x, n 2 0 for i in [,3,5,99] do L_zwei[n=i] evalf(sqrt((/pi*int((f(x-y(x,n=i^2,x= *pi,4 od; L_zwei n = L_zwei n = L_zwei n = L_zwei n = Example: h(x 3

4 h(xpiecewise(x0 and x<pi,pi,xpi and x<2*pi,x-pi, x2*pi and x<3*pi,pi; h( x 0 < x and x < x < x and x < 2 2 < x and x < 3 plot(%,x=0..3*pi,color=blac,labels=[x,h]; alias(h=heaviside: HEAVISIDE[h]convert(h(x,H; HEAVISIDE h H( x 2 H ( x + x H ( x x H( 2 + x + 2 H( 2 + x H( 3 + x plot(%,x=0..3*pi,color=blac, title="heaviside Function for h(x"; A[0]value(subs(f(x=h(x,a[0]; 3 A 0 2 A[]simplify(value(subs(f(x=h(x,a[]: A[]subs({sin(*Pi=0,cos(*Pi=(-^,(cos(*Pi^2=},%; (- A 2 B[]simplify(value(subs(f(x=h(x,b[]: B[]subs({sin(*Pi=0,cos(*Pi=(-^,(cos(*Pi^2=},%; B (- y(x,na[0]/2+sum(a[]*cos(*x+b[]*sin(*x,=..n; n 3 y ( x, n + 4 ( (- cos( x (- sin( x = 2 4

5 for i in [,2,3] do y(x,n=ia[0]/2+sum(a[]*cos(*x+b[]*sin(*x,=..i od; 3 2 cos( x y ( x, n = + + sin( x cos( x y ( x, n = sin( x sin( 2 x cos( x 2 cos( 3 x y ( x, n = sin( x sin( 2 x + + sin( 3 x alias(h=heaviside,co=color: p[]plot({y(x,n=,y(x,n=2,y(x,n=3},x=0..3*pi,co=blac: p[2]plot(.72*h(x-3*pi,x=0..3.0*pi,co=blac: p[3]plot(h(x,x=0..3*pi, title="approximations with n = [, 2, 3]" : plots[display]({seq(p[],=..3}; for i in [,2,50] do y(x,n=ia[0]/2+sum(a[]*cos(*x+b[]*sin(*x,=..i od: p[4]plot({y(x,n=,y(x,n=2,y(x,n=50},x=0..3*pi,co=blac: p[5]plot(.72*h(x-3*pi,x=0..3.0*pi,co=blac, title="approximations with n = [, 2, 50]": plots[display]({p[4],p[5]}; Smoothing of the FOURIER-Series y(x, n=3: g(,nsin(pi*/n/(pi*/n; # smoothing factor 5

6 sin N N g (, N G(x,n,NA[0]/2+ sum(g(appa,nu*(a[appa]*cos(appa*x+b[appa]*sin(appa*x, appa=..n; # smoothing function n 3 G ( x, n, N + 4 g ( κ, Ν ( A κ cos( κ x + B κ sin( κ x κ = g(,4subs(n=4,g(,n; # N n + 4 sin 4 g (, 4 G(x,n=3,N=4subs({n=3,Nu=4,appa=,g(appa,Nu=g(,4, A[appa]=A[],B[appa]=B[]},G(x,n,N; 3 4 sin ( (- cos( x ( 3 4 G ( x, n = 3, N = = alias(h=heaviside,co=color: p[6]plot({y(x,n=3,g(x,n=3,n=4},x=0..3*pi,co=blac: p[7]plot(.72*h(x-3*pi,x=0..3.0*pi,co=blac: p[8]plot(h(x,x=0..3*pi, title="approximation with n = 3 and its Smoothing": plots[display](seq(p[],=6..8; - sin( x Smoothing of the FOURIER-Series y(x, n = 2: g(,3subs(n=3,g(,n; 3 sin 3 g (, 3 G(x,n=2,N=3evalf(A[0]/2+ sum(g(,3*(a[]*cos(*x+b[]*sin(*x,=..2,4; G ( x, n = 2, N = cos( x sin( x sin( 2. x 6

7 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=blac, title="approximation with n = 2 and its Smoothing": plots[display]({p[7],p[9]}; The above Figures illustrate the good approximations to the given function h(x. Example: K(x K(xpiecewise(x-Pi and x<0,3,x0 and x</2,0, x/2 and x<pi,/x; 3 < x and x < 0 K( x 0 0 < x and x < 2 < x and x < x 2 plot(%,x=-pi..pi,color=blac,labels=[x,k]; alpha[0](/pi*int(f(x,x=-pi..pi; α 0 f( x alpha[](/pi*int(f(x*cos(*x,x=-pi..pi; α f( x cos( x beta[](/pi*int(f(x*sin(*x,x=-pi..pi; β f( x sin( x 7

8 Alpha[0]value(subs(f(x=K(x,alpha[0]; 3 + ln( 2 + ln( Α 0 Alpha[]simplify(value(subs(f(x=K(x,alpha[]; 3 sin( + Ci 2 Ci( Α A[]subs(sin(*Pi=0,%; Ci 2 Ci( A BETA[]simplify(value(subs(f(x=K(x,beta[]; 3 cos( 3 Si + 2 Si( BETA BETA[]subs(cos(*Pi=(-^,%; 3( - 3 Si + 2 Si( BETA y(x,nalpha[0]/2+ sum(alpha[]*cos(*x+beta[]*sin(*x,=..n: y(x,nsubs(sin(*pi=0,%; 3 + ln( 2 + ln( y ( x, n 2 n Ci 2 Ci( ( cos x 3( - 3 Si + 2 Si( sin ( x + + = y(x,subs(n=,y(x,n: Y(x,evalf(%,4; Y ( x, cos( x.477 sin( x y(x,3subs(n=3,y(x,n: Y(x,3evalf(%,4; Y ( x, cos( x.477 sin( x 0.46 cos( 2. x sin( 2. x cos( 3. x sin( 3. x y(x,49subs(n=49,y(x,n: Y(x,49evalf(%: plot({y(x,,y(x,3,y(x,49},x=-pi..pi,co=blac, title="approximations with n = [, 3, 49]"; 8

9 L_two[n]sqrt((/2/Pi*Int((K(x_-Y(x,n^2,x=-Pi..Pi; L_two n 2 2 ( K( x_ Y ( x, n 2 for i in [,3,49] do L_two[n=i]evalf(sqrt((/2/Pi*int((K(x-Y(x,i^2,x=-3..3,4 od; L_two n = L_two n = L_two n = Example: M(x restart: M(xpiecewise(-Pi<x and x<0,, x0 and x</2,2, x/2 and x<pi,/x; < x and x < 0 M( x 2 0 < x and x < 2 < x and x < x 2 plot(%,x=-pi..pi,color=blac,labels=[x,m]; 9

10 a[0](/pi*int(f(x,x=-pi..pi; a 0 f( x a[](/pi*int(f(x*cos(*x,x=-pi..pi; a f( x cos( x b[](/pi*int(f(x*sin(*x,x=-pi..pi; b f( x sin( x A[0]value(subs(f(x=M(x,a[0]; + + ln( 2 + ln( A 0 A[]simplify(value(subs(f(x=M(x,a[]; sin( 2 sin + Ci 2 2 Ci( A A[]subs(sin(Pi*=0,%; 2 sin + Ci 2 2 Ci( A B[]simplify(value(subs(f(x=M(x,b[]; cos( + 2 cos Si Si( B B[]subs(cos(Pi*=(-^,%; (- + 2 cos Si Si( B for i in [,2,3,0] do y(x,n=ievalf(a[0]/2+ sum(a[]*cos(*x+b[]*sin(*x,=..i,4 od; y ( x, n = cos( x sin( x y ( x, n = cos( x sin( x cos( 2. x sin( 2. x y ( x, n = cos( x sin( x cos( 2. x sin( 2. x cos( 3. x sin( 3. x y ( x, n = cos( x sin( x cos( 2. x sin( 2. x cos( 3. x sin( 3. x cos( 4. x sin( 4. x cos( 5. x sin( 5. x cos( 6. x sin( 6. x cos( 7. x sin( 7. x cos( 8. x sin( 8. x cos( 9. x sin( 9. x cos( 0. x sin( 0. x 0

11 alias(h=heaviside,th=thicness,co=color: p[]plot({seq(y(x,n=i,i=..3},x=-pi..pi,co=blac: p[2]plot(m(x,x=-pi..pi,th=2,co=blac: p[3]plot(0.79*h(x+pi,x=-.00*pi *pi,co=blac: p[4]plot(0.796*h(x-pi,x=0.999*pi...00*pi,co=blac, title="approximations with n = [, 2, 3]": plots[display]({seq(p[],=..4}; y(x,n=99evalf(a[0]/2+ sum(a[]*cos(*x+b[]*sin(*x,=..99,4: plot({m(x,y(x,n=99},x=-pi..pi,color=blac, title="m(x and y(x, n = 99"; L_two[n]sqrt((/2/Pi*Int((M(x_-y(x,n^2,x=-Pi..Pi; L_two n 2 2 ( M( x_ y ( x, n 2 for i in [,2,3,99] do L_two[n=i]evalf(sqrt((/2/Pi*int((M(x-y(x,n=i^2, x=-3..3,4 od; L_two n = L_two n =

12 L_two n = L_two n = Smoothing of the FOURIER-Series y(x, n = 3: g(,nn*sin(*pi/n//pi; N sin N g (, N g(,4subs(n=4,%; 4 sin 4 g (, 4 G(x,n=3,N=4evalf(A[0]/2+ sum(g(,4*(a[]*cos(*x+b[]*sin(*x,=..3,4; G ( x, n = 3, N = cos( x 0.34 sin( x cos( 2. x sin( 2. x cos( 3. x sin( 3. x plot({m(x,y(x,n=3,g(x,n=3,n=4},x=-pi..pi,color=blac, title="smoothing of the FOURIER-Series y(x, n = 3"; g(,subs(n=,g(,n; sin g (, G(x,n=0,N=evalf(A[0]/2+ sum(g(,*(a[]*cos(*x+b[]*sin(*x,=..0,4; G ( x, n = 0, N = cos( x sin( x cos( 2. x sin( 2. x cos( 3. x sin( 3. x cos( 4. x 2

13 sin( 4. x cos( 5. x sin( 5. x cos( 6. x sin( 6. x cos( 7. x sin( 7. x cos( 8. x sin( 8. x cos( 9. x sin( 9. x cos( 0. x sin( 0. x plot({m(x,y(x,n=0,g(x,n=0,n=},x=-pi..pi,color=blac, title="fourier-series y(x, n = 0 and its Smoothing"; Résumé In this worsheet it has been illustrated that Maplesoft furnishes powerful tools in finding best approximations to several piecewise continuous functions. 3