Functional Approximation through Finite Fourier Series 

INSTITUTO DE ESTUDIOS SUPERIORES DE TAMAULIPAS
MEXICO, MMVI 

Prepared by David Macias Ferrer
E-mail: david.macias@iest.edu.mx
Madero City, Mexico
URL: http://www.geocities.com/dmacias_iest/MyPage.html 

Goals 

• To aproximate a Piecewise Continuous Function through Trigonometric Polynomials commonly called Fourier Partial Sums or Finite Fourier Series.

 

• To show the convergence of these aproximation via Bessel's Inequality and using Maple spreadsheets.

 

• To verify the Weiersstras's Theorem applied to case.

 

• To show the powerful Maple 8 graphics tools to visualize the application of Weierstrass's Theorem.

 

 

Fourier Partial Sums 

The theory of approximation of functions is one of the central branches in mathematical analysis and has been developed over a number of decades. In 1822 the French mathematician Jean Baptiste Joseph Fourier (1768-1830) had completed his important essay "Th?orie Analytique de la Chaleur". Although in his work, the current "Fourier series" don't appear, Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable. In 1900 the Hungarian mathematician Lip?t Fej?r (1880-1959) published a fundamental summation theorem for Fourier series;  he worked on power series and on potential theory. Much of his work is on Fourier series and their singularities but he also contributed to Approximation Theory. Another important contribution to Approximation Theory was the work of the russian mathematician Pafnuty Lvovich Chebyshev (1821-1894) "Th?orie des M?canismes Connus sous le nom de Parall?logrammes" published in 1854. It was in this work that his famous Chebyshev Orthogonals Polynomials appeared for the first time. In this work the notion is found.   

We will use the trigonometric polynomials approximation of the form:    to particular case of a periodic piecewise continuous function defined on closed interval .  

 

>

restart:

 

The piecewise continuous function is given by: 

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

 

(2.1)

 

the graphic is: 

>	plot(f(x),x=-Pi..Pi,color=blue);

 

 

We will calculate the Fourier Coefficients: 

>	cos(n*Pi):=(-1)^n;

 

(2.2)

 

>	sin(n*Pi):=0;

 

(2.3)

 

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

 

(2.4)

 

>	a[n]:=1/Pi*int((f(x)*cos(n*x)),x=-Pi..Pi);

 

(2.5)

 

>	simplify(%,trig);

 

(2.6)

 

which may rewritten as:  

>	a[n]:=(5/(n^2*Pi))*((-1)^n-1);

 

(2.7)

 

>	b[n]:=1/Pi*int((f(x)*sin(n*x)),x=-Pi..Pi);

 

(2.8)

 

>	b[n]:=(-1)^(n+1)/n;

 

(2.9)

 

The following theorem is verified 

Theorem.- Let be a piecewise continuous function on the Fourier coefficients associated satisfy the following relationships: 

 

  and    

with the Maple commands the previous theorem is verified: 

>	limit('a[n]','n'=infinity)=limit(a[n],n=infinity);

 

(2.10)

 

>	limit('b[n]','n'=infinity)=limit(b[n],n=infinity);

 

(2.11)

 

The Finite Fourier Series associated to function is: 

 

 

Exist several types of convergence criteria for functional approximations. One of the most useful for Fourier Approximations is . This clearly implies the following result: 

Theorem. Bessel's Inequality. Let be a function defined on such that has a finite integral on . If and are the Fourier coefficient of the function , then we have: 

 

In particular the series is convergent. We will use this criterion but using the inequality: 

 

this is, the Fourier partial series for . 

Using the Maple command we have: 

>	1/Pi*Int(('f(x)^2'),x=-Pi..Pi)>='a[0]^2'/2+sum('a[n]^2+b[n]^2','n'=1..N);

 

(2.12)

 

where: 

>	1/Pi*Int((f(x)^2),x=-Pi..Pi)=1/Pi*int((f(x)^2),x=-Pi..Pi);

 

(2.13)

 

>	I[B]=evalf(1/Pi*int((f(x)^2),x=-Pi..Pi));

 

(2.14)

 

 

with Fourier coefficients we will use a spreadsheet to show the convergence of Fourier partial series 

 

 

 

In this spreadsheet for ,     and . Although the convergence is slow 

 

 

For n = 1000, we have: 

>	abs((a[0]^2/2+Sum('a[n]^2+b[n]^2','n' = 1 .. 1000))-(1/Pi)*Int(f(x)^2,x = -Pi .. Pi));

 

(2.15)

 

>

evalf(%);

 

(2.16)

 

 

Weierstrass Approximation Theorem 

Theorem. (Weierstrass).- If is a continuous real-valued function on and if any given, then there exist a polynomial on such that: 

 

 

 

for all on . In our problem represent a trigonometric polynomial and  . 

 

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

 

(3.1)

 

>	g1:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 3);

 

(3.2)

 

>	g2:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 10);

 

(3.3)

 

>	plot([f(x),g1(x),g2(x)],x=-Pi..Pi,linestyle=[1,3,3],color=[red,blue,black], legend=["Function","Fourier Series for N=3","Fourier Series for N=10"],thickness=[2,2,2]);

 

 

>

 

>	g3:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 50);

 

(3.4)

 

>	g4:=x->5/4*Pi+sum(5/(n^2*Pi)*((-1)^n-1)*cos(n*x)+(-1)^(n+1)/n*sin(n*x),n = 1 .. 100);

 

(3.5)

 

>	plot([f(x),g3(x)],x=-Pi..Pi,linestyle=[1,3],color=[red,blue], legend=["Function","Fourier Series for N=50"],thickness=[2,2]);

 

 

>

 

 

Is evident that each decreases when n increases, thie is 

 

 

where  

 

Bibliography 

 

Piskunov, N., "C?lculo Diferencial e Integral", Volume II, Spanish Edition, Mir Mosc? Editorial, Mosc?, Russian, 1986 

 

Copyright 2006 

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.