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Monday, 25 March 2013

MORE ON Convergence of Fourier Series

Most of the results presented for -periodic functions extend easily to functions 2L-periodic functions. So we only discuss the case of -periodic functions.

Definition. The function f(x) defined on [a,b], is said to be piecewise continuous if and only if, there exits a partition  of [a,b] such that
(1)
f(x) is continuous on [a,b] except may be for the points xi,
(2)
the right-limit and left-limit of f(x) at the points xi exist.
We will say that f(x) is piecewise smooth if and only if f(x) as well as its derivatives are piecewise continuous.

Recall that  is a partition of [a,b] if

a = x1 < x2 < ..< xn-1 < xn = b.

All the known results on the sum, product, and the quotient are valid for piecewise smooth functions. Except for the fundamental theorem of Calculus which needs to be modified. Indeed, we have let f(x) be piecewise smooth function on the open interval a < x < b. There is no reason for f(x) and f'(x) to be defined at the end-points a and b. But if we denote the left-limit and right-limit of f(x) at a point x0 by

the fundamental theorem of Calculus translates into

Before we state the fundamental result on convergence of Fourier series, we need some intermediary results.

Result 1. If f(x) and f'(x) are piecewise continuous on [a,b], then

and

Proof.

Remark. Recall that our initial problem is to approximate a function globally (on an interval versus Taylor approximations which are local). In this context, the approximation of f(x) will be done via the Fourier polynomials

These Fourier polynomials will be called the Fourier partial sums. Since

we obtain

Set

We have the following result:

Result 2. We have

Proof.

Using the result above, we get

This formula is quite interesting since it gives the Fourier polynomials of f(x) without the coefficients.

Definition. The Dirichlet kernel is defined by

The function DN(x) is continuous and periodic, with  as its period. Using the formula above, we get

Now we are ready to state and prove the fundamental result on convergence of Fourier series, due to Dirichlet.

Theorem. Let f(x) be a function, which is twice differentiable, such that f(x), f'(x), and f''(x) are piecewise continuous on the interval . Then, for any , the sequence of Fourier partial sums converges , as n tends to .

Recall that the notation f(x+) (resp. f(x-)) represent the right-limit and left-limit respectively of f at the point x. Let us associate to f the new function S(f) defined by

The conclusion of the Theorem above translates into

Proof.

Example. Show that

This function satisfies the assumptions of the main Theorem. Before, we use the Thoerem's conclusion, let us find its Fourier series. We have

Easy calculations give

The Theorem's conclusion gives the desired identity.