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

Answer. Set

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.

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Author: M.A. Khamsi

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