It turns out that many interesting problems in Fourier analysis do not depend
on the pointwise convergence of Fourier series. Actually, Fourier series is
based on another type of convergence, called 𝔏² convergence. Of
course pointwise convergence is important for Fourier series; however, it is
more convenient ti use Cesàro summation, which is a topic of next section.
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Introduction to Linear Algebra with Mathematica
One of the main issues of Fourier series applications is a possiblity of
restoring the function from its Fourier series. This ill-posed problem is related to the
convergence of Fourier series of a periodic function. This topic is a field known as classical harmonic analysis, a branch of pure mathematics.
Historically, Fourier series was the first example of expansion of a function
with respect to eigenfunctions corresponding to some Sturm--Liouville problem.
Previously, Taylor series were mostly in use and their expansions require a
usage of pointwise convergence,
uniform convergence, and absolute convergence. Taylor's series coefficients
are defined by derivatives at one point---the center of convergence, so only
infenisimal information is used for their determination. In opposite,
eigenfunction expansions are based on the integration over whole interval.
This caused development of another kinds of convergence: 𝔏² spaces, summability, and the the Cesàro mean.
The Fourier series is an example of an eigenfunction
expansion:
The first convergence theorem was proved by the German
mathematician Peter Gustav Lejeune Dirichlet (1805--1859) in 1829.
Theorem (Dirichlet):
Let f(x) be a periodic function with period T = 2ℓ
and satisfy the following conditions (that are usually referred to as
Dirichlet's conditions):
f(x) must be of bounded variation (which means that the function has finite number of local maximum and minimum points and it monotonically increases/decreases between them) in any given bounded interval;
f(x) must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite.
The absolutely integrability condition is a sufficient condition to guarantee existence of Fourier coefficients. The last two conditions of Dichlet's theorem are imposed to apply the second mean value theorem:
Second mean value Theorem:
Let g(x) be a bounded, increasing real valued function defined on the interval [𝑎,b], that is continuous at 𝑎 and b; and f(x) a Riemann integrable function on [𝑎,b]. Then there exists a real number c∈[𝑎,b] such that
\[
\int_a^b g(x)\,f(x)\,{\text d} x = g(a+0) \int_a^c f(x)\,{\text d} x + g(b-0) \int_c^b f(x)\,{\text d} x .
\]
In section on Fourier series, a class of functions,
called piecewise smooth, was introduced. Such functions satisfy Dirichlet's conditions.
Lemma 1:
If f is integrable on the interval [-ℓ,ℓ], then
Lemma 2:
Let f be a continuous periodic function. If its Fourier series converges absolutely, then the Fourier series converges to f uniformly.
; ⧫
Theorem 3:
For piecewise smooth f(x), the Fourier series of f(x) is continuous
and converges to f(x) for −ℓ ≤ x ≤ ℓ if and only if f(x) is
continuous and f(−ℓ) = f(ℓ).
⧫
Theorem 4:
A Fourier series for a 2ℓ-periodic function f(x) that is continuous can be differentiated term by
term if its derivative f'(x) satisfies Dirichlet's conditions.
⧫
Order of Growth
Theorem 4 (Riemann lemma):
Let function f∈𝔏¹ be absolutely integrable on an interval of length T = 2ℓ and 𝑎_{n}, b_{n} be the Fourier coefficients of f. Then
Hewitt, E. and Hewitt, R.E., The Gibbs--Wilbraham phenomenon: An episode in Fourier analysis, Archive for History of Exact Science, 1979, Vol. 21, No. 2, pp. 129--160.
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