Trigonometric Fourier Series - Fourier Series and Integral Transforms

University Mathematics Handbook (2015)

XIII. Fourier Series and Integral Transforms

Chapter 1. Trigonometric Fourier Series

1.1 Basic Concepts

a. Let be the inner product space of piecewise continuous functions defined on and with values in :

can be extended periodically on all such that .

b. For the inner product .

c. The norm of is .

d. The sequence of functions

is an orthonormal system on (see X.13.3).

1.2 Fourier Series

a. The series

when

is a Fourier series corresponding to . and are the Fourier coefficients of .

b. A corresponding Fourier series to defines periodic function on , with a period , that is, , and , except, perhaps, the points of discontinuity of .

c. Example: Given , Fourier coefficients are

and the Fourier series is .

The following illustration shows functions and , which is a partial sum of 's Fourier series:

d. Uniqueness theorem: Let . If the Fourier series corresponding to is equal to that of , then in each except, perhaps, a finite number of points.

e. For every trigonometric polynomial , the expression attains its minimum when and are the Fourier coefficients of on .

f. If is an even function, then

g. If is an odd function , then

1.3 Convergence of Fourier Series

a. Dirichlet theorem: If and has one-sided derivatives (see IV.2.3, 4.6), in every point on , then, for every , the corresponding Fourier series of converges to , and at the ends of interval , the series converges to .

b. Riemann-Lebesgue lemma: If and are Fourier coefficients of , then .

c. Bessel's inequality: If and are Fourier coefficients of , then,

d. Parseval's identity: For every , there holds

when and are Fourier coefficients of .

e. Generalized Parseval's identity: For every ,

when are Fourier coefficients of , and are Fourier coefficients of .

1.4 Uniform Convergence

a. If function is continuous on , , and , then Fourier series corresponding to uniformly converges to on every interval of .

b. Let and let be all the jump points of on . If is a subinterval of containing no point , then Fourier series of uniformly converges to on .

c. If and , then for every there holds

and the series uniformly converges on .

d. If is continuous on , , and is piecewise continuous on , then, the differentiation term by term of Fourier series corresponding to , results in Fourier series of :