Fourier series

Introduction

Now that you know how to find Fourier transforms of functions on ℝ, you can quickly learn to find Fourier transforms of functions on Tp, i.e., to construct the Fourier series

when f is given. In principle, you can always obtain F by evaluating the integrals from the analysis equation

but this is often quite tedious. We will present several other methods for finding these coefficients. You can then select the procedure that requires the least amount of work!

You will recall from your study of Chapter 1 that the synthesis equation (1) for f on Tp can be written as the analysis equation

for F on ℤ. In view of this duality, every Fourier series (1) simultaneously tells us that

Direct integration

You can evaluate the integrals (2) with the techniques from elementary calculus when the function f is a linear combination of segments of

You will use the integration by parts formula

for such calculations. Here q(-1), q(-2), … are successive antiderivatives of

q(x) = e2πkx/p or cos(2πkx/p) or sin(2πkx/p), k=±1, ±2, ….

When f is a polynomial, the integrated term will eventually disappear from the right side of (4), and the resulting identity,

is known as Kronecker's rule. The k = 0 integral is usually done separately.