Book contents
- Frontmatter
- Contents
- Preface
- 1 Fourier's representation for functions on, Tp, and ℙN
- 2 Convolution of functions on, Tp, and ℙN
- 3 The calculus for finding Fourier transforms of functions on ℝ
- 4 The calculus for finding Fourier transforms of functions on Tp, and ℙN
- 5 Operator identities associated with Fourier analysis
- 6 The fast Fourier transform
- 7 Generalized functions on ℝ
- 8 Sampling
- 9 Partial differential equations
- 10 Wavelets
- 11 Musical tones
- 12 Probability
- Appendices
- Index
4 - The calculus for finding Fourier transforms of functions on Tp, and ℙN
Published online by Cambridge University Press: 01 September 2010
- Frontmatter
- Contents
- Preface
- 1 Fourier's representation for functions on, Tp, and ℙN
- 2 Convolution of functions on, Tp, and ℙN
- 3 The calculus for finding Fourier transforms of functions on ℝ
- 4 The calculus for finding Fourier transforms of functions on Tp, and ℙN
- 5 Operator identities associated with Fourier analysis
- 6 The fast Fourier transform
- 7 Generalized functions on ℝ
- 8 Sampling
- 9 Partial differential equations
- 10 Wavelets
- 11 Musical tones
- 12 Probability
- Appendices
- Index
Summary
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.
- Type
- Chapter
- Information
- A First Course in Fourier Analysis , pp. 173 - 238Publisher: Cambridge University PressPrint publication year: 2008