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1 - Fourier's representation for functions on, Tp, and ℙN

Published online by Cambridge University Press:  01 September 2010

David W. Kammler
Affiliation:
Southern Illinois University, Carbondale
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Summary

Synthesis and analysis equations

Introduction

In mathematics we often try to synthesize a rather arbitrary function f using a suitable linear combination of certain elementary basis functions. For example, the power functions 1, x, x2, … serve as such basis functions when we synthesize f using the power series representation

f(x) = a0 + a1x + a2x2 + …. (1)

The coefficient ak that specifies the amount of the basis function xk needed in the recipe (1) for constructing f is given by the well-known Maclaurin formula

from elementary calculus. Since the equations for a0, a1, a2,… can be used only in cases where f, f′, f″, … are defined at x = 0, we see that not all functions can be synthesized in this way. The class of analytic functions that do have such power series representations is a large and important one, however, and like Newton [who with justifiable pride referred to the representation (1) as “my method”], you have undoubtedly made use of such power series to evaluate functions, to construct antiderivatives, to compute definite integrals, to solve differential equations, to justify discretization procedures of numerical analysis, etc.

Fourier's representation (developed a century and a half after Newton's) uses as basis functions the complex exponentials

eisx≔cos(2πsx) + i·sin(2πsx), (2)

where s is a real frequency parameter that serves to specify the rate of oscillation, and i2 = -1. When we graph this complex exponential, i.e., when we graph …

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Publisher: Cambridge University Press
Print publication year: 2008

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