Harmonic Analysis

Carlos A. Berenstein and Roger Gay
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Carlos A. Berenstein: University of Maryland, Mathematics Department and Institute for Systems Research
Roger Gay: Université de Bordeaux I, Centre de Recherche en Mathématiques

Chapter Chapter 6 in Complex Analysis and Special Topics in Harmonic Analysis, 1995, pp 353-469 from Springer

Abstract: Abstract The origins of Harmonic Analysis lie in the work of Euler [Eu] and the Bernoullis who proposed to write periodic functions in terms of the exponentials e inx , n ∈ ℤ, in their study of the vibrating string. It is known that every C∞-function which is 2π-periodic in the real line has an expansion of the form En $$\sum\nolimits_{n = - \infty }^{ + \infty } {{a_n}{e^{inx}}} $$ (we remind the reader one can estimate these coefficients a n very precisely, and that we do not need to restrict ourselves to C∞-functions). It was the work of Fourier [Fo] on heat conduction that showed, once and for all, the importance and the interest of such expansions, and since then they have been called Fourier expansions. It is clear that another way of saying that a function f is periodic with period τ is to say that f satisfies the convolution equation $$\left( {{\delta _\tau } - \delta } \right) * f = 0$$

Keywords: Entire Function; Meromorphic Function; Invariant Subspace; Exponential Type; Common Zero (search for similar items in EconPapers)
Date: 1995
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DOI: 10.1007/978-1-4613-8445-8_6

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