Fourier Integral

Roald M. Trigub and Eduard S. Bellinsky
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Roald M. Trigub: Donetsk National University
Eduard S. Bellinsky: University of West Indies

Chapter Chapter 3 in Fourier Analysis and Approximation of Functions, 2004, pp 67-104 from Springer

Abstract: Abstract If f : ℝℶℂ, then its Fourier transform is defined as (3.0.1) $$ \hat f\left( y \right) = \left( {2\pi } \right){ - ^{1/2}}\int_{ - \infty }^\infty {f\left( x \right){e^{ - iyx}}dx} = {\left( {2\pi } \right)^{1/2}}\int_{ - \infty }^\infty {f\left( x \right)\cos yxdx - i{{\left( {2\pi } \right)}^{1/2}}\int_{ - \infty }^\infty {f\left( x \right)\sin yxdx,} } $$ provided this integral converges in some sense. This is an analog of the Fourier coefficients of a periodic function.

Keywords: Entire Function; Exponential Type; Inversion Formula; Riesz Basis; Algebraic Polynomial (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-1-4020-2876-2_3

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