Titlebook: Harmonic Analysis; Henry Helson Book 2010Latest edition Hindustan Book Agency (India) 2010

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The Fourier Integral,The Fourier integral was introduced in Sections 2 and 3 of Chapter 1, and some results were proved analogous to those already known for Fourier series. Now the Fourier integral is our subject. First the things we know will be summarized.

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Hardy Spaces,For 1 ≤ . ≤ ∞, .(.) is the subspace of .(.) consisting of . such that .(.) = 0 for all . < 0. This subspace is closed in .(.), and *-closed if . > 1 (when .(.) is a dual space). The functions of .(.) have Fourier series.. Thus the harmonic extension. is actually analytic.

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Hindustan Book Agency (India) 2010

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Fourier Series and Integrals,we replace Lebesgue measure . on the interval (0, 2.) by .(.) = ./2.. We shall generally omit the limits of integration when the measure is .; they are always 0 and 2., or another interval of the same length.

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Translation,e Fourier transform to multiplication by exponentials. Thus much of Chapter 4 was about such subspaces. The first objective of this chapter is to characterize the closed subspaces of .(.) invariant under all translations, or under translations to the right. These results are analogous to theorems of Chapter 4 on the circle.

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