An Introduction to Harmonic Analysis
This concrete approach to harmonic analysis begins with the circle group T and deals with classical Fourier series in the first five chapters, turning to the real line in chapter six, locally compact abelian groups in chapter seven, and commutative Banach algebras in final chapter.
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analytic AP(R arbitrary assume Banach space bounded function chapter clearly compact support complex numbers condition consequently consider constant contains continuous function convolution Corollary defined DEFINITION denote dense dual element equivalent EXERCISES FOR SECTION Fejer kernel Fejer's finite number Fourier coefficients Fourier series Fourier transform Fourier-Stieltjes coefficients Fourier-Stieltjes transform function F Haar measure Hahn-Banach theorem Hausdorff space hence Hint holomorphic homeomorphism homogeneous Banach space identified implies inequality infinitely differentiable interval lacunary LCA group Lebesgue lemma Let F linear functional LP(R mapping maximal ideal maximal ideal space multiplicative linear functional neighborhood norm obtain open set orthogonal Parseval's formula Plancherel's theorem pointwise Poisson integral positive measure proof is complete prove real numbers real valued Remark satisfying self-adjoint sequence set of divergence Show spectral synthesis subalgebra subset subspace summability kernel tends to zero trigonometric polynomial trigonometric series uniformly continuous valid vanishes weak-star topology write
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