A First Course in Harmonic Analysis
This primer in harmonic analysis gives a lean and stream-lined introduction to the central concepts of this beautiful theory. In contrast to other books on the topic, A First Course in Harmonic Analysis is entirely based on the Riemann integral and metric spaces instead of the more demanding Lebesgue integral and abstract topology. Nevertheless, almost all proofs are given in full and all central concepts are presented clearly. This book introduces Fourier analysis, leading up to the Poisson Summation Formula, as well as the techniques used in harmonic analysis of noncommutative groups.
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The Fourier Transform
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assume C+(G called Cauchy sequence Chapter characters complete complex vector space compute continuous function continuously differentiable convergence theorem converges pointwise converges uniformly convolution countable define denote dense Examples Exercises Exercise fcez finite abelian group follows Fourier coefficients Fourier series Fourier transform function f G CC(G G N let given gives GLn(C group homomorphism Haar integral hence Hilbert space Hint implies inner product interval 0,1 invariant integral isometry isomorphic L2-norm LCA group Lemma Let f Let G Lie algebra locally compact locally uniformly Matn(C metric space natural number neighborhood nonnegative norm oo J—oo OO open set orthonormal basis path connected periodic function Plancherel's theorem pre-Hilbert space Proof Proposition Q.E.D. Theorem Re(s real numbers Riemann integrable satisfies sequence xn Show subset subspace tends to zero Theory topology triangle inequality unitary representation vector space xn converges