By Anton Deitmar

This e-book is a primer in harmonic research at the undergraduate point. It supplies a lean and streamlined advent to the crucial options of this gorgeous and utile idea. unlike different books at the subject, a primary path in Harmonic research is solely according to the Riemann imperative and metric areas rather than the extra tough Lebesgue crucial and summary topology. however, just about all proofs are given in complete and all imperative strategies are provided sincerely. the 1st target of this e-book is to supply an advent to Fourier research, prime as much as the Poisson Summation formulation. the second one goal is to make the reader conscious of the truth that either imperative incarnations of Fourier thought, the Fourier sequence and the Fourier remodel, are distinct instances of a extra common conception bobbing up within the context of in the neighborhood compact abelian teams. The 3rd target of this booklet is to introduce the reader to the suggestions utilized in harmonic research of noncommutative teams. those suggestions are defined within the context of matrix teams as a relevant instance.

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**Sample text**

The claim now follows by the dominated converge nce theorem . D. 5 Th e Fourier tran sform restricted to S gives a bijec tion of th e set S. D . inversion theorem . 6 Let f( x) = e- 1rX . 2 j = Th en f E Sand f. 3 the fun ction f is, up to scalar multiples, the un ique solut ion of t he differential equat ion f ' (x ) = -27rxf(x). 5. PLANCHEREL 'S THEOREM By induction one deduces t hat for every natural number n there is 2 2 a polyn omial P n (x) such t hat f

Let A be a finite abelian group. A character X of A is a group homomorphism X : A ~ 1l' to the unit torus, so X is a map satisfying x(ab) = x(a)x(b) for every a, b E A. Let A. be the set of all characters of A . 57 A. Deitmar, A First Course in Harmonic Analysis © Springer Science+Business Media New York 2002 CHAPTER 4. 2 The point wise product (X, "7) makes A an abelian group. We call dual, of A . N X"7 with A the dual group, or Pontryag in P roof: We have to show t hat X"7 is a char act er when X and "7 are.

These are called the e2 -spaces. Let S be an arbitrary set . Let 2 (S) be the set of functions f : S ---+ C satisfying e IIfl1 2 = L If(s)1 2 < 00. s ES The fact that the sum is finite actually means that all but countably many of the f(s) are zero, and that the sum over those countably many converges absolutely. 4) is L If(s)1 2 sES Note that if S is a finite set, then the convergence condition is vacuous, and so 2 (S) then consists of the finite-dimensional complex vector space of all maps from S to C.