By Anton Deitmar
This e-book is a primer in harmonic research at the undergraduate point. It offers a lean and streamlined advent to the vital suggestions of this gorgeous and utile thought. unlike different books at the subject, a primary path in Harmonic research is fullyyt in keeping with the Riemann fundamental and metric areas rather than the extra difficult Lebesgue crucial and summary topology. however, just about all proofs are given in complete and all relevant thoughts are awarded sincerely. the 1st goal of this e-book is to supply an creation to Fourier research, major as much as the Poisson Summation formulation. the second one objective is to make the reader conscious of the truth that either critical incarnations of Fourier thought, the Fourier sequence and the Fourier remodel, are distinctive circumstances of a extra common idea coming up within the context of in the neighborhood compact abelian teams. The 3rd target of this ebook is to introduce the reader to the recommendations utilized in harmonic research of noncommutative teams. those ideas are defined within the context of matrix teams as a important instance.
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Additional info for A First Course in Harmonic Analysis
The following lemma will be used later on. 4 Let V be a pre-Hilbert space with completion H, and let H' be a Hilbert subspace of H containing V. Then H' = H. Proof: Let h E H. Then there is a sequence V n in V converging to h. It follows that (vn ) must be a Cauchy sequence in V C H', which is therefore convergent in H', and hence its limit h lies in H' . D. 4 Fourier Series Revisited In the previous chapter we saw that the space C(IR/Z) with the inner product (1, g) = f(x)g(x)dx is a pre-Hilbert space.
1 (Main theorem on finite abelian groups) Any finite abelian group A is a product A = Al X A2 X ... X A k of cyclic groups. 1. D. 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.
Some of the main results of the first chapter can be summarized in the following theorem. 1 The exponentials ek(x) = e27rikx, k E Z, form an orthonormal basis (ek)kEz of the Hilbert space L 2(IR/Z). 1. Hence the (ek) form an orthonormal system. Let H be the space of all series of the form I:kEZ Ckek with I: kEZ ICkl2 < 00, which CHAPTER 2. HILBERT SPACES 34 therefore conver ges in L 2 (JR j Z). Then the map I:: kEZ Cke k H ( Ck) kE Z gives an isomorphism to f2(Z), and henc e H is a Hilbert subspac e of L 2 (JRj Z).
A First Course in Harmonic Analysis by Anton Deitmar