Discrete Mathematics

Download PDF by Yitzhak Katznelson: An introduction to harmonic analysis

By Yitzhak Katznelson

ISBN-10: 0521543592

ISBN-13: 9780521543590

ISBN-10: 0521838290

ISBN-13: 9780521838290

Whilst the 1st variation of Katznelson's e-book seemed again in 1968 (when i used to be a student), it quickly turned the mentioned, and universally used, reference quantity for a standard instruments of harmonic research: Fourier sequence, Fourier transforms, Fourier analysis/synthesis, the mathematics of time-frequency filtering, causality rules, H^p-spaces, and many of the incarnations of Norbert Wiener's rules at the Fourier remodel within the complicated area, Paley-Wiener, spectral idea, and extra. you could decide up the necessities during this wonderful e-book. Now, decades later, I occasionaly ask starting scholars what their favourite reference is on such things as that, and typically, it's Katznelson. due to Dover, it's at the shelf of so much collage bookstores, and priced below US$ 10.

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1 ; we now claim Lemma. Assume that f, g ∈ A(T). Then f g ∈ A(T) and fg A(T) ≤ f A(T) P ROOF : We have f (t) = fˆ(n)eint , both series converge absolutely: g A(T) . gˆ(n)eint and since g(t) = fˆ(k)ˆ g (m)ei(k+m)t . f (t)g(t) = k m Collecting the terms for which k + m = n we obtain fˆ(k)ˆ g (n − k)eint f (t)g(t) = n so that f g(n) = k k fˆ(k)ˆ g (n − k); hence |fˆ(k)||ˆ g (n − k)| = |f g(n)| ≤ n |fˆ(k)| |ˆ g (n)| . 2 Not every continuous function on T has an absolutely convergent Fourier series, and those that have cannot† be characterized by smoothness conditions (see exercise 5 of this section).

Then 1 2π ∞ fˆ(n)ˆ g (n) . f (t)g(t)dt = n=−∞ I. F OURIER S ERIES ON T 31 We denote by 2 the space of all square summable sequences {an }∞ −∞ , (that is, such that |an |2 < ∞). With pointwise addition and scalar 1 multiplication, and with the norm |an |2 2 and the inner product ∞ {an }, {bn } = −∞ an¯bn , 2 is a Hilbert space. 5 amounts to the statement that the correspondence f → {fˆ(n)} is an isometry between L2 (T) and 2 . EXERCISES FOR SECTION 5 1. Let {ϕn }N n=1 be an orthogonal system in a Hilbert space H.

The mapping f → (1/2π)f (t)dt is an isometric embedding of L1 (T) in M (T). The Fourier coefficients of (1/2π)f (t)dt are precisely fˆ(n), hence a Fourier series is a Fourier-Stieltjes series. An example of a measure that is not obtained as (1/2π)f (t)dt is the so-called Dirac measure; it is the measure δ of mass one concentrated at t = 0. δ can also be defined by f, δ = f (0) for all f ∈ C(T). We denote by δτ , τ ∈ T, the unit mass concentrated at τ . Thus δ = δ0 and f, δτ = f (τ ) for all τ ∈ T.

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An introduction to harmonic analysis by Yitzhak Katznelson

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