By Kenneth Hoffman

This vintage of natural arithmetic bargains a rigorous research of Hardy areas and the invariant subspace challenge. Its hugely readable therapy of complicated services, harmonic research, and useful research is appropriate for complex undergraduates and graduate scholars. The textual content positive aspects a hundred hard routines. 1962 edition.

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B) Show that f E L~(O, 21r). (c) Is f of bounded variation on (0, 21r)? Is it (a) Verify that L~(O, 21r)? (d) We wish to determine the Fourier series expansion of f: ;o + Lancosnx. f(x) = n=l Compute an, n 2: 1, by noticing that the integral In= 111" cot ~ sinnxdx does not depend on n (compute In- In-l)· (e) Determine the value of ao and prove that 00 L ~ cosnx =-In (2sin ~), x E (0, 21r). 8 Let x be a real parameter and Iet j(t) = f be defined by exeit. (a) What is the period of f? Show that Cn(f) {~n = n!

N as N--+ +oo. Hereis an example. Take f (t) = { +1 if 0 ~ t < 1f' -1 if 1f ~ t < 21f. By writing the exponentials in terms of sines, we have the following approximations for N = 1, 3, 5: 4 . sin3t); 3 1 . 1 . 4 . ;:(smt + "3 sm3t + "5 sm5t). 5. 3. fl(t) = ~sint. N tends to we have following important general result. f as N increases. 4), tends to otherwise, f in L~(O, a) as N--+ +oo. Expressed 32 Lesson 4. 4. fs(t) y = ~(sint+ isin3t). 5. j5(t) = ~(sint + i sin3t + i sin5t). The proof of this theorem requires more background than is available in these early lessons.

1. (d) Show that k 2: 2. (e) Compute the Fourier series of h and use it to determine the Fourier series of h for all k 2: 2. 10 Let f be the 27r-periodic function defined on [-7r,7r) by f(x) = cosh(ax), a > 0. (a) Show that the Fourier seriesoff converges uniformly to (b) Compute the expansion of f f. in a series of cosines. _ [coth(7ra)- 1l'a - 1~ L.... a 2 + n 2 2a a E R\{0}. n=l (d) Justify the term-by-term differentiation of the series for fand show that . n 2 sinh(a7r) ~( 1)n+1 . 7l' sm ax = xE(-7r,7r).

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Banach spaces of analytic functions by Kenneth Hoffman
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