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Remarks on Problem Set 3Math 201B: Winter 20111. Suppose thatP∞n=0cnis a series of complex numbers with partial sumssn=nXk=0ck.The series is Borel summable with Borel sum s if the following limit exists:s = limx→+∞e−x ∞Xn=0snxnn!!.(b) For what complex numbers a ∈ C is the geometric series∞Xn=0anBorel summable? What is its Borel sum? For what a ∈ C is this seriesCes`aro summable? Abel summable?(c) Do you get anything useful from the Borel summation of a Fourier series?Remarks.• As (b) illustrates, Borel summation can give the analytic continuationof a power series outside its radius of convergence, in this case fromthe disc |a| < 1 to the half-plane <a < 1. Roughly speaking, Borelsummation is a more powerful — but cruder — method of summingdivergent series than Abel or Ces`aro summation, which can only suma power series on the boundary |a| = 1 of its disc of convergence.• Borel summation has been used to re-sum divergent perturbation seriesthat arise in quantum field theory and from various PDEs.• (c) The short answer is no. A longer answer is in the paper by Moore.2. Let A(T) denote the space of integrable functions whose Fourier coeffi-cients are absolutely convergent. That is, f ∈ A(T) ifXn∈Zˆf(n)< ∞.1(a) If f ∈ A(T), show that f ∈ C(T). Also show that f ∈ A(T) if and onlyif f = g ∗ h for some functions g, h ∈ L2(T).(b) If f, g ∈ A(T), show that fg ∈ A(T) and expresscfg in terms ofˆf, ˆg.Optional question!(c) Give an example of a function f ∈ C(T) such that f /∈ A(T).Remarks.• (a) The space A(T) of functions with absolutely convergent Fourierseries is not so easy to characterize explicitly.• As (b) shows, A(T) is an algebra with respect to the pointwise prod-uct. This algebra maps under the Fourier transform to `1(Z) with thediscrete convolution product: ifˆf : Z → C, ˆg : Z → C belong to `1(Z)thenˆf ∗ ˆg : Z → C in `1(Z) is defined byˆf ∗ ˆg(k) =Xn∈Zˆf(k − n)ˆg(n).This is dual to the fact that L1(T) is an algebra with respect to theconvolution product, and the Fourier transform maps L1(T) — anal-ogous to `1(Z) — to a sequence subspace of c0(Z) — analogous to thesubspace A(T) of C(T), with the pointwise product.• (c) Continuous functions with non-absolutely convergent Fourier co-efficients are constructed in Zygmund’s book on Trigonometric Seriesc.f. Theorem 4.2 in Chapter V. One example isf(x) =∞Xn=1ein log nneinx.Even though this Fourier series is not absolutely convergent, it con-verges uniformly to f , and f is not only continuous but H¨older con-tinuous with exponent 1/2, meaning that|f(x) − f(y)| ≤ C|x − y|1/2.There are also continuous function whose Fourier series diverge at apoint, or on a set of measure zero, as well as continuous functionswhose Fourier series converge pointwise everywhere but do not con-verge uniformly c.f. Section 1, Chapter VIII of Zygmund.23. Let D = {z ∈ C : |z| < 1} denote the unit disc in the complex plane. TheHardy space H2(D) is the space of functions with a power series expansionF (z) =∞Xn=0cnzn(1)such that∞Xn=0|cn|2< ∞. (2)If F ∈ H2(D), show thatkF k2H2= sup0<r<112πZ2π0Freiθ2dθ < ∞.Show conversely that if F : D → C is a holomorphic function such thatsup0<r<112πZ2π0Freiθ2dθ < ∞then F ∈ H2(D).Remarks.• More generally, if 0 < p < ∞, the Hardy space Hp(D) is defined tobe the set of analytic functions F : D → C that satisfy the followinggrowth condition at the boundary of D:sup0<r<1Z2π0Freiθpdθ < ∞.The Hardy space H∞(D) consists of the bounded analytic functionson D. The only one of these Hardy spaces that is a Hilbert space isH2(D).• For 1 < p ≤ ∞, the Hardy spaces are essentially equivalent to Lp(T),but this is no longer true for 0 < p ≤ 1. A great deal of effort inharmonic analysis has been made to understand the structure of func-tions in these Hardy spaces, originally by the use of complex-variablemethods and subsequently by the use of real-variable methods (whichextend to functions of more than one

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