Math Problems Vault

Problem 14.1

Prove Lemma 14.1.

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Remark.

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Lemma 14.1: Given $f \in L^{1}(\mathbf{T})$ and $N \in \mathbb {N}$ , we have $S_{N}^{\mathrm{o}} f = M_{N} S_{N} M_{-N} f$ .

Problem 14.2

Finish the proof of Theorem 14.2.

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Problem 14.3

(a)(b)

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(a)

Suppose that $f \in C(\mathbf{T}), f(0) = 0$ , and $f$ is differentiable at $x = 0$ . Show that $g(x) = f(x) /\left(e^{-2 \pi i x}-1\right) \in C(\mathbf{T})$ and

\sum _{k = -N}^{N} \widehat{f}(k) = \widehat{g}(N+1)-\widehat{g}(-N)

Conclude that the symmetric partial sums of the Fourier series for $f$ converge at the point $x = 0$ to $f(0)$ .

(b)

Show that if $f \in C(\mathbf{T})$ is differentiable at a point $x \in \mathbf{T}$ , then

\lim _{N \rightarrow \infty } \sum _{k = -N}^{N} \widehat{f}(k) e^{2 \pi i k x} = f(x)

Problem 14.4

(a)(b)

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(a)

Suppose that $g \in L^{1}(\mathbf{T})$ . Show that $g$ is real valued if and only if $\overline{\widehat{g}(n)} = \widehat{g}(-n)$ for $n \in \mathbb {Z}$ .

(b)

Show that if $f \in A(\mathbf{T})$ is real valued, then its conjugate function $H f$ is also real valued.

Problem 14.5

Prove Theorem 14.6.

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Remark.

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Theorem 14.6: The Fourier algebra $A(\mathbf{T})$ admits conjugation, and if $f \in A(\mathbf{T})$ , then the twisted partial sums $S_{N}^{\mathrm{t}} f$ converge to $H f$ in $L^{p}$ -norm for each $1 \leq p \leq \infty$ : for all $f \in A(\mathbf{T})$ and $1 \leq p \leq \infty$ , $\lim_{N \rightarrow \infty }\left\| H f-S_{N}^{\mathrm{t}} f\right\|_{L^{p}} = 0$ .

Problem 14.6

Finish the proof of Theorem 14.7.

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Problem 14.7

Given $1 < p < \infty$ , show that conjugation is bounded on $L^{p}(\mathbf{T})$ if and only if it is bounded on $L^{p^{\prime }}(\mathbf{T})$ .

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