M. Riesz theorem on conjugate functions

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Problem 2.17.24 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

The conformal mapping of $U$ onto a vertical strip shows that M. Riesz's theorem on conjugate functions cannot be extended to $p = \infty$ . Deduce that it cannot be extended to $p = 1$ either.

Problem 2.17.25 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

(a)(b)(c)(d)

Suppose $1 < p < \infty$ , and associate with each $f \in L^{p}(T)$ its Fourier coefficients

\hat{f}(n) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } f\left(e^{i t}\right) e^{-i n t} d t \quad (n = 0, \pm 1, \pm 2, \ldots )

Deduce the following statements from Theorem 17.26:

(a)

To each $f \in L^{p}(T)$ there corresponds a function $g \in L^{p}(T)$ such that $\hat{g}(n) = \hat{f}(n)$ for $n \geq 0$ but $\hat{g}(n) = 0$ for all $n < 0$ . In fact, there is a constant $C$ , depending only on $p$ , such that

\left\| g\right\| _{p} \leq C\left\| f\right\| _{p}

The mapping $f \rightarrow g$ is thus a bounded linear projection of $L^{p}(T)$ into $L^{p}(T)$ . The Fourier series of $g$ is obtained from that of $f$ by deleting the terms with $n < 0$ .

(b)

Show that the same is true if we delete the terms with $n < k$ , where $k$ is any given integer.

(c)

Deduce from (b) that the partial sums $s_{n}$ of the Fourier series of any $f \in L^{p}(T)$ form a bounded sequence in $L^{p}(T)$ . Conclude further that we actually have

\lim _{n \rightarrow \infty }\left\| f-s_{n}\right\| _{p} = 0

(d)

If $f \in L^{p}(T)$ and if

F(z) = \sum _{n = 0}^{\infty } \hat{f}(n) z^{n}

then $F \in H^{p}$ , and every $F \in H^{p}$ is so obtained. Thus the projection mentioned in (a) may be regarded as a mapping of $L^{p}(T)$ onto $H^{p}$ .

Problem 2.17.26 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

Show that there is a much simpler proof of Theorem 17.26 if $p = 2$ , and find the best value of $A_{2}$ .

References

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.