Reproducing kernel and point-evaluation functionals on H^2

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Problem 37 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 4 "Analytic Functions"

If $\left\{ e_{j}\right\}$ is an orthonormal basis for a functional Hilbert space $\mathbf{H}$ , then the kernel function $K$ of $\mathbf{H}$ is given by

K(x, y) = \sum _{j} e_{j}(x) e_{j}(y)^{*}.

What are the kernel functions of $A^{2}$ and of $\tilde{H}^{2}$ ?

Remark.

If $\mathbf{H}$ is a functional Hilbert space, over $X$ say, then the linear functional $f \mapsto f(y)$ on $\mathbf{H}$ is bounded for each $y$ in $X$ , and, consequently, there exists, for each $y$ in $X$ , an element $K_{y}$ of $\mathbf{H}$ such that $f(y) = \left\langle f, K_{y} \right\rangle$ for all $f$ . The function $K$ on $X \times X$ , defined by $K(x, y) = K_{y}(x)$ , is called the kernel function or the reproducing kernel of $\mathbf{H}$ . The kernel functions of $A^{2}$ and of $\tilde{H}^{2}$ are known, respectively, as the Bergman kernel and the Szegő kernel. If $f \in H^{2}$ , its image $\tilde{f}$ in $\tilde{H}^{2}$ (the set of those functions analytic in the unit disc whose series of Taylor coefficients is square-summable) may be spoken of as the extension of $f$ into the interior.

Problem 39 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 4 "Analytic Functions"

The extension mapping $f \mapsto \tilde{f}$ (from $H^{2}$ to $\tilde{H}^{2}$ ) is continuous not only in the Hilbert space sense, but also in the sense appropriate to analytic functions. That is: if $f_{n} \to f$ in $H^{2}$ , then $\tilde{f}_{n}(z) \to \tilde{f}(z)$ for $\left|z\right| < 1$ , and, in fact, the convergence is uniform on each disc $\left\{ z : \left|z\right| \leq r\right\}$ , $0 < r < 1$ .

Problem 40 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 4 "Analytic Functions"

If an element $f$ of $H^{2}$ is such that the corresponding analytic function $\tilde{f}$ in $\tilde{H}^{2}$ is bounded, then $f$ is bounded (i.e., $f \in H^{\infty }$ ).

Problem 41 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 4 "Analytic Functions"

If $f \in H^{\infty }$ , does it follow that $\tilde{f}$ is bounded?

Problem 42 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 4 "Analytic Functions"

Is the mapping $f \mapsto \tilde{f}$ multiplicative?

Problem 2.17.10 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

Fix $\alpha \in U$ . Prove that the mapping $f \rightarrow f(\alpha )$ is a bounded linear functional on $H^{2}$ . Since $H^{2}$ is a Hilbert space, this functional can be represented as an inner product with some $g \in H^{2}$ . Find this $g$ .

Problem 2.17.11 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.17 "$H^p$ Spaces"

Fix $\alpha \in U$ . How large can $\left|f^{\prime }(\alpha )\right|$ be if $\left\| f\right\|_{2} \leq 1$ ? Find the extremal functions. Do the same for $f^{(n)}(\alpha )$ .

References

[Hal82] Halmos P.R. A Hilbert Space Problem Book. 2nd ed., Springer-Verlag, 1982.

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