Analytic Functions

A region $D$ is a non-empty open connected subset of the complex plane. Let $\mu$ be planar Lebesgue measure in $D$, and let $A^{2}(D)$ be the set of all complex-valued functions that are analytic throughout $D$ and square-integrable with respect to $\mu$. The set $A^{2}(D)$ is a vector space with respect to pointwise addition and scalar multiplication, and it is an inner-product space with respect to the inner product defined by

When $D$ is the open unit disc, $D = \left\{ z : \left|z\right| < 1\right\}$, the corresponding function space is denoted simply by $A^{2}$.

Let $C$ be the unit circle in the complex plane, $C = \left\{ z : \left|z\right| = 1\right\}$, and let $\mu$ be Lebesgue measure (the extension of arc length) on the Borel sets of $C$, normalized so that $\mu (C) = 1$. If $e_{n}(z) = z^{n}$ for $\left|z\right| = 1$ ($n = 0, \pm 1, \pm 2, \ldots$), then the functions $e_{n}$ form an orthonormal basis for $L^{2}(\mu )$. The space $H^{2}$ is, by definition, the subspace of $L^{2}$ spanned by the $e_{n}$'s with $n \geq 0$; equivalently $H^{2}$ is the orthogonal complement in $L^{2}$ of $\left\{ e_{-1}, e_{-2}, e_{-3}, \ldots \right\}$. The elements of $H^{2}$ are called the analytic elements of $L^{2}$. All statements such as "$f = 0$" or "$f$ is real" are to be interpreted as holding almost everywhere.

A functional Hilbert space is a Hilbert space $\mathbf{H}$ of complex-valued functions on a (non-empty) set $X$; the Hilbert space structure of $\mathbf{H}$ is related to $X$ in two ways. It is required that (1) if $f$ and $g$ are in $\mathbf{H}$ and if $\alpha$ and $\beta$ are scalars, then $(\alpha f + \beta g)(x) = \alpha f(x) + \beta g(x)$ for each $x$ in $X$, i.e., the evaluation functionals on $\mathbf{H}$ are linear, and (2) to each $x$ in $X$ there corresponds a positive constant $\gamma_{x}$, such that $\left|f(x)\right| \leq \gamma_{x} \left\| f\right\|$ for all $f$ in $\mathbf{H}$, i.e., the evaluation functionals on $\mathbf{H}$ are bounded.

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.

If $f$ is an element of a functional Hilbert space $\mathbf{H}$, the complex conjugate $f^{*}$ may fail to belong to $\mathbf{H}$; the spaces $H^{2}$ and $\tilde{H}^{2}$ yield examples. A functional Hilbert space is called self-conjugate if it is closed under the formation of complex conjugates; an example is the sequence space $l^{2}$. A more sophisticated example is the set of all square-integrable complex harmonic functions in the unit disc (the functions of the form $u + iv$, where each of $u$ and $v$ is the real part of some analytic function).

Whenever the answer is yes for all $f$ in the space, then a routine polarization argument shows that $\left\langle f^{*}, g^{*} \right\rangle = \left\langle f, g \right\rangle^{*}$ for all $f$ and $g$.

The relation between $u$ and $v$ is expressed by saying that they are conjugate functions; alternatively, $v$ is the Hilbert transform of $u$.