By James Brown, Ruel Churchill

Complicated Variables and purposes, 8E

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**Extra info for Complex variables and applications**

**Sample text**

3. Background Facts. Suppose that H is a RKHS on X with kernel functions {kx }x∈X and multiplier algebra M (H). The following are elementary Hilbert space results. 2. Suppose f ∈ H, f ≤ 1, x ∈ H. The maximum possible value of Re f (z) (and hence also of |f (z)|) is the value kz = kz (z)1/2 attained by the unique function f = kˆz . 3. There is a unique Fz,w ∈ H with Fz,w 0 which maximizes Re Fz,w (w). It is given by Fz,w (·) = kw (·) − kz −2 H ≤ 1 and Fz,w (z) = kw (z) kz (·) 2 kw 1− kˆz , kˆw Fz,w (w) = kw 1− kˆz , kˆw .

To make the example computationally tractable we suppose w is radial; w(z) = v(|z|). We deﬁne the weighted Bergman space H2,w by the norming function f 2 H2,w 2 = D |f (z)| w(z)dxdy. The space H2,w is a RKHS on the disk and is an equivalent renorming of H2 but is not related by rescaling. One way to see this is to note that, because the densities 1 and w(z) are both radial, in both cases the monomials are a complete orthogonal set. Thus, in both cases, the kernel function restricted to the diagonal is of the form ∞ |z|2n 2 K(z, z) = 2 = a0 + a1 |z| + · · ·.

2] Mati Abel, Gelfand-Mazur algebras. In ”Topological vector spaces, algebras and related areas. Pitman Research Notes in Math. , Harlow, 1994”, 116–129. [3] Mati Abel, Advertive topological algebras. General topological algebras (Tartu, 1999), 14–24, Math. Stud. (Tartu) 1, Est. Math. , Tartu, 2001. [4] Mati Abel, Descriptions of the topological radical in topological algebras. General topological algebras (Tartu, 1999), 25–31, Math. Stud. (Tartu) 1, Est. Math. , Tartu, 2001. [5] Mati Abel, Inductive limits of Gelfand-Mazur algebras.