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Currently displaying 1 – 2 of 2

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Zeros of random functions in Bergman spaces

Joel H. Shapiro — 1979

Annales de l'institut Fourier

Suppose $μ$ is a finite positive rotation invariant Borel measure on the open unit disc $Δ$ , and that the unit circle lies in the closed support of $μ$ . For $0 < p < \infty$ the $A_{μ}^{p}$ is the collection of functions in $L^{p} (μ)$ holomorphic on $Δ$ . We show that whenever a Gaussian power series $f (z) = Σ ζ_{n} a_{n} z^{n}$ almost surely lies in $A_{μ}^{p}$ but not in $⋃_{q > p} A_{μ}^{p}$ , then almost surely: a) the zero set $Z (f)$ of $f$ is not contained in any $A_{μ}^{q}$ zero set ( $q > p$ , and b) $Z (f + 1) \cup Z (f - 1)$ is not contained in any $A_{μ}^{q}$ zero set.

Putnam's Theorem, Alexander's Spectral Area Estimate, and VMO.

Sheldon Axler; Shapiro Joel H. — 1985

Mathematische Annalen

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