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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 < the Bergman space 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 ) Z ( f - 1 ) is not contained in any A μ q zero set.

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