Zeros of random functions in Bergman spaces

Joel H. Shapiro

Annales de l'institut Fourier (1979)

  • Volume: 29, Issue: 4, page 159-171
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Shapiro, Joel H.. "Zeros of random functions in Bergman spaces." Annales de l'institut Fourier 29.4 (1979): 159-171. <http://eudml.org/doc/74429>.

@article{Shapiro1979,
abstract = {Suppose $\mu $ is a finite positive rotation invariant Borel measure on the open unit disc $\Delta $, and that the unit circle lies in the closed support of $\mu $. For $0&lt; p&lt; \infty $ the Bergman space$A^p_\mu $ is the collection of functions in $L^p(\mu )$ holomorphic on $\Delta $. We show that whenever a Gaussian power series $f(z) = \Sigma \zeta _na_nz^n$ almost surely lies in $A^p_\mu $ but not in $\bigcup _\{q&gt;p\} A^p_\mu $, then almost surely: a) the zero set $Z(f)$ of $f$ is not contained in any $A^q_\mu $ zero set ($q&gt;p$, and b) $Z(f+1)\cup Z(f-1)$ is not contained in any $A^q_\mu $ zero set.},
author = {Shapiro, Joel H.},
journal = {Annales de l'institut Fourier},
keywords = {Random Functions; Bergman Spaces; Rotation Invariant Borel Measure; Gaussian Power Series},
language = {eng},
number = {4},
pages = {159-171},
publisher = {Association des Annales de l'Institut Fourier},
title = {Zeros of random functions in Bergman spaces},
url = {http://eudml.org/doc/74429},
volume = {29},
year = {1979},
}

TY - JOUR
AU - Shapiro, Joel H.
TI - Zeros of random functions in Bergman spaces
JO - Annales de l'institut Fourier
PY - 1979
PB - Association des Annales de l'Institut Fourier
VL - 29
IS - 4
SP - 159
EP - 171
AB - Suppose $\mu $ is a finite positive rotation invariant Borel measure on the open unit disc $\Delta $, and that the unit circle lies in the closed support of $\mu $. For $0&lt; p&lt; \infty $ the Bergman space$A^p_\mu $ is the collection of functions in $L^p(\mu )$ holomorphic on $\Delta $. We show that whenever a Gaussian power series $f(z) = \Sigma \zeta _na_nz^n$ almost surely lies in $A^p_\mu $ but not in $\bigcup _{q&gt;p} A^p_\mu $, then almost surely: a) the zero set $Z(f)$ of $f$ is not contained in any $A^q_\mu $ zero set ($q&gt;p$, and b) $Z(f+1)\cup Z(f-1)$ is not contained in any $A^q_\mu $ zero set.
LA - eng
KW - Random Functions; Bergman Spaces; Rotation Invariant Borel Measure; Gaussian Power Series
UR - http://eudml.org/doc/74429
ER -

References

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  1. [1] X. FERNIQUE, Intégrabilité des vecteurs gaussiens, C.R. Acad. Sci., Paris, 270 (1970), 1698-1699. Zbl0206.19002MR42 #1170
  2. [2] Ch. HOROWITZ, Zeros of functions in Bergman spaces, Duke Math. J., 41 (1974), 693-710. Zbl0293.30035MR50 #10215
  3. [3] J.P. KAHANE, Some Random Series of Functions, D.C. Heath and Co., Lexington, MA, 1968. Zbl0192.53801MR40 #8095
  4. [4] W. RUDIN, Zeros of holomorphic functions in balls, Indag. Math., 38 (1976), 57-65. Zbl0319.32003MR52 #14347
  5. [5] W. RUDIN, Principles of Real Analysis, 3rd ed., McGraw-Hill, New York, 1976. Zbl0346.26002
  6. [6] W. RUDIN, Real and Complex Analysis, McGraw-Hill, New York, 1974. Zbl0278.26001MR49 #8783
  7. [7] J. H. SHAPIRO, Zeros of functions in weighted Bergman spaces, Michigan Math. J., 24 (1977), 243-256. Zbl0376.30009MR57 #3404

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