Zeros of functions in weighted spaces with mixed norm

2013 ◽  
Vol 94 (1-2) ◽  
pp. 266-280 ◽  
Author(s):  
E. A. Sevast’yanov ◽  
A. A. Dolgoborodov
Keyword(s):  
Weighted Spaces ◽  
Mixed Norm ◽  
Zeros Of Functions

scholarly journals Zeros of Functions in Bergman-Type Hilbert Spaces of Dirichlet Series

2016 ◽  
Vol 119 (2) ◽  
pp. 237
Author(s):  
Ole Fredrik Brevig
Keyword(s):  
Hilbert Space ◽  
Real Number ◽  
Dirichlet Series ◽  
Hilbert Spaces ◽  
Hardy Spaces ◽  
Weighted Spaces ◽  
Number Of Divisors ◽  
Zeros Of Functions

For a real number $\alpha$ the Hilbert space $\mathscr{D}_\alpha$ consists of those Dirichlet series $\sum_{n=1}^\infty a_n/n^s$ for which $\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha < \infty$, where $d(n)$ denotes the number of divisors of $n$. We extend a theorem of Seip on the bounded zero sequences of functions in $\mathscr{D}_\alpha$ to the case $\alpha>0$. Generalizations to other weighted spaces of Dirichlet series are also discussed, as are partial results on the zeros of functions in the Hardy spaces of Dirichlet series $\mathscr{H}^p$, for $1\leq p <2$.

scholarly journals HARDY–BLOCH TYPE SPACES AND LACUNARY SERIES ON THE POLYDISK

2007 ◽  
Vol 49 (2) ◽  
pp. 345-356 ◽  
Author(s):  
K. L. AVETISYAN
Keyword(s):  
Lacunary Series ◽  
Weighted Spaces ◽  
Mixed Norm ◽  
Fractional Operators ◽  
Mixed Norm Spaces ◽  
Sharp Estimates ◽  
Type Spaces ◽  
Unit Polydisk ◽  
The Mean

AbstractWe extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of $\C^n$. Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h(p, α), h(p, log(α)) of Hardy–Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixed norm spaces and naturally arise as images under some fractional operators.

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