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Higher-Order, Polar and Sz.-Nagy’s Generalized Derivatives of Random Polynomials with Independent and Identically Distributed Zeros on the Unit Circle

Higher-Order, Polar and Sz.-Nagy’s Generalized Derivatives of Random Polynomials with Independent... For random polynomials with independent and identically distributed (i.i.d.) zeros following any common probability distribution $$\mu $$ μ with support contained in the unit circle, the empirical measures of the zeros of their first and higher-order derivatives will be proved to converge weakly to $$\mu $$ μ almost surely (a.s.). This, in particular, completes a recent work of Subramanian on the first-order derivative case where $$\mu $$ μ was assumed to be non-uniform. The same almost sure weak convergence will also be shown for polar and Sz.-Nagy’s generalized derivatives, assuming some mild conditions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Higher-Order, Polar and Sz.-Nagy’s Generalized Derivatives of Random Polynomials with Independent and Identically Distributed Zeros on the Unit Circle

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References (30)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-014-0097-4
Publisher site
See Article on Publisher Site

Abstract

For random polynomials with independent and identically distributed (i.i.d.) zeros following any common probability distribution $$\mu $$ μ with support contained in the unit circle, the empirical measures of the zeros of their first and higher-order derivatives will be proved to converge weakly to $$\mu $$ μ almost surely (a.s.). This, in particular, completes a recent work of Subramanian on the first-order derivative case where $$\mu $$ μ was assumed to be non-uniform. The same almost sure weak convergence will also be shown for polar and Sz.-Nagy’s generalized derivatives, assuming some mild conditions.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Jan 21, 2015

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