scholarly journals Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

2017 ◽  
Vol 48 (4) ◽  
pp. 459-471 ◽  
Author(s):  
Turgay Bayraktar
Keyword(s):  
Orthogonal Polynomials ◽  
Expected Number ◽  
Linear Combinations ◽  
Real Roots ◽  
Number Of Real Roots

scholarly journals Expected number of real zeros for random orthogonal polynomials

2016 ◽  
Vol 164 (1) ◽  
pp. 47-66 ◽  
Author(s):  
DORON S. LUBINSKY ◽  
IGOR E. PRITSKER ◽  
XIAOJU XIE
Keyword(s):  
Orthogonal Polynomials ◽  
Real Line ◽  
Asymptotic Relation ◽  
Expected Number ◽  
Linear Combinations ◽  
The Real ◽  
Real Zeros ◽  
Freud Weight ◽  
Number Of Real Zeros ◽  
Local Results

AbstractWe study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only (2/π + o(1))logn expected real zeros in terms of the degree n. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight defined on the whole real line, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution.

scholarly journals Number of real roots of a random trigonometric polynomial

1992 ◽  
Vol 5 (4) ◽  
pp. 307-313 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Trigonometric Polynomial ◽  
Expected Number ◽  
Real Roots ◽  
Identical Distribution ◽  
Random Trigonometric Polynomial ◽  
Number Of Real Roots ◽  
Normally Distributed

We study the expected number of real roots of the random equation g1cosθ+g2cos2θ+…+gncosnθ=K where the coefficients gj's are normally distributed, but not necessarily all identical. It is shown that although this expected number is independent of the means of gj, (j=1,2,…,n), it will depend on their variances. The previous works in this direction considered the identical distribution for the coefficients.

scholarly journals On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Takashi Uno
Keyword(s):  
Lower Bound ◽  
Algebraic Equation ◽  
Random Variables ◽  
Real Roots ◽  
Number Of Real Roots

We estimate a lower bound for the number of real roots of a random alegebraic equation whose random coeffcients are dependent normal random variables.

scholarly journals Asymptotic variance of the number of real roots of random polynomial systems

2018 ◽  
Vol 146 (12) ◽  
pp. 5437-5449 ◽  
Author(s):  
D. Armentano ◽  
J-M. Azaïs ◽  
F. Dalmao ◽  
J. R. León
Keyword(s):  
Asymptotic Variance ◽  
Polynomial Systems ◽  
Random Polynomial ◽  
Real Roots ◽  
Number Of Real Roots
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