On minimum modulus of trigonometric polynomials with random coefficients

1997 ◽  
Vol 61 (3) ◽  
pp. 369-373
Author(s):  
A. G. Karapetyan
Keyword(s):  
Random Coefficients ◽  
Trigonometric Polynomials ◽  
Minimum Modulus

On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients

1984 ◽  
Vol 16 (4) ◽  
pp. 819-842 ◽  
Author(s):  
K. F. Turkman ◽  
A. M. Walker
Keyword(s):  
Moving Average ◽  
Random Coefficients ◽  
Distribution Functions ◽  
Asymptotic Distributions ◽  
Trigonometric Polynomials ◽  
Stationary Gaussian Sequence ◽  
Moving Average Representation ◽  
Standard Normal ◽  
Average Representation ◽  
Mutually Independent

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ < x <∞.It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

scholarly journals On Zeros of Self-Reciprocal Random Algebraic Polynomials

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
K. Farahmand
Keyword(s):  
Matrix Theory ◽  
Random Coefficients ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Level Crossings ◽  
Random Algebraic Polynomials ◽  
The Relationship ◽  
Normal Standard ◽  
Independent Identically Distributed

This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(θ)=∑j=0N−1{αN−jcos(j+1/2)θ+βN−jsin(j+1/2)θ}, where αj and βj, j=0,1,2,…, N−1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+θ/2). We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.

On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients

1984 ◽  
Vol 16 (04) ◽  
pp. 819-842 ◽  
Author(s):  
K. F. Turkman ◽  
A. M. Walker
Keyword(s):  
Moving Average ◽  
Random Coefficients ◽  
Distribution Functions ◽  
Asymptotic Distributions ◽  
Trigonometric Polynomials ◽  
Stationary Gaussian Sequence ◽  
Moving Average Representation ◽  
Standard Normal ◽  
Average Representation ◽  
Mutually Independent

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let X n(λ) and Y n(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {X n(λ)}, (Y n(λ)}, {I n(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x ], —∞ &lt; x &lt;∞. It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

Approximation of Unbounded Functions by Trigonometric Polynomials

2017 ◽  
Vol 13 (4) ◽  
pp. 106-116
Author(s):  
Alaa A. Auad ◽  
◽  
Mousa M. Khrajan
Keyword(s):  
Trigonometric Polynomials

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

Sign in / Sign up

Export Citation Format

Share Document