scholarly journals Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients

2001 ◽  
Vol 262 (2) ◽  
pp. 554-563
Author(s):  
K. Farahmand ◽  
A. Grigorash
Keyword(s):  
Random Coefficients ◽  
Trigonometric Polynomials ◽  
Standard Normal ◽  
Complex Zeros

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.

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.

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.

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.

Sebaran Peluang Acak Kontinu, Distribusi Normal, Distribusi Normal Baku, Distribusi T, Distribusi Chi Square, dan Distribusi F

2020 ◽  
Author(s):  
Ahmad Sudi Pratikno
Keyword(s):  
Hypothesis Test ◽  
Standard Normal Distribution ◽  
Process Data ◽  
Nominal Data ◽  
Chi Square ◽  
Stable Curve ◽  
Standard Normal ◽  
Research Findings ◽  
F Distribution ◽  
T Distribution

In statistics, there are various terms that may feel unfamiliar to researcher who is not accustomed to discussing it. However, despite all of many functions and benefits that we can get as researchers to process data, it will later be interpreted into a conclusion. And then researcher can digest and understand the research findings. The distribution of continuous random opportunities illustrates obtaining opportunities with some detection of time, weather, and other data obtained from the field. The standard normal distribution represents a stable curve with zero mean and standard deviation 1, while the t distribution is used as a statistical test in the hypothesis test. Chi square deals with the comparative test on two variables with a nominal data scale, while the f distribution is often used in the ANOVA test and regression analysis.

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