scholarly journals Necessary and Sufficient Conditions for Sample Continuity of Random Fourier Series and of Harmonic Infinitely Divisible Processes

1992 ◽  
Vol 20 (1) ◽  
pp. 1-28 ◽  
Author(s):  
M. Talagrand
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinitely Divisible ◽  
Infinitely Divisible Processes ◽  
Necessary And Sufficient ◽  
Random Fourier Series

Uniform and 𝐿-Convergence of Logarithmic Means of Double Walsh–Fourier Series

2005 ◽  
Vol 12 (1) ◽  
pp. 75-88
Author(s):  
György Gát ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Function Space ◽  
Modulus Of Continuity ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Two Dimensional ◽  
Logarithmic Means ◽  
Convergence And Divergence ◽  
Necessary And Sufficient ◽  
Series Of Functions

Abstract We discuss some convergence and divergence properties of twodimensional (Nörlund) logarithmic means of two-dimensional Walsh–Fourier series of functions both in the uniform and in the Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function, and also the function space.

Max-infinite divisibility

1977 ◽  
Vol 14 (02) ◽  
pp. 309-319 ◽  
Author(s):  
A. A. Balkema ◽  
S. I. Resnick
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Infinite Divisibility ◽  
Necessary And Sufficient Conditions ◽  
Infinitely Divisible ◽  
Extremal Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if F t is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.

Summability of general orthonormal Fourier series

2015 ◽  
Vol 52 (4) ◽  
pp. 511-536
Author(s):  
L. Gogoladze ◽  
V. Tsagareishvili
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Orthonormal System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

S. Banach in [1] proved that for any function f ∈ L2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0. D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0. In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φn(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

Max-infinite divisibility

1977 ◽  
Vol 14 (2) ◽  
pp. 309-319 ◽  
Author(s):  
A. A. Balkema ◽  
S. I. Resnick
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Infinite Divisibility ◽  
Necessary And Sufficient Conditions ◽  
Infinitely Divisible ◽  
Extremal Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if Ft is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.

scholarly journals Applications of Hankel and Regular Matrices in Fourier Series

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Abdullah Alotaibi ◽  
M. Mursaleen
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Hankel Matrix ◽  
Necessary And Sufficient Conditions ◽  
Regular Matrix ◽  
Conjugate Fourier Series ◽  
Necessary And Sufficient

Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and conjugate Fourier series.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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