scholarly journals Infinite combinatorics in function spaces: Category methods

2009 ◽  
Vol 86 (100) ◽  
pp. 55-73 ◽  
Author(s):  
N.H. Bingham ◽  
A.J. Ostaszewski
Keyword(s):  
Function Spaces ◽  
Embedding Theorem ◽  
Regularly Varying Functions ◽  
Infinite Combinatorics ◽  
Regularly Varying ◽  
Special Case ◽  
Null Set

The infinite combinatorics here give statements in which, from some sequence, an infinite subsequence will satisfy some condition - for example, belong to some specified set. Our results give such statements generically - that is, for 'nearly all' points, or as we shall say, for quasi all points - all off a null set in the measure case, or all off a meagre set in the category case. The prototypical result here goes back to Kestelman in 1947 and to Borwein and Ditor in the measure case, and can be extended to the category case also. Our main result is what we call the Category Embedding Theorem, which contains the Kestelman-Borwein-Ditor Theorem as a special case. Our main contribution is to obtain function wise rather than point wise versions of such results. We thus subsume results in a number of recent and related areas, concerning e.g., additive, subadditive, convex and regularly varying functions.

Pseudo-Regularly Varying Functions and Generalized Renewal Processes

2018 ◽  
Author(s):  
Valeriĭ V. Buldygin ◽  
Karl-Heinz Indlekofer ◽  
Oleg I. Klesov ◽  
Josef G. Steinebach
Keyword(s):  
Renewal Processes ◽  
Regularly Varying Functions ◽  
Regularly Varying

A Multiplicative Analogue of Schur's Tauberian Theorem

2003 ◽  
Vol 46 (3) ◽  
pp. 473-480 ◽  
Author(s):  
Karen Yeats
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Regularly Varying Functions ◽  
Regularly Varying

AbstractA theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations

2015 ◽  
Vol 145 (5) ◽  
pp. 1007-1028 ◽  
Author(s):  
Jaroslav Jaroš ◽  
Kusano Takaŝi
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Nonlinear Differential Equations ◽  
Radial Symmetry ◽  
Second Order ◽  
Accurate Information ◽  
Regularly Varying Functions ◽  
Cyclic System ◽  
Regularly Varying ◽  
Image Position

The n-dimensional cyclic system of second-order nonlinear differential equationsis analysed in the framework of regular variation. Under the assumption that αi and βi are positive constants such that α1 … αn > β1 … βn and pi and qi are regularly varying functions, it is shown that the situation in which the system possesses decreasing regularly varying solutions of negative indices can be completely characterized, and moreover that the asymptotic behaviour of such solutions is governed by a unique formula describing their order of decay precisely. Examples are presented to demonstrate that the main results for the system can be applied effectively to some classes of partial differential equations with radial symmetry to provide new accurate information about the existence and the asymptotic behaviour of their radial positive strongly decreasing solutions.

scholarly journals Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1236
Author(s):  
Vojtěch Růžička
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Type Equation ◽  
Real Numbers ◽  
Regularly Varying Functions ◽  
Varying Coefficients ◽  
Even Order ◽  
Regularly Varying ◽  
Nonoscillation Criterion ◽  
Linear Differential

We establish nonoscillation criterion for the even order half-linear differential equation (−1)nfn(t)Φx(n)(n)+∑l=1n(−1)n−lβn−lfn−l(t)Φx(n−l)(n−l)=0, where β0,β1,…,βn−1 are real numbers, n∈N, Φ(s)=sp−1sgns for s∈R, p∈(1,∞) and fn−l is a regularly varying (at infinity) function of the index α−lp for l=0,1,…,n and α∈R. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms fn(t) and fn−l(t) are replaced by the tα and tα−lp, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.

scholarly journals New Classes of Weighted Hölder-Zygmund Spaces and the Wavelet Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Stevan Pilipović ◽  
Dušan Rakić ◽  
Jasson Vindas
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Elementary Proof ◽  
Left Inverse ◽  
Regularly Varying Functions ◽  
Continuity Theorem ◽  
New Class ◽  
Zygmund Spaces ◽  
Regularly Varying

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces𝒮0(ℝn)and𝒮(ℍn+1). We then introduce and study a new class of weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.

scholarly journals Principal components analysis of regularly varying functions

Bernoulli ◽  
2019 ◽  
Vol 25 (4B) ◽  
pp. 3864-3882
Author(s):  
Piotr Kokoszka ◽  
Stilian Stoev ◽  
Qian Xiong
Keyword(s):  
Principal Components Analysis ◽  
Principal Components ◽  
Regularly Varying Functions ◽  
Components Analysis ◽  
Regularly Varying
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