SETS OF UNIQUENESS OF SERIES OF STOCHASTICALLY INDEPENDENT FUNCTIONS

2002 ◽  
Vol 45 (3) ◽  
pp. 557-563 ◽  
Author(s):  
Francisco J. Freniche ◽  
Ricardo Ríos-Collantes-de-Terán
Keyword(s):  
Positive Measure ◽  
Subject Classification ◽  
Secondary 60G50 ◽  
Best Constant ◽  
Rademacher System ◽  
Independent Functions ◽  
Set Of Uniqueness ◽  
Sets Of Uniqueness ◽  
Uniformly Bounded

AbstractIt is shown that, for every sequence $(f_n)$ of stochastically independent functions defined on $[0,1]$—of mean zero and variance one, uniformly bounded by $M$—if the series $\sum_{n=1}^\infty a_nf_n$ converges to some constant on a set of positive measure, then there are only finitely many non-null coefficients $a_n$, extending similar results by Stechkin and Ul’yanov on the Rademacher system. The best constant $C_M$ is computed such that for every such sequence $(f_n)$ any set of measure strictly less than $C_M$ is a set of uniqueness for $(f_n)$.AMS 2000 Mathematics subject classification:Primary 42C25. Secondary 60G50

Perfect Sets of Uniqueness on the Group 2ω

1982 ◽  
Vol 34 (3) ◽  
pp. 759-764 ◽  
Author(s):  
Kaoru Yoneda
Keyword(s):  
Haar Measure ◽  
Measure Zero ◽  
Countable Subset ◽  
Walsh Series ◽  
Dyadic Group ◽  
Set Of Uniqueness ◽  
Image Position ◽  
Sets Of Uniqueness

Let ω0, ω1, … denote the Walsh-Paley functions and let G denote the dyadic group introduced by Fine [3]. Recall that a subset E of G is said to be a set of uniqueness if the zero series is the only Walsh series ∑ akωk which satisfiesA subset E of G which is not a set of uniqueness is called a set of multiplicity.It is known that any subset of G of positive Haar measure is a set of multiplicity [5] and that any countable subset of G is a set of uniqueness [2]. As far as uncountable subsets of Haar measure zero are concerned, both possibilities present themselves. Indeed, among perfect subsets of G of Haar measure zero there are sets of multiplicity [1] and there are sets of uniqueness [5].

Sets of uniqueness for trigonometric series and integrals

1950 ◽  
Vol 46 (4) ◽  
pp. 538-548 ◽  
Author(s):  
R. Henstock
Keyword(s):  
Trigonometric Series ◽  
Set Of Uniqueness ◽  
Point Set ◽  
Image Position ◽  
Sets Of Uniqueness

It is well known that, if a trigonometric seriesconverges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.

Sets of Uniqueness for Univalent Functions

2000 ◽  
Vol 43 (1) ◽  
pp. 105-107
Author(s):  
Marius Overholt
Keyword(s):  
Univalent Functions ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Set Of Uniqueness ◽  
Sets Of Uniqueness

AbstractWe observe that any set of uniqueness for the Dirichlet space 𝐷 is a set of uniqueness for the class S of normalized univalent holomorphic functions.

Classical Tauberian Theorems for the (C; 1; 1) Summability Method

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Subject Classification ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Littlewood Theorem

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

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