scholarly journals Affine Bessel sequences and Nikishin’s example

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 963-966 ◽  
Author(s):  
V.A. Mironov ◽  
A.M. Sarsenbi ◽  
P.A. Terekhin
Keyword(s):  
Hilbert Space ◽  
Periodic Function ◽  
Spectral Theory ◽  
Continuous Periodic Function ◽  
Affine Systems ◽  
Convergence In Measure ◽  
Affine System ◽  
Bessel Sequences

We study affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. We construct a non-Besselian affine system fun(x)g1 n=0 generated by continuous periodic function u(x). The result is based on Nikishin?s example concerning convergence in measure. We also show that affine systems fun(x)g1 n=0 generated by any Lipchitz function u(x) are Besselian.

scholarly journals Applications of Affine Systems of Walsh Type to Generate Smooth Basis

2021 ◽  
pp. 4875-4884
Author(s):  
Khaled Hadi ◽  
Saad Nagy
Keyword(s):  
Generating Function ◽  
Riesz Basis ◽  
Walsh System ◽  
Continuous Periodic Function ◽  
Affine Systems ◽  
Rademacher System ◽  
Almost Everywhere ◽  
Affine System ◽  
Smooth Basis ◽  
Differential Properties

The question on affine Riesz basis of Walsh affine systems is considered. An affine Riesz basis is constructed, generated by a continuous periodic function that belongs to the space on the real line, which has a derivative almost everywhere; in connection with the construction of this example, we note that the functions of the classical Walsh system suffer a discontinuity and their derivatives almost vanish everywhere. A method of regularization (improvement of differential properties) of the generating function of Walsh affine system is proposed, and a criterion for an affine Riesz basis for a regularized generating function that can be represented as a sum of a series in the Rademacher system is obtained.

Spectral Theory for the Differential Equation Lu= λ Mu

1958 ◽  
Vol 10 ◽  
pp. 431-446 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Spectral Theory ◽  
Real Line ◽  
Differential Operators ◽  
Identity Operator ◽  
The Real ◽  
Ordinary Differential Operators ◽  
Extensive Bibliography

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L 2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

scholarly journals Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space

1988 ◽  
Vol 31 (1) ◽  
pp. 127-144 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Hilbert Space ◽  
Spectral Theory ◽  
Linear Operators ◽  
Coupled Systems ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Image Position

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operatorswhere λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

On the existence of periodic solutions of a certain third-order differential equation

1960 ◽  
Vol 56 (4) ◽  
pp. 381-389 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Initial Conditions ◽  
Continuous Periodic Function ◽  
Third Order ◽  
Third Order Differential Equation ◽  
Existence Of Periodic Solutions ◽  
Image Position

In this paper we shall be concerned with the differential equationin which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:under which every solution x(t) of (1) satisfieswhere t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),or, in the case (3),In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).

scholarly journals Absolutely convergent Fourier series and generalized Zygmund classes of functions

2009 ◽  
Vol 104 (1) ◽  
pp. 124
Author(s):  
Ferenc Móricz
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Fourier Coefficients ◽  
Sufficient Conditions ◽  
Modulus Of Smoothness ◽  
Slowly Varying Function ◽  
Continuous Periodic Function ◽  
Convergent Fourier Series ◽  
Period 2 ◽  
Order Of Magnitude

We investigate the order of magnitude of the modulus of smoothness of a function $f$ with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that $f$ belongs to one of the generalized Zygmund classes $(\mathrm{Zyg}(\alpha, L)$ and $(\mathrm{Zyg} (\alpha, 1/L)$, where $0\le \alpha\le 2$ and $L= L(x)$ is a positive, nondecreasing, slowly varying function and such that $L(x) \to \infty$ as $x\to \infty$. A continuous periodic function $f$ with period $2\pi$ is said to belong to the class $(\mathrm{Zyg} (\alpha, L)$ if 26740 |f(x+h) - 2f(x) + f(x-h)| \le C h^\alpha L\left(\frac{1}{h}\right)\qquad \text{for all $x\in \mathsf T$ and $h>0$}, 26740 where the constant $C$ does not depend on $x$ and $h$; and the class $(\mathrm{Zyg} (\alpha, 1/L)$ is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of $f$ are all nonnegative.

Spectral theory of dissipative q-Sturm-Liouville problems

2014 ◽  
Vol 51 (3) ◽  
pp. 366-383
Author(s):  
Aytekin Eryilmaz ◽  
Hüseyin Tuna
Keyword(s):  
Hilbert Space ◽  
Characteristic Function ◽  
Spectral Theory ◽  
Difference Operator ◽  
Functional Model ◽  
Dissipative Operator ◽  
Eigenvalues And Eigenvectors ◽  
Maximal Dissipative Operator ◽  
Sturm Liouville ◽  
Liouville Operators

This paper is devoted to studying a q-analogue of Sturm-Liouville operators. We formulate a dissipative q-difference operator in a Hilbert space. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which is based on the method of Pavlov and define its characteristic function. Finally, we prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative q-Sturm-Liouville difference operator.

Sign in / Sign up

Export Citation Format

Share Document