Approximation of integrable functions by linear methods almost everywhere

1975 ◽  
Vol 18 (1) ◽  
pp. 628-636
Author(s):  
T. V. Radoslavova
Keyword(s):  
Almost Everywhere ◽  
Integrable Functions

Convergence of Marcinkiewicz Means of Integrable Functions with Respect to Two-Dimensional Vilenkin Systems

2004 ◽  
Vol 11 (3) ◽  
pp. 467-478
Author(s):  
György Gát
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Integrable Function ◽  
Two Dimensional ◽  
Vilenkin Groups ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Marcinkiewicz Means

Abstract We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛𝑓 converges to 𝑓 almost everywhere.

scholarly journals Walsh-Fourier series with coefficients of generalized bounded variation

1989 ◽  
Vol 47 (3) ◽  
pp. 458-465 ◽  
Author(s):  
F. Móricz
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Real Numbers ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Integrable Functions

AbstractWe extend in different ways the class of null sequences of real numbers that are of bounded variation and study the Walsh-Fourier series of integrable functions on the interval [(0, 1) with such coefficients. We prove almost everywhere convergence as well as convergence in the pseu dometric of Lr(0, 1) for 0 < r < 1.

Two Results Concerning the Zeros of Functions with Finite Dirichlet Integral

1969 ◽  
Vol 21 ◽  
pp. 312-316 ◽  
Author(s):  
James G. Caughran
Keyword(s):  
Riemann Surface ◽  
Boundary Values ◽  
Hardy Class ◽  
Dirichlet Integral ◽  
Finite Area ◽  
Taylor Coefficients ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Zeros Of Functions ◽  
Image Position

A function f, analytic in the unit disk, is said to have finite Dirichlet integral if1Geometrically, this is equivalent to f mapping the disk onto a Riemann surface of finite area. The class of Dirichlet integrable functions will be denoted by . The condition above can be restated in terms of Taylor coefficients; if f(z) = Σanzn, then if and only if Σn|an|2 < ∞. Thus, is contained in the Hardy class H2.In particular, every such function has boundary valuesalmost everywhere and log |f(eiθ)| ∊ L1(dθ).The zeros zn of a function must satisfy the Blaschke conditionand f(s) = B(z)F(z), where F(z) has no zeros andis the Blaschke product with zeros zn; see (5).

scholarly journals Biconcave-function characterisations of UMD and Hilbert spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Jinsik Mok Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Banach Space ◽  
Unit Interval ◽  
Martingale Differences ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Image Position ◽  
Umd Banach Spaces

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only iffor all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 andIf the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

scholarly journals Almost everywhere convergence of Cesàro means of two variable Walsh – Fourier series with varying parameteres

2021 ◽  
Vol 73 (3) ◽  
pp. 291-307
Author(s):  
A. A. Abu Joudeh ◽  
G. G´at
Keyword(s):  
Fourier Series ◽  
Maximal Operator ◽  
Weak Type ◽  
Partial Sums ◽  
Cesàro Means ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Walsh Group ◽  
Cesáro Means

UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type . Moreover, the -means of the function converge a.e. to for , where is the Walsh group for some sequences .

scholarly journals Expansions of Functions Based on Rational Orthogonal Basis with Nonnegative Instantaneous Frequencies

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xiaona Cui ◽  
Suxia Yao
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Fast Algorithm ◽  
Orthogonal Basis ◽  
Basis Functions ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Instantaneous Frequencies ◽  
Square Integrable Functions

We consider in this paper expansions of functions based on the rational orthogonal basis for the space of square integrable functions. The basis functions have nonnegative instantaneous frequencies so that the expansions make physical sense. We discuss the almost everywhere convergence of the expansions and develop a fast algorithm for computing the coefficients arising in the expansions by combining the characterization of the coefficients with the fast Fourier transform.

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