ON A UNIQUENESS THEOREM FOR THE FRANKLIN SYSTEM

2018 ◽  
Vol 52 (2 (246)) ◽  
pp. 93-100
Author(s):  
K.A. Navasardyan
Keyword(s):  
Uniqueness Theorem ◽  
Partial Sums ◽  
Uniqueness Theorems ◽  
Franklin System ◽  
Almost Everywhere ◽  
Franklin Series

In this paper we prove that there exist a nontrivial Franklin series and a sequence $ M_n $ such that the partial sums $ S_{M_n} (x) $ of that series converge to 0 almost everywhere and $ \lambda \cdot \text{mes} \{ x : \sup\limits_{n}{\left| S_{M_n} (x) \right|} > \lambda \} \to 0 $ as $ \lambda \to +\infty $. This shows that the boundedness assumption of the ratio $ \dfrac{ M_{n+1}}{M_n} $, used for the proofs of uniqueness theorems in earlier papers, can not be omitted.

scholarly journals Generalized localization for spherical partial sums of multiple Fourier series

2019 ◽  
Vol 489 (1) ◽  
pp. 7-10
Author(s):  
R. R. Ashurov
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Localization Principle ◽  
Open Set ◽  
Almost Everywhere

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - ​everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

scholarly journals A uniqueness theorem of algebraically non-degenerate meromorphic maps into PN(C)

1976 ◽  
Vol 64 ◽  
pp. 117-147 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Projective Space ◽  
Uniqueness Theorem ◽  
Complex Projective Space ◽  
Meromorphic Functions ◽  
Uniqueness Theorems ◽  
Meromorphic Maps ◽  
Complex Projective

In the previous paper [3], the author generalized the uniqueness theorems of meromorphic functions given by G. Pólya in [5] and R. Nevanlinna in [4] to the case of meromorphic maps of Cn into the N- dimensional complex projective space PN(C).

On a Uniqueness Theorem

1979 ◽  
Vol 31 (5) ◽  
pp. 1072-1076
Author(s):  
Mikio Niimura
Keyword(s):  
Riemann Surface ◽  
Uniqueness Theorem ◽  
Analytic Functions ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Uniqueness Theorems ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
Hyperbolic Riemann Surfaces

The classical uniqueness theorems of Riesz and Koebe show an important characteristic of boundary behavior of analytic functions in the unit disc. The purpose of this note is to discuss these uniqueness theorems on hyperbolic Riemann surfaces. It will be necessary to give additional hypotheses because Riemann surfaces can be very nasty. So, in this note the Wiener compactification will be used as ideal boundary of Riemann surfaces. The Theorem, Corollaries 1, 2 and 3 are of Riesz type, Riesz-Nevanlinna type, Koebe type and Koebe-Nevanlinna type respectively. Corollaries 4 and 5 are general forms of Corollaries 2 and 3 respectively.Let f be a mapping from an open Riemann surface R into a Riemann surface R′.

Uniqueness Theorems for Multiple Franklin Series Converging over Rectangles

2021 ◽  
Vol 109 (1-2) ◽  
pp. 208-217
Author(s):  
G. G. Gevorkyan ◽  
L. A. Hakobyan
Keyword(s):  
Uniqueness Theorems ◽  
Franklin Series

Uniform estimates for families of singular integrals and double Fourier series

1986 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Elena Prestini
Keyword(s):  
Fourier Series ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Double Fourier Series ◽  
Variable Coefficients ◽  
Partial Sums ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Almost Everywhere ◽  
The One

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

CONVERGENCE ALMOST EVERYWHERE OF CERTAIN PARTIAL SUMS OF FOURIER INTEGRALS

2003 ◽  
Vol 35 (02) ◽  
pp. 225-228 ◽  
Author(s):  
ANTHONY CARBERY ◽  
DIRK GORGES ◽  
GIANFRANCO MARLETTA ◽  
CHRISTOPH THIELE
Keyword(s):  
Partial Sums ◽  
Convergence Almost Everywhere ◽  
Almost Everywhere ◽  
Fourier Integrals
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