scholarly journals Partial best approximations and the absolute Cesaro summability of multiple Fourier series

2021 ◽  
Vol 103 (3) ◽  
pp. 4-12
Author(s):  
S. Bitimkhan ◽  
◽  
D.T. Alibieva ◽  
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Multiple Fourier Series ◽  
Dimensional Case ◽  
Cesàro Summability ◽  
Best Approximations ◽  
One Dimensional ◽  
Cesaro Summability ◽  
The Absolute ◽  
The One

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1–4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f ∈ Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ- summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1 = ... = βs = 1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f ∈ Lq(I2) are obtained in the cases β1 = 1/q', −1 < β2 < 1/q'(in the second theorem), 1/q'< β1 < +∞, β2 = 1/q', (in the third theorem), −1 < β1 < 1/q', 1/q' < β2 < +∞ (in the fourth theorem).

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

Higher Dimensional Asymptotic Cycles

2003 ◽  
Vol 55 (3) ◽  
pp. 636-648 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Stationary Points ◽  
Dimensional Case ◽  
One Dimensional ◽  
Smooth Flow ◽  
Higher Dimensional ◽  
The One

AbstractGiven a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure τ, Sullivan has defined an element of Hp(Mn; R). This generalized the notion of a μ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure μ. In this one-dimensional case there was a natural 1—1 correspondence between transversal invariant measures τ and invariant measures μ when one had a smooth flow without stationary points.For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

scholarly journals The Absolute Cesaro Summability of the Successively Derived Allied Series of a Fourier Series

1950 ◽  
Vol 8 (4) ◽  
pp. 163-176
Author(s):  
R. Mohanty
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
The Absolute ◽  
Image Position

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series byThen the allied series is

The absolute generalized harmonic-Cesàro summability of a Fourier series

1994 ◽  
Vol 20 (3) ◽  
pp. 173-183
Author(s):  
Prem Chandra ◽  
G. D. Dikshit
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
The Absolute

scholarly journals Some sufficient conditions for the absolute Cesàro summability of series

1939 ◽  
Vol 6 (1) ◽  
pp. 51-56 ◽  
Author(s):  
J. M. Hyslop
Keyword(s):  
Sufficient Conditions ◽  
Cesàro Summability ◽  
Cesàro Mean ◽  
Cesaro Summability ◽  
Cesaro Mean ◽  
The Absolute ◽  
Image Position

denote the n-th Cesàro mean of order k for the series aΣan, that is,whereand let

scholarly journals Alternating and variable controls for the wave equation

2020 ◽  
Vol 26 ◽  
pp. 38 ◽  
Author(s):  
Antonio Agresti ◽  
Daniele Andreucci ◽  
Paola Loreti
Keyword(s):  
Wave Equation ◽  
Sufficient Conditions ◽  
Dimensional Case ◽  
Equivalent Condition ◽  
Open Problems ◽  
One Dimensional ◽  
Constant Rate ◽  
Exact Observability ◽  
Single Exchange ◽  
The One

The present article discusses the exact observability of the wave equation when the observation subset of the boundary is variable in time. In the one-dimensional case, we prove an equivalent condition for the exact observability, which takes into account only the location in time of the observation. To this end we use Fourier series. Then we investigate the two specific cases of single exchange of the control position, and of exchange at a constant rate. In the multi-dimensional case, we analyse sufficient conditions for the exact observability relying on the multiplier method. In the last section, the multi-dimensional results are applied to specific settings and some connections between the one and multi-dimensional case are discussed; furthermore some open problems are presented.

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