Applications of general monotone sequences to strong approximation by Fourier series

We consider the strong means of Fourier series generated by infinite nonnegative triangular matrices and prove some estimates of such means in the case of a matrix with rows stating the sequences from the class GM(₅β)$GM\left({}_5\beta \right)$. Our theorems correspond to the results of L. Leindler [L. Leindler, A note on strong approximation of Fourier series, Anal. Math. 29 (2003) 195–199] and essentially extend the result of S.M. Mazhar and V. Totik [S.M. Mazhar, V. Totik, Approximation of continuous functions by T -means of Fourier series, J. Approx. Theory, 60 (1990) 174–182].     

A survey of certain results on strong approximation by orthogonal series

This is a survey of results in a particular direction of the theory of strong approximation by orthogonal series, related mostly with author's contributions to the subject.     

Pointwise simultaneous approximation by left gamma quasi-interpolants

Recently some classical operator quasi-interpolants were introduced to obtain much faster convergence.We consider left Gamma quasi-interpolants and give a pointwise simultaneous approximation equivalence theorem with ω2r(4)λ(f,f)∞ by means of unified the classical modulus and Ditzian-Totick modulus.     

On the rate of pointwise strong summability of orthogonal expansions

There is introduced a modificated local modulus of continuity as a measure of pointwise strong summability of expansion with respect to polynomial-like orthonormal system. The approximation versions and generalizations of the some special cases of the known results of K. Tandori [6], [7] and G. Freud [4] are obtained.     

On strong approximation applied to post-widder operators

We introduce modified Post-Widder operators in polynomial weighted spaces of differentiable functions and we study strong approximation for them.     

Isomorphism of complete local noetherian rings and strong approximation

About a year ago Angus Macintyre raised the following question. Let and be complete local noetherian rings with maximal ideals and such that is isomorphic to for every . Does it follow that and are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.

     

A note on strong approximation of Fourier series

We generalize a theorem of Mazhar and Totik in two directions. We extend the set of their summability matrix and consider strong approximation instead of ordinary one.     

左Bernstein逆插值算子的点态逼近性质

利用光滑模ω2φrλ(f,t)给出了左Bernste in逆插值算子的逼近等价定理.     

Embedding relations connected with strong approximation of Fourier series

We consider the embedding relation between the class W ^q H _β ^ω , including only odd functions and a set of functions defined via the strong means of Fourier series of odd continuous functions. We establish an improvement of a recent theorem of Le and Zhou [Math. Inequal. Appl. 11(4) (2008) 749–756] which is a generalization of Tikhonov’s results [Anal. Math. 31 (2005) 183–194]. We also extend the Leindler theorem [Anal. Math. 31 (2005) 175–182] concerning sequences of Fourier coefficients.     

Additional results on the strong approximation of Fourier series

Первая цель работы — п олучить достаточное условие, выраженное в термина х некоторых свойств сильной аппр оксимации функций, пр и котором она принадлежит клас суW ^r H ^ω без каких-либо ограни чений на модуль непре рывностиω. Вторая теорема также дает достаточное усл овие для того, чтобы функция пр инадлежалаW ^r H ^ω α, гдеH ^ω α обозначает ра сширенный класс Липш ица.     

Pointwise weighted approximation by Bernstein operators

We consider the pointwise weighted approximation by Bernstein operators. The related weight functions are Jacobi weights . We obtain approximation equivalence theorems, using a weighted modulus of smoothness , which is an extension of~[5].     

Application of the absolute Euler method to some series related to Fourier series and its conjugate series

We study the absolute Euler summability problem of some series associated with Fourier series and its conjugate series generalizing some known results in the literature. Also, it is shown that absolute Euler summability of rth derived Fourier series and rth derived conjugate series can be ensured under local conditions.     

On the uniform strong approximation of Marcinkiewicz type for multivariable continuous functions

The rate of uniform strong approximation of Marcinkiewicz type for multivariable continuous func-tions is obtained in this paper as follows:‖1/k 1 k∑j=0|Sj(f)- f|q‖≤C/k 1 k∑j=0 Eqj(f),where Sj (f) denotes the square partial Fourier sum of f and Ej (f) denotes the square best approximation of f by trigonometric polynomials of degree (j, j, … ,j),j = 0,1, 2,….     

Pointwise convergence of vector-valued Fourier series

We prove a vector-valued version of Carleson’s theorem: let $Y=[X,H]_\theta $ be a complex interpolation space between an unconditionality of martingale differences (UMD) space $X$ and a Hilbert space $H$ . For $p\in (1,\infty )$ and $f\in L^p(\mathbb T ;Y)$ , the partial sums of the Fourier series of $f$ converge to $f$ pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form $Y=[X,H]_\theta $ . In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.