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We consider the Lipschitz class functions on [0, 1] and special series of their Fourier coefficients with respect to general orthonormal systems (ONS). The convergence of classical Fourier series (trigonometric, Haar, Walsh systems) of Lip 1 class functions is a trivial problem and is well known. But general Fourier series, as it is known, even for the function f (x) = 1 does not converge. On the other hand, we show that such series do not converge with respect to general ONSs. In the paper we find the special conditions on the functions φ n of the system (φ n) such that the above-mentioned series are convergent for any Lipschitz class function. The obtained result is the best possible. [ABSTRACT FROM AUTHOR]

Recently, Devaiya and Srivastava [3] studied the T r -strong convergence of numerical sequences and Fourier series using a lower triangular matrix T = (b m , n) , and generalized the results of Kórus [8]. The main objective of this paper is to introduce [ T r , G , u , q ] -strongly convergent sequence spaces for r ∈ N , and defined by a sequence of modulus functions. We also provide a relationship between [ T , G , u , q ] and [ T r , G , u , q ] -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for T r -strong convergent sequences, statistical convergence and Fourier series using the idea of [ T r , G , u , q ] -strongly convergent sequence spaces. [ABSTRACT FROM AUTHOR]

Srivastava and Singh [9] generalized the idea of Λ -strong convergence given by Móricz [6] to T-strong convergence of numerical sequences and Fourier series. In this present work, we introduce the idea of Tr-strong convergence r ∈ N , by generalizing the idea of T-strong convergence. This paper also generalizes some results of Kórus [5], as shown in conclusion. Additionally, we provide a relationship between T-strong convergence, T2-strong convergence, ⋯ , Tr-strong convergence, and ordinary convergence. We also demonstrate that this idea can be used to explain the Tr-convergence of Fourier series in the C-metric and the Lp-metric. [ABSTRACT FROM AUTHOR]

A Sjölin–Soria–Antonov type extrapolation theorem for locally compact σ -compact non-discrete Hausdorff groups is proved. Applying this result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class L log + L log + log + log + L . [ABSTRACT FROM AUTHOR]

We introduce vast generalizations of the Hardy–Berndt sums. They involve higher-order Euler and/or Bernoulli functions, in which the variables are affected by certain linear shifts. By employing the Fourier series technique we derive linear relations for these sums. In particular, these relations yield reciprocity formulas for Carlitz, Rademacher, Mikolás and Apostol type generalizations of the Hardy–Berndt sums, and give rise to generalizations for some Goldberg's three-term relations. We also present an elementary proof for the Mikolás' linear relation and a reciprocity formula in terms of the generation function. [ABSTRACT FROM AUTHOR]

We generalize an important property of trigonometric series to the case of series by orthonormal spline systems corresponding to the dyadic sequence of grid points. We prove that Ciesielski series cannot diverge to infinity on a set of positive measure. [ABSTRACT FROM AUTHOR]

Let p (·) T n → (0 , ∞) be a variable exponent function satisfying the globally log-Hölder condition and 0 < q ≤ ∞ . We introduce the periodic variable Hardy and Hardy–Lorentz spaces H p (·) (T d) and H p (·) , q (T d) and prove their atomic decompositions. A general summability method, the so called θ -summability is considered for multi-dimensional Fourier series. Under some conditions on θ , it is proved that the maximal operator of the θ -means is bounded from H p (·) (T d) to L p (·) (T d) and from H p (·) , q (T d) to L p (·) , q (T d) . This implies some norm and almost everywhere convergence results for the summability means. The Riesz, Bochner–Riesz, Weierstrass, Picard and Bessel summations are investigated as special cases. [ABSTRACT FROM AUTHOR]

We give some estimates for discrepancies of irrational rotations with several large partial quotients, and report unusual aspects of behavior of discrepancies caused by several large partial quotients. [ABSTRACT FROM AUTHOR]

We obtain a sufficient condition for the weighted β-absolute convergence of the multiple Fourier series ∑k1∈Z⋯∑kN∈Zγk1,…,kN|f^(k1,…,kN)|β,where 0<β<2 and γk1,…,kN is a weight sequence, of a function f of the class (Λ1,…,ΛN)BV(p1,…,pN)([0,2π]N). [ABSTRACT FROM AUTHOR]

We show that it is equiconsistent with $${\mathsf{ZF}}$$ that Fodor's lemma fails everywhere, and furthermore that the club filter on every regular cardinal is not even $${\sigma}$$ -complete. Moreover, these failures can be controlled in a very precise manner. [ABSTRACT FROM AUTHOR]

We investigate Marcinkiewicz-Zygmund type inequalities for multivariate polynomials on various compact domains in $${\mathbb{R}^d}$$ . These inequalities provide a basic tool for the discretization of the L norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz-Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein-Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains. [ABSTRACT FROM AUTHOR]

The assertion that the Salem test [5] for the uniform convergence of a trigonometric Fourier series is improvable, is proved. In particular, an example of a continuous function, which does not fulfill the condition of the Salem test but satisfies the condition of the generalized Salem test [10], is constructed. Besides, the theorem which improves Golubov's [3,4] result for continuous functions of two variables, is given. [ABSTRACT FROM AUTHOR]

Under some conditions on $${\theta}$$ , we characterize the set of convergence of the Marcinkiewicz- $${\theta\mbox{-}}$$ means of a function $${f \in L_1(\mathbb{T}^d)}$$ . More exactly, the $${\theta\mbox{-}}$$ means converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of $${f \in L_p(\mathbb{T}^d)}$$ , whenever $${1 < p < \infty}$$ . The $${\theta\mbox{-}}$$ summability includes the Fejér, Abel, Cesàro and some other summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong $${\theta\mbox{-}}$$ summability. [ABSTRACT FROM AUTHOR]

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all $${L^{p}}$$ spaces, $${1\leq p < \infty}$$ . We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation. [ABSTRACT FROM AUTHOR]

Recently, Kórus [7] studied the $${\Lambda^{2}}$$ -strong convergence of numerical sequences. By using the idea of this paper, we introduce $${[\Lambda^{2}, F, u, p]}$$ -strongly convergent sequence spaces defined by a sequence of modulus functions. We also make an effort to study some inclusion relations, topological and geometric properties of these spaces. Some characterizations for strong convergent sequences are given. Finally, we study statistical convergence over these spaces and problems related to Fourier series. [ABSTRACT FROM AUTHOR]

This note exhibits, as an example, that integral-type conditions can be replaced by sequence-type assumptions in four theorems pertaining to P. Chandra's theorem and its generalizations. [ABSTRACT FROM AUTHOR]

We estimate the truncated double trigonometric series in Lebesgue spaces with mixed norms in terms of the $${p^{\rm th} -q^{\rm th}}$$ power finite double sums of its coefficients. We obtain these estimates for all possible values of the exponents involved and we provide examples of matrices in $${\mathbb{c}^{M \times N}}$$ that maximize some of them up to a constant independent of M and N. We also handle the more general case where the oscillatory factors in the sum terms are not orthogonal in either variable. [ABSTRACT FROM AUTHOR]

We prove the results generalized and improved of some theorems of X. Z. Krasniqi [] on pointwise approximation of integrable functions and their conjugate by matrix means of Fourier series and its conjugate. [ABSTRACT FROM AUTHOR]

We investigate Cesàro summability of bivariate integrable functions on the 2-adic group. We prove the a.e. convergence of 2-adic Cesàro means $${\sigma^{\alpha, \beta}_{n, m} f \rightarrow f}$$ as $${n, m \rightarrow \infty}$$ for functions $${f \in\,L\,{\rm log}^{+}\,L(\mathbb{I}^2)}$$ and $${\alpha, \beta > 0}$$ . Then it is shown that this convergence result can not be improved in the Pringsheim sense, that is, $${L\,{\rm log}^{+}\,L}$$ is the maximal convergence space for $${\sigma^{1, 1}_{n, m}}$$ when there are no conditions for the indices except that they tend to infinity. We prove that for all measurable functions $${\delta : [o, \infty) \rightarrow [o, \infty)}$$ for which $${{\rm lim}_{t \rightarrow \infty} \delta(t) = 0}$$ there is a function $${f \in\,L\,\log^{+}\,L\, \delta(L)}$$ with lim sup $${|\sigma_{2^{n_{1}}, 2^{n_{2}}} f(x)| = +\infty}$$ a.e. as $${{\rm min}\,\{n_{1}, n_{2}\} \rightarrow \infty}$$ . [ABSTRACT FROM AUTHOR]

We continue to investigate the application of the Mean Value Bounded Variation (MVBV) concept in real sense, and prove the necessary and sufficient condition for L-convergence of trigonometric series. [ABSTRACT FROM AUTHOR]

We introduce a new kind of double sequences named LBVDS and generalize Móricz's theorem. [ABSTRACT FROM AUTHOR]

This note presents two versions of an interesting theorem due to Le and Zhou [4], utilizing a new condition of bounded variation type for a trigonometric series to have an asymptotic sum. [ABSTRACT FROM AUTHOR]

As main result we prove that Fejér means of Walsh-Fourier series are uniformly bounded operators from H to H (0< p≦1/2). [ABSTRACT FROM AUTHOR]

We generalize a theorem of Leindler while the series is under the condition of MVBVS ∗, and investigate the relation between the best approximation of functions and its Fourier coefficients under the given $L_{2\pi}^{p}$ norm. In the second theorem, we add a new condition to get the same result as in Theorem B. [ABSTRACT FROM AUTHOR]

We introduce the Λ 2-strong convergence of numerical sequences and with it we generalize the concept of Λ-strong convergence of the results published by F. Móricz [2]. [ABSTRACT FROM AUTHOR]

The second author recently introduced modified Walsh-Dirichlet kernels which generated filtered Walsh-Fourier series. The purpose of this article is to show that those filtered Walsh series can be generated by a partial ordering on the natural numbers which may be of interest in its own right. [ABSTRACT FROM AUTHOR]

We prove the everywhere divergence of series for sequences a and ρ satisfying some extremal conditions. These results generalize some well known examples of everywhere divergent power and trigonometric series. [ABSTRACT FROM AUTHOR]

This paper is a recapitulation of the work of L. Szili and P. Vértesi [4] on multivariate Fourier series with triangular type partial sums. Namely, we give another proof for the corresponding lower estimation, which, in a way, is more direct than the previous one in [4]. [ABSTRACT FROM AUTHOR]

We prove that certain means of the ( C, α,..., α)-means ( α=1/ p−1) of the d-dimensional trigonometric Fourier series are uniformly bounded operators from the Hardy space H to H (1≦ p≦2). As a consequence we obtain strong summability theorems concerning ( C, α,..., α)-means. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh-Fourier series of bivariate integrable functions. That is, let $a = (a_{1}, a_{2})\:\mathbb{N} \to \mathbb{N}^{2}$ such that a( n+1)≧ δsup a( n) ( j=1,2, n∈ℕ) for some δ>0 and a(+∞)= a(+∞)=+∞. Then for each integrable function f∈ L( I) we have the a.e. relation $\lim_{n\to\infty}\sigma_{a_{1}(n), a_{2}(n)}f = f$. It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh-Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8]. [ABSTRACT FROM AUTHOR]

A theorem of Ferenc Lukács [4] states that the partial sums of conjugate Fourier series of periodic Lebesgue integrable functions f diverge at logarithmic rate at the points of discontinuity of first kind of f. F. Móricz [5] proved an analogous theorem for the rectangular partial sums of bivariate functions. The present paper proves analogues of Móricz's theorem for generalized Cesàro means and for positive linear means. [ABSTRACT FROM AUTHOR]

By employing new ideas and techniques, we will refigure out the whole frame of L-approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement f∈ L but directly consider the sine or cosine series. Secondly, to achieve nontrivial generalizations in complex spaces, we use a one-sided condition with some kind of balance conditions. In addition, a conjecture raised in [9] is disproved in Section 3. [ABSTRACT FROM AUTHOR]

Let f∈ L be a real-valued even function with its Fourier series $\frac {a_{0}}{2}+\sum\limits_{n=1}^{\infty}{a_{n}}\cos nx$ , and let S( f, x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence { a} is decreasing and $\lim\limits_{n\to\infty} a_{n} =0$ , then $\lim\limits_{n\to\infty} \|{f-S_{n}(f)}\|_{L}=0$ if and only if $\lim\limits_{n\to\infty} a_{n}\log n=0$ . Later, the monotonicity condition set on { a} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM condition in real space. In this paper, we give a complete generalization to the complex space. [ABSTRACT FROM AUTHOR]

It is proved that the maximal operator of the triangular Cesàro means of a two-dimensional Fourier series is bounded from the periodic Hardy space $H_{p}(\mathbb{T}^{2})$ to $L_{p}(\mathbb{T}^{2})$ for all 2/(2+ α)< p≦∞ and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular Cesàro means of a function $f \in L_{1}(\mathbb{T}^{2})$ converge a.e. to f. [ABSTRACT FROM AUTHOR]

Pointwise estimates of the deviation T f(⋅)− f(⋅) in terms of moduli of continuity $\bar{w}_{\cdot}f$ and w f are proved. Analog results on norm approximation with remarks and corollaries are also given. In the results essentially weaker conditions than those in [2, Theorem 1, p. 437] are used. [ABSTRACT FROM AUTHOR]

We study L-integrability (1< p<∞) of a sum φ of trigonometric series under the assumptions that the sequence of coefficients of φ belongs to the class $\overline{\mathrm{GM}}_{\theta}^{r}$. Then we discuss the relations between the properties of φ and the properties of the sequence ( λ)∈GM( β, r), and deduce an estimate for modulus of continuity of φ in L norm. [ABSTRACT FROM AUTHOR]

We consider the double Walsh orthonormal system on the unit square $\mathbb{I}^{2}$ , where { w( x)} is the ordinary Walsh system on the unit interval $\mathbb{I}:=[0,1)$ in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh-Fourier series of a function $f \in L^{p}(\mathbb{I}^{2})$ for some 1< p≦2. More generally, we give best possible sufficient conditions for the finiteness of the double series where { a} is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0< r<2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of f. [ABSTRACT FROM AUTHOR]

In 1953 Nash [7] introduced the class of functions Φ. In this paper the behaviour of generalized Cesàro ( C, α)-means ( α∈(−1,0)) of trigonometric Fourier series of the classes H∩Φ in the space of continuous functions is studied. The sharpness of the results obtained is shown. [ABSTRACT FROM AUTHOR]

We extend the Dezern-Waterman version of the Lebesgue test for convergence of Vilenkin-Fourier series to the case of unbounded Vilenkin groups. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the limit distribution of $(\cos 2\pi n_{1}x+\cdots +\cos 2\pi n_{N}x)/\sqrt{N}$ on the unit interval when the increasing sequence { n} has bounded gaps, i.e., 1≤ n− n= O(1). By Bobkov-Götze [4], it was proved that the limiting variance must be less than 1/2 in this case. They proved that the centered Gaussian distribution with variance 1/4 together with mixtures of Gaussian distributions belonging to a huge class can be limit distributions. In this paper it is proved that any Gaussian distribution with variance less than 1/2 can be a limit distribution. [ABSTRACT FROM AUTHOR]