Abstract: The main objective of this paper is a study of some new multidimensional Hilbert type inequalities with a general homogeneous kernel. We derive a pair of equivalent inequalities, and also establish the conditions under which the constant factors included in the obtained inequalities are the best possible. Some applications in particular settings are also considered.

Abstract: Реэюме Получены точные в смысле порядка оценки для поперечников Фурье классов типа Никольского-Бесова \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$B_{pq}^{sm} (\mathbb{T}^k )$$ \end{document} и Лиэоркина-Трибеля \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L_{pq}^{sm} (\mathbb{T}^k )$$ \end{document} в метрике \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \u...

Abstract: It is proved that for any dimension n ≥ 2, L(ln+ L) n−1 is the widest integral class in which the almost everywhere convergence of spherical partial sums of multiple Fourier-Haar series is provided. Moreover,it is shown that the divergence effects of rectangular and spherical general terms of multiple Fourier-Haar series can be achieved simultaneously on a set of full measure by an appropriate rearrangement of values of arbitrary summable function f not belonging to L(ln+ L) n−1.

Abstract: A b s t r a c t . Several sufficient conditions for generalized absolute convergence of single and double Vilenkin–Fourier series of bounded type are obtained. These conditions

Abstract: We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallee Poussin sums on the sets CβqHω of Poisson integrals of functions from the class Hω generated by convex upwards moduli of continuity ω(t) which satisfy the condition ω(t)/t → ∞ as t → 0. As an implication, a solution of the Kolmogorov-Nikol’skii problem for de la Vallee Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes Hα, 0 < α < 1, is obtained.

Abstract: A b s t r a c t . We introduce a new kind of double sequences named MVBVDS and some new classes of weight functions to study the weighted integrability of the double trigonometric series. Several results of Chen, Marzuq, M´, Ram and Singh Bhatia (see [2]–[10]) are generalized and some new results are established.

Abstract: Реэюме Получены точные неравенства типа Джексона-Стечкина для ос-редненных с весом модулей непрерывности m-го (m ∈ ℕ) порядка. Для классов функций, определенных при помоши мажорант и укаэанных осредненных величин, вычислены точные эначения раэличных n-поперечников при выполнении определенных ограничений на мажоранты.

Abstract: Our aim in the present paper is to introduce non-homogeneous Herz-Sobolev spaces with variable exponent and to obtain a wavelet characterization and an unconditional basis of them using wavelets with proper decay and compact support. Our method is based on the extrapolation theorem and duality argument.

Abstract: In this paper, we are interested in the Laguerre hypergroup \(\mathbb{K} = [0,\infty ) \times \mathbb{R}\) which is the fundamental manifold of the radial function space for the Heisenberg group So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup Open image in new window which can be topologically identified with the so-called Heisenberg fan, the subset of ℝ2: $$\bigcup\limits_{j \in \mathbb{N}} {\left\{ {(\lambda ,\mu ) \in \mathbb{R}^2 :\mu = \left| \lambda \right|(2j + \alpha + 1),\lambda e 0} \right\} \cup \left\{ {(0,\mu ) \in \mathbb{R}^2 :\mu \geqslant 0} \right\}} ,$$ by means of which the maximal function is investigated For 1 < p ≤ ∞, the Lp(Open image in new window)-boundedness and weak L1(Open image in new window)-boundedness result for the maximal function is obtained

Abstract: Получены точные неравенства типа Джексона-Стечкина для ос-редненных с весом модулей непрерывности m-го (m ∈ ℕ) порядка. Для классов функций, определенных при помоши мажорант и укаэанных осредненных величин, вычислены точные эначения раэличных n-поперечников при выполнении определенных ограничений на мажоранты.

Abstract: Получено необходимое условие полноты системы экспонент $$e(\Lambda ) = \left\{ {e^{ - \lambda _n t} :\lambda _n \in \left\{ {z:0 < \operatorname{Re} z < A \in \mathbb{R}^ + ,0 < \operatorname{Im} z < 2\pi } \right\}} \right\}$$ в прострaнстве квaдрaтично интегрируемых функций со степенным весом tα при −1 < α < 0.

Abstract: Our aim is to find the source why the logarithm sequences play the crucial role in the L1-convergence of sine series. We define three new classes of sequences; one of them has the character of the logarithm sequences, the other two are the extensions of the class defined by Zhou and named Logarithm Rest Bounded Variation Sequences. In terms of these classes, extended analogues of Zhou’s theorems are proved.

Abstract: It is established that Karagulyan’s exact estimate of the divergence rate of strong integral means of summable functions is extendable to strong means of additive functions of intervals having bounded variation Furthermore, it is proved that each function defined on [0, 1]n with bounded variation in the sense of Hardy has a strong gradient at almost every point (this strengthens the corresponding result of Burkill and Haslam-Jones on the differentiability almost everywhere), whereas the same is not true for functions with bounded variation in the sense of Arzela

Abstract: A b s t r a c t . In this paper an extension of a Holder-type inequality given in (C. E. M. Pearce and J. Pecaric, On an extension of Holder's inequality, Bull. Austral. Math. Soc., 51(1995), 453-458) is improved using log-convexity. Furthermore, new Cauchy-type means are defined and their monotonicity property is proven.

Abstract: Резюме В работе рассмотрены поперечники по Колмогорову классов 2п-периодических функций многих переменных, мажоранта смещанных модулей непрерывности которых содержит как степенные, так и логарифмические множители.

Abstract: В работе рассмотрены поперечники по Колмогорову классов 2п-периодических функций многих переменных, мажоранта смещанных модулей непрерывности которых содержит как степенные, так и логарифмические множители.

Abstract: The extension of the coefficient test of Menshov and Kaczmarz ensuring the almost everywhere (C, 1, 1)-((C, 1, 0) or (C, 0, 1)) summability of double series with respect to block-orthonormal systems is studied.

Abstract: It is established that Karagulyan’s exact estimate of the divergence rate of strong integral means of summable functions is extendable to strong means of additive functions of intervals having bounded variation. Furthermore, it is proved that each function defined on [0, 1] n with bounded variation in the sense of Hardy has a strong gradient at almost every point (this strengthens the corresponding result of Burkill and Haslam-Jones on the differentiability almost everywhere), whereas the same is not true for functions with bounded variation in the sense of Arzela.

Abstract: We consider the triangular summability of two-dimensional Fourier transforms, and show that the maximal operator of the triangular-θ-means of a tempered distribution is bounded from Hp(ℝ2) to Lp(ℝ2) for all 2/(2 + α) < p ≤ ∞; consequently, it is of weak type (1,1), where 0 < α ≤ 1 is depending only on θ. As a consequence, we obtain that the triangular-θ-means of a function f ∈ L1(ℝ2) converge to f a.e. Norm convergence is also considered, and similar results are shown for the conjugate functions. Some special cases of the triangular-θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejer, de la Vallee-Poussin, Rogosinski, and Riesz summations.