The Menšov-Paley theorem and related results for double orthogonal series

клАссИЧЕскИЕ тЕОРЕМ ы Ф. РИссА, шлИ И МЕНьшО ВА (сМ., НАпРИМЕР, [3. стР. 102, 121 И 189]) РАспРОстРАНьУтсь, И пРИ ЁтОМ В ОБОБЩЕННОИ пОстАНОВкЕ МАРпИНкЕ ВИЧА И жИгМУНДА [1], с ОДНОМЕРНОгО НА ДВУМ ЕРНыИ слУЧАИ. пУсть {?_i(x)} (i=1, 2,...) И (ψ_k(y) (k=1, 2,...) — ДВЕ сИстЕМы ФУНкцИИ, ОРтО НОРМИРОВАННыЕ НА (НЕ О БьжАтЕльНО кОНЕЧНых) ИНтЕРВАлАх (А, ь) И (с, d), сООтВЕтстВЕННО. пР ЕДпОлАгАЕтсь, ЧтО ∥?_i¦|_v≦M_i И ψ_{k v}≦N_k Дль НЕкОтОРОгОv>2, гДЕМ _iИN _k кОНЕЧНы. ИжУ Ч АУтсь сВОИстВА схОДИ МОстИ ДВОИНых РьДОВ \(\mathop \Sigma \limits_{\iota = 1}^\infty \mathop \Sigma \limits_{\kappa = 1}^\infty c_{ik} \varphi _i (x)\psi _k (y).\) ОсНОВНОИ РЕжУльтАт с ОстОИт В слЕДУУЩЕМ Ут ВЕРжДЕНИИ. тЕОРЕМА 5. пУсть 2u ВыпО лНЕНО УслОВИЕ (3.2).ЕслИ Р ьД (3.3)схОДИтсь, тО РьД (1.16)схОДИтсь пОЧтИ ВсУД У пРИ пРОИжВОльНОИ пЕРЕстАНОВкЕ ЕгО ЧлЕ НОВ. ЕслИ (1.17) —пРОИжВОл ьНАь пЕРЕстАНОВкА РьДА (1.16),mО ВыпОлНЕНА ОцЕНкА (3.4),пР ИЧЕМ ВЕлИЧИНА Ag _q,v ^* жАВИсИт тОлькО От q И v. тЕОРЕМА 5 Дльv=∞ ьВльЕт сь ДВУМЕРНыМ АНАлОгО М тЕОРЕМы МЕНьшОВА—пЁлИ.     

On the divergence of double Fourier–Walsh and Fourier–Walsh–Kaczmarz series of continuous functions

Acta Mathematica Hungarica - We prove that there exists a continuous function on $$[0,1]^2$$ , with a certain smoothness, whose double Fourier–Walsh series diverges on a set of positive...     

Pointwise convergence of Marcinkiewicz-Fejér means of double Vilenkin-Fourier series

In this paper we give a characterization of points at which the Marcinkiewicz-Fejér means of double Vilenkin-Fourier series converge.     

On sharp estimates of the convergence of double Fourier–Bessel series

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier–Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier–Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented.     

Universality properties of Walsh–Fourier series

It is shown that Walsh–Fourier series of \(W\) -continuous functions can have maximal sets of limit functions on small subsets of the unit interval.     

λ-Convergence of multiple Walsh–Paley series and sets of uniqueness

Mathematical Notes - λ-convergent multiple Walsh–Paley series on a multidimensional dyadic group are studied. It is proved that, for all λ > 1, any arbitrary finite union of...     

The exponent of convergence of poincaré series

This paper continues the systematic study of the exponent of convergence (G) of a Fuchsian groupG begun byA. F. Beardon. The object is to show that in various senses (G) is a continuous function ofG. In view of the incompleteness of our knowledge about (G) considerable attention is paid to illustrative examples.     

On Λ2-strong convergence of numerical sequences and Fourier series

We introduce the Λ²-strong convergence of numerical sequences and with it we generalize the concept of Λ-strong convergence of the results published by F. Móricz [2].     

On the convergence and divergence of multiple Fourier series with respect to the Walsh—Paley system in the spacesC andL

Работа касается вопр осов сходимости и рас ходимости кратных рядов Фурье п о системе Уолша-Пэли в метрикахС и L. Из доказанных тео рем следует, в частности, ч то еишf∈Е (Е=С или E=L) и существует н атуральноеi ₀ (1≦i ₀ ≦N) так ое, что и $$\begin{gathered} \omega (\delta _{i_0 } ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _{i_0 } }}}}} \right) (\delta _{i_0 } \to 0) \hfill \\ \omega (\delta _k ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _k }}}}} \right), k \ne i_0 (\delta _k \to 0), k = 1,2, ...,N, \hfill \\ \end{gathered}$$ тоN-кратный ряд Фурье функцииf по системе У олша-Пэли сходится по Прингсхе йму в смысле метрики пространств аЕ. Доказано также, что вы шеотмеченное утверж дение неусиляемо в метрикеL не только для системы Уолша, но и для некоторого класс а ОНС, ограниченных в совок упности.     

The convergence of padé approximants and the size of the power series coefficients

Let f be a power series ∑a_izⁱ with complex coefficients. The (n. n) Pade approximant to f is a rational function P/Q where P and Q are polynomials, Q(z) ? 0, of degree ≦ n such that f(z)Q(z)-P(z) = Az²ⁿ⁺¹ + higher degree terms. It is proved that if the coefficients a_i satisfy a certain growth condition, then a corresponding subsequence of the sequence of (n, n) Pade approximants converges to f in the region where the power series f converges, except on an exceptional set E having a certain Hausdorff measure 0. It is also proved that the result is best possible in the sense that we may have divergence on E. In particular,there exists an entire function f such that the sequence of (ny n) Pade approximants diverges everywhere (except at 0)     

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L ₂ = L ₂((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P _n (x)Q _n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ?? L ₂ by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ?? L ₂ in the polynomials P _n (x)Q _m (y), n, m = 0, 1, ..., and its ??hyperbolic?? partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity ?? _k (A, h) of a function f ?? L ₂ are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.     

The radius of convergence of Poincaré series of loop spaces

LetR _S (resp.R _A) be the radius of convergence of the Poincaré series of a loop space (S) (resp. of the Betti-Poincaré series of a noetherian connected graded commutative algebraA over a field of characteristic zero).IfS is a finite 1-connected CW-complex, the rational homotopy Lie algebra ofS is finite dimensional if and only ifR _S-1. OtherwiseR _S<1.There is an easily computable upper bound (usually less than 1) forR _S ifS is formal or coformal.On the other handR _A=+ if and only ifA is a polynomial algebra andR _A=1 if and only ifA is a complete intersection (Golod and Gulliksen conjecture). OtherwiseR _A<1 and the sequence dim Tor _p ^H grows exponentially withp.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.     

On the mean convergence of Fourier–Jacobi series

The convergence of Fourier–Jacobi series in the spaces L _p,A,B is studied in the case where the Lebesgue constants are unbounded.     

On the properties of the strong Cesàro summability and strong convergence of series

Говорят, что ряд \(\mathop \sum \limits_{k = 0}^\infty a_k \) сумм ируется к s в смысле (С, gа), gа >?1, если $$\sigma _n^{(k)} - s = o(1),n \to \infty ,$$ в смысле [C,α] _λ , α<0, λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {\sigma _k^{(\alpha - 1)} - s} \right|^\lambda = o(1),n \to \infty ,$$ и в смысле [C,0] _λ , λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {(k + 1)(s_k - 1) - k(s_{k - 1} - 1)} \right|^\lambda = o(1),n \to \infty ,$$ где σ _n ^(α) обозначаетn-ое ч езаровское среднее р яда. Суммируемость [C,α] _λ , α>?1, λ ≧1 о значает, что $$\mathop \sum \limits_{k = 0}^\infty k^{\lambda - 1} \left| {\sigma _k^{(\alpha )} - \sigma _{k - 1}^{(\alpha )} } \right|^\lambda< \infty .$$ В данной статье содер жится продолжение ис следований свойств [C,α] _λ -суммиру емо сти, которые начали Винн, Х ислоп, Флетт, Танович-М иллер и автор, в частности свя зей между указанными методами суммирования. Наконец, даны некотор ые простые приложени я к вопросам суммируемости ортог ональных рядов.