Approximation by Nörlund means of double fourier series to continuous functions in two variables

We study the rate of uniform approximation by Nörlund means of the rectangular partial sums of double Fourier series of continuous functionsf(x, y), 2π-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness defined by $$\begin{gathered} \omega _2 \left( {f,\delta _1 ,\delta _2 } \right) = \mathop {\sup }\limits_{x,y} \mathop {\sup }\limits_{\left| u \right| \leqslant \delta _1 ,\left| v \right| \leqslant \delta _2 } \left| {f\left( {x + u,y + v} \right)} \right. + f\left( {x + u,y - v} \right) + f\left( {x - u,y + v} \right) \hfill \\ + \left. {f\left( {x - u,y - v} \right) + 4f\left( {x,y} \right)} \right| for \delta _1 ,\delta _2 \geqslant 0. \hfill \\ \end{gathered} $$ As a special case we obtain the rate of uniform approximation to functionsf(x,y) in Lip({α, β}), the Lipschitz class, and inZ({α, β}), the Zygmund class of ordersα andβ, 0<α,β ≤ l, as well as the rate of uniform approximation to the conjugate functions \(\tilde f^{(1,0)} (x,y), \tilde f^{(0,1)} (x,y)\) and \(\tilde f^{(1,1)} (x,y)\) .     

An improved Menshov-Rademacher theorem

We study the a.e. convergence of orthogonal series defined over a general measure space. We give sufficient conditions which contain the Menshov-Rademacher theorem as an endpoint case. These conditions turn out to be necessary in the particular case where the measure space is the unit interval and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality.

     

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 convergence in a restricted sense of multiple series

Z ^d — k=(k ₁, ...,k _d) k _j,d1.d- (8), . . a _k s _m= a _k s, >0 N, min (m ₁,...,m _d)N, ¦s _m–s¦. , , >0 N, min (m ₁,...,m _d)N min (n ₁,...,n _d)N, ¦s _m–s _n. . , (8) , >0 N, max (b ₁,...,b _d) N, m — Z ^d , m1, ¦s(b, m)¦ where      

Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejér on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.