On Szalay's theorem of double series

[1] . , [1] .     

The order of magnitude of the arithmetic means of double orthogonal series

, - , (C, 1,1), (C, 1,0) (C, 0,1)- . , , . - .

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Dedicated to Academician S. M. Nikol'skii on his 80th birthday     

A matrix generalization of a theorem of SzegŐ

— -B T . , T, T. — T, .., d=1. - . Cere: 0<p< exp logd=inf ¦1–t¦ ^p d, t , t(0)=0., . . . . . , . [1]. , . p=2 . - . p=2 [6, 8]. — p.     

On power series with positive coefficients

M. . , . , ^p () (). , , .     

On a problem of B. I. Golubov

. . . : s_n(x) — , _n-(, 1)- n L ². , . , : a _k l ² _n () [0,1] , (*) , (**) . a _k l ² u _n () [0,1] , (**), (*) .     

Thep-adic differential-integral type operator

, p- , , , , , , . —p- - — , . , , .. , .     

WT measure vector spaces

n- M WT- , M n–1 . . WT- . .     

On ortbogonal polynomial bases in the spacesC andL

, >0 C L - ( ) {Q _n(x)} , Q _n (x)–v _n n ¹⁺ nn ₀ (). , =0.     

Perfect splines of least uniform deviation

(v _k) ₁ ⁿ , 1v _kr, v₁+...+v_n r, p(;r) — =(x ₁,v ₁),...,(x _n,v _n), v _1+...+v _n-r ^(r(;·)=1. , (^*;t) [, ] (;t) _{x 1}<... n v ₁ ...,v _n. (^*;t) .     

Generalized localization and convergence tests for double trigonometric Fourier series of functions fromL p,p>1

. E , f(x)L _p (T ^N ),P1,f(x)=0 E (E— ^N =[-, ]^N) E , . , .

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In closing the author thanks V. A. Il'in and . A. Alimov for their constant attention paid to the present work.     

On contiguity of probability measures corresponding to semimartingales

, (ⁿ), - (P ⁿ ), P ⁿ (A ⁿ )>0P ⁿ (A ⁿ )0,n. [15] - , . , P ⁿ P ⁿ T ⁿ T ⁿ .     

On the growth order of partial sums of non-orthogonal series

a _k f _k , f _k L ₂, w-, (2), w(n) — . a _k f _k N {a _k }l ₂, {a _k }l ₂ ( 1, 2, 1a, 2a). ( 2) [8]. , {a _k } w-.     

Fourier transforms of functions in Herz spaces on certain groups

G— - {G _n } _n =– , G _n =G G _n ={0}. G. K(,p,q;G) K(,p,q;) - G , . , G - (. . sup {order (G _n /G _n +1):=0, ± 1, ...<), K(.,p,q;G) L ^{p/(p–p–1),q} () L ^{p/(p–p–1),q} () K(-,p,q; ), 1<p2, 0<<1/p=1–1/p, 0<q. . . . .     

A comparison of uniform and discrete polynomial approximation

- , . .     

Riesz representation theorem and weak compactness

. , .     

On the notion of measurability

. , . , . , , - , . , , - .

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I wish to record my thanks to Dr. Kummer for his constant encouragement and many helpful suggestions in preparing this paper.     

The absolute generalized harmonic-Cesàro summability of a Fourier series

[8] . , , [2] , . - , , , . .     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.     

Some expansion formulae for Fox'sH-function of two variables involving associated Legendre functions

H- .     

Trigonometric Cesàro bases in the spaces of functions integrable with power weight

(2), k1, >0, L _p (0,), 1p L =C. , , p, k, (C, )- L _p (0,), , , {sinnx} _n ^=k/ (C, )- L _p (0,) |x|^p . , 1p, {x sinnx} _n=k , k2 2k–2–1/p<2k–1/p, (C, )- L ^p [0,] , >–(p–1)/.