On power series with positive coefficients

M. . , . , ^p () (). , , .     

Lacunary (0; 0, 1) interpolation on the roots of Jacobi polynomials and their derivatives,respectively. I (Existence,explicit formulae,unicity)

(0; 0, 1) , {x _k <x _k ^* <x _k+1} _k=1 ^n–1 {x _{k k=1} ⁿ }., I, , _n (x)=P _n ^{(, )} (x)–n- , =, n3 . , x ₀=+1 x _n+1= –1. II .

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To the memory of Paul Erds

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The research was supported by the Hungarian National Foundation for Scientific Research under Grant # T 914 244.     

A comparison of uniform and discrete polynomial approximation

- , . .     

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.     

WT measure vector spaces

n- M WT- , M n–1 . . WT- . .     

The absolute generalized harmonic-Cesàro summability of a Fourier series

[8] . , , [2] , . - , , , . .     

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

H- .     

On extremal subspaces and approximation of periodic functions by splines of minimal defect

. . , 320 625 . 72      

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 Szalay's theorem of double series

[1] . , [1] .     

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)/.     

Thep-adic differential-integral type operator

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

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

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-.     

Summability of multiple Fourier series. Growth of Lebesgue constants

. , . . , H ₂ . , .     

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.     

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. . . . .     

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     

Hardy spaces of predictable martingales

, P _p Q _p .

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This research was partially supported by the Hungarian Scientific Foundation.     

An analogue of a Hardy-Littlewood theorem for Laplace transform and its applications

G(z) — g(t), Rez>0. , 1 pq<, <–1, G(x) L ^p (R ⁺,x dx) L ^q (R ⁺,x dx), =q–1–q(+1)/p. - - , .