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 , . , .

---

---

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

On power series with positive coefficients

M. . , . , ^p () (). , , .     

Some remarks on the approximation in the strong sense

. , , , , . , . , , .

---

---

On the 70th birthday of Professor S. M. Nikol'skii     

On convergence and summability of Fourier series

. , , –1<<0. .

---

---

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- . .     

A generalization of the Heine limit for functions which converge on a base

. , , . . . .     

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 .

---

---

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

Rate of approximation by rectangular partial sums of double orthogonal series

( ) . .

---

---

Dedicated to Professor K. Tandori on his seventieth birthday

---

This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.     

Summability of multiple Fourier series. Growth of Lebesgue constants

. , . . , H ₂ . , .     

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 .     

The absolute generalized harmonic-Cesàro summability of a Fourier series

[8] . , , [2] , . - , , , . .     

Blocks of orthogonal random variables and the strong law of large numbers

(X _k ),k=1,2,... — _k ² >1; (X _k ) , E(X _k X _t )=0 p k<>(p+1) (p,k,l=1, 2, ...) , , ,

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 .

---

---

To the memory of Paul Erds

---

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

A test of convergence of Fourier series with respect to multiplicative systems,analogous to the Jordan test

, {p _n} _n=0 (p₀=1, _n2 n2). : x f(t) V(G)

On the interrelation of the generalized Besov-Nikol'skii and Weyl-Nikol'skii classes of functions

, , , . .

---

---

Dedicated to Professors K. Tandori and L. Leindler on the occasion of their anniversaries

---

This work was completed in support of the Russian Foundation of Fundamental Research (Project # 96-01-00094) and of the International Scientific Foundation (Grant # NCI-300).     

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) .     

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-.     

The trace to subsets ofR n of Besov spaces in the general case

R ⁿ. , , , F R ⁿ, F , R ⁿ R ⁿ . ^p,q (Rⁿ), >0, 1, q, — ( ) Rⁿ. , ^p,q (Rⁿ) ^F Rⁿ. , q B ^p,q (F), = – (n–)/, >0, — « », ad — F, . , . : , F=R ^d,F— « » F— R ⁿ, « », F. .

---

---

This work has been supported in part by the Swedish Natural Science Research Council.