# The convergence of double Fourier-Haar series over homothetic copies of sets

## Abstract

The paper is concerned with the convergence of double Fourier- Haar series with partial sums taken over homothetic copies of a given bounded set W⊂R{sub +}{sup 2} containing the intersection of some neighbourhood of the origin with R{sub +}{sup 2}. It is proved that for a set W from a fairly broad class (in particular, for convex W) there are two alternatives: either the Fourier-Haar series of an arbitrary function f∈L([0,1]{sup 2}) converges almost everywhere or Lln{sup +} L([0,1]{sup 2}) is the best integral class in which the double Fourier-Haar series converges almost everywhere. Furthermore, a characteristic property is obtained, which distinguishes which of the two alternatives is realized for a given W. Bibliography: 12 titles. (paper)

- Authors:

- Tseriteli State University, Kutaisi, Georgia (United States)

- Publication Date:

- OSTI Identifier:
- 22364932

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Sbornik. Mathematics; Journal Volume: 205; Journal Issue: 7; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CALCULATION METHODS; CONVERGENCE; FUNCTIONS; INTEGRALS; MATHEMATICAL SOLUTIONS

### Citation Formats

```
Oniani, G. G., E-mail: oniani@atsu.edu.ge.
```*The convergence of double Fourier-Haar series over homothetic copies of sets*. United States: N. p., 2014.
Web. doi:10.1070/SM2014V205N07ABEH004406.

```
Oniani, G. G., E-mail: oniani@atsu.edu.ge.
```*The convergence of double Fourier-Haar series over homothetic copies of sets*. United States. doi:10.1070/SM2014V205N07ABEH004406.

```
Oniani, G. G., E-mail: oniani@atsu.edu.ge. Thu .
"The convergence of double Fourier-Haar series over homothetic copies of sets". United States.
doi:10.1070/SM2014V205N07ABEH004406.
```

```
@article{osti_22364932,
```

title = {The convergence of double Fourier-Haar series over homothetic copies of sets},

author = {Oniani, G. G., E-mail: oniani@atsu.edu.ge},

abstractNote = {The paper is concerned with the convergence of double Fourier- Haar series with partial sums taken over homothetic copies of a given bounded set W⊂R{sub +}{sup 2} containing the intersection of some neighbourhood of the origin with R{sub +}{sup 2}. It is proved that for a set W from a fairly broad class (in particular, for convex W) there are two alternatives: either the Fourier-Haar series of an arbitrary function f∈L([0,1]{sup 2}) converges almost everywhere or Lln{sup +} L([0,1]{sup 2}) is the best integral class in which the double Fourier-Haar series converges almost everywhere. Furthermore, a characteristic property is obtained, which distinguishes which of the two alternatives is realized for a given W. Bibliography: 12 titles. (paper)},

doi = {10.1070/SM2014V205N07ABEH004406},

journal = {Sbornik. Mathematics},

number = 7,

volume = 205,

place = {United States},

year = {Thu Jul 31 00:00:00 EDT 2014},

month = {Thu Jul 31 00:00:00 EDT 2014}

}