# Two-dimensional Waterman classes and u-convergence of Fourier series

## Abstract

New results on the u-convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than BV(T{sup 2}) ensures even the uniform boundedness of the u-sums of the double Fourier series of functions in this class. On the other hand, the concept of u(K)-convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions f(x,y) belonging to the class {lambda}{sub 1/2}BV(T{sup 2}), where {lambda}{sub a}={l_brace}n{sup 1/2}/(ln(n+1)){sup a}{r_brace}{sub n=1}{sup {infinity}}, the corresponding u(K)-partial sums are uniformly bounded, while if f(x,y) element of {lambda}{sub a}BV(T{sup 2}), where a<1/2, then the double Fourier series of f(x,y) is u(K)-convergent everywhere.

- Authors:

- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21202881

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 190; Journal Issue: 7; Other Information: DOI: 10.1070/SM1999v190n07ABEH000414; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONVERGENCE; COORDINATES; FUNCTIONS; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
D'yachenko, M I.
```*Two-dimensional Waterman classes and u-convergence of Fourier series*. United States: N. p., 1999.
Web. doi:10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
D'yachenko, M I.
```*Two-dimensional Waterman classes and u-convergence of Fourier series*. United States. doi:10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
D'yachenko, M I. Tue .
"Two-dimensional Waterman classes and u-convergence of Fourier series". United States. doi:10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21202881,
```

title = {Two-dimensional Waterman classes and u-convergence of Fourier series},

author = {D'yachenko, M I},

abstractNote = {New results on the u-convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than BV(T{sup 2}) ensures even the uniform boundedness of the u-sums of the double Fourier series of functions in this class. On the other hand, the concept of u(K)-convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions f(x,y) belonging to the class {lambda}{sub 1/2}BV(T{sup 2}), where {lambda}{sub a}={l_brace}n{sup 1/2}/(ln(n+1)){sup a}{r_brace}{sub n=1}{sup {infinity}}, the corresponding u(K)-partial sums are uniformly bounded, while if f(x,y) element of {lambda}{sub a}BV(T{sup 2}), where a<1/2, then the double Fourier series of f(x,y) is u(K)-convergent everywhere.},

doi = {10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 7,

volume = 190,

place = {United States},

year = {1999},

month = {8}

}