### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 15 (2021)

# About the convergence of the Fourier transform

### UDK: 517.51

### Pages: 1 - 21

### DOI: 10.31029/demr.15.1

The main result is the proof of the theorems, the results of which one can characterize as a weak form of the formula for the inversion of the bi-dimmensional Fourier transform. Sufficient conditions on a function are obtained for a weak (of degree $r$) convergence of bi-dimmensional Fourier transform for a function $f(x;y)$. These conditions have an integral form and describe the behavior of the function near the border of a rectangle. A similar theorem is proved, in which the Fourier transform of a function $f$ is replaced by the Fourier transform of another function $g$, the norm of the central difference of which does not exceed the norm of the central difference of $f$.
The principal objective is to study the behavior of the Fourier transform of $g$ and $f$.

**Keywords: **
Two-dimensional Fourier transform, Riemann-Lebesgue theorem.