On representation of aperiodic functions of \(\Lambda\)-bounded variation by the Fourier integral (Q1406386)

The main result of the paper is as follows. Let the function \( f\in L({\mathbb R}^2)\cap \Lambda \text{BV}({\mathbb R}^2)\), where \( \Lambda\in {\mathbb K}\). Then at every regular point \( (x,t)\in {\mathbb R}^2 \) of the function \(f\) \[ \lim\limits_{M, N\to\infty} \sigma_{M,N}(x,y) = f^*(x,y). \] In these relation \[ \begin{gathered} \sigma_{M,N}(x,y) = \frac{1}{2\pi} \int\limits_{-M}^M \int\limits_{-N}^N \hat f(s,t)e^{i(sx+ty)} dt ds,\\ \hat f(x,y) = \frac{1}{2\pi}\iint\limits_{{\mathbb R}^2} f(x+s,y+t) \frac{\sin Ms}{s} \frac{\sin Nt}{t} dt ds,\\ f^*(x,y) = \frac{f(x+0,y+0) + f(x+0,y-0) + f(x-0,y+0) + f(x-0,y-0)}{4}. \end{gathered} \] {}.