Mirakvyan singular integral operators and approximation of unbounded continuous functions (Q2765626)

Using the multiplier-enlargement method and the classical appropriate functions \(1,x,x^2\), a theorem for approximating unbounded continuous functions by some modified positive linear operators is indicated.NEWLINENEWLINENEWLINEAs an example of possible applications of this theorem, the authors discussed the Mirakvyan singular integral operators in details. The main result embodied in Theorem 3.1 yields a sufficient condition ensuring that the limit relation NEWLINE\[NEWLINE\lim_{n\to\infty} M_n\bigl[f(\alpha_nt); \alpha_n^{-1}x \bigr]= f(x),\quad \text{a .e. on }(-\infty, \infty)NEWLINE\]NEWLINE holds uniformly for any continuous function \(f(x)\) defined on \((-\infty,\infty)\), where \(\alpha_n \uparrow \infty\), \(\beta_n\downarrow 0\) are real sequences and NEWLINE\[NEWLINEM_n\bigl[f(t); x\bigr]: ={1\over\Delta_n} \int^\beta_\alpha f(t)\bigl[\psi(t-x) \bigr]^{-1}dt(-r<\alpha <\beta <r)NEWLINE\]NEWLINE is the Mirakvyan singular integral operator.NEWLINENEWLINENEWLINEThe above theorem is proved on the base of a known result given by \textit{R. Wang} [Mathematica (cluj), 28(1), 131-136, (1963; Zbl 0133.31105)]. Some related results can be found in the paper by \textit{S. Eisenberg} and \textit{B. Wood} [SIAM J. Numer. Anal., 9, 266-276 (1972; Zbl 0242.41009)].