Approximation of functions and Mihesan operators

Abstract

<p style='text-indent:20px;'>For real numbers <inline-formula><tex-math id="M1">\begin{document}$ a,q\geq 0 $\end{document}</tex-math></inline-formula> and a weight <inline-formula><tex-math id="M2">\begin{document}$ \varrho(x) = 1/(1+x)^q $\end{document}</tex-math></inline-formula>, the author provides necessary and sufficient conditions for a function <inline-formula><tex-math id="M3">\begin{document}$ f\in C[0,\infty) $\end{document}</tex-math></inline-formula> in order to <inline-formula><tex-math id="M4">\begin{document}$ \sup_{x\geq 0}\mid \varrho(x)(B_n^a(f,x)-f(x))\mid \to 0 $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M5">\begin{document}$ n\to \infty $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ B_n^a(f) $\end{document}</tex-math></inline-formula> is the Mihesan operator. In particular, it is proved that Mihesan operators behave similar to the classical Baskakov operators.</p>

Directs estimates and a Voronovskaja-type formula for Mihesan operators

“…Theorem 2.1. (see [4]) If a ≥ 0 and q ≥ 0 are real numbers, there exists a constant M q (a) such that…”

“…Proposition 2.1. (see [4]) If a > 0, r ∈ [0, 1], there exists a constant C such that for each integer n > 1 and each x ≥ 0,…”

“…It is known a similar result for a = 0. We can use the arguments in [4] to verify that, if a ≥ 0 and r ∈ [0, 2], there exists a constant C = C(a, r) such that, for n > 2,…”

Fourier Series Approximation in Besov Spaces

“…However, the use of structural properties can lead to better results. For the latest developments in the feld of approximation by Fourier series and linear operators, one can see [17,18]. Te Besov spaces, being on the top of the L p -spaces, are good at encoding the smoothness properties of their functions.…”