Abstract

In this paper, we define a new generalization of Mellin-Gauss-Weierstrass operators that preserve logarithmic functions. We compute logarithmic moments of the new operators and describe the behavior of the modified operators in some weighted spaces. The weighted approximation properties of operators including weighted approximation and weighted quantitative type approximation properties, using weighted logarithmic modulus of continuity, are presented. Using the Mellin-Gauss-Weierstrass kernel p(⋅,⋅)$p(·,·)$$p(·,·)$ as a logarithmic probability density, we study the associated information potential, the expected value E[log p(·,·)]E[log p(⋅,⋅)]$E[\log \hspace{0.17em}p(·,·)]$ and the variance $\mathit{Var}[\log \hspace{0.17em}p(·,·)]$.

摘要

在本文中，我们定义了保留对数函数的 Mellin-Gauss-Weierstrass 算子的新推广。我们计算新算子的对数矩，并描述修改算子在某些加权空间中的行为。给出了算子的加权逼近性质，包括使用加权对数连续性模的加权逼近和加权定量型逼近性质。使用 Mellin-Gauss-Weierstrass 内核p ( · , · )$p（·,·）$$p(·,·)$作为对数概率密度，我们研究相关的信息潜力，即期望值E [ log p ( · , · ) ] 乙[ log p ( ⋅ , ⋅ ) ] $E[\log \hspace{0.17em}p(·,·)]$和方差$\mathit{瓦尔}[\mathrm{日志}\hspace{0.17em}p（·,·）]$。