In this paper, we construct two classes of rational function operators using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.

The maximum entropy method for the Hausdorff moment problem suffers from ill conditioning as it uses monomial basis {1,x,x2,...,xn}. The maximum entropy method for the Chebyshev moment probelm was studied to overcome this drawback in. In this paper we review and modify the maximum entropy method for the Hausdorff and Chebyshev moment problems studied in and present the maximum entropy method for the Legendre moment problem. We also give the algorithms of converting the Hausdorff moments into the Chebyshev and Lengendre moments, respectively, and utilizing the corresponding maximum entropy method.

Let S: [0, 1]→[0, 1] be a chaotic map and let f* be a stationary density of the Frobenius-Perron operator PS: L1→L1 associated with S. We develop a numerical algorithm for approximating f*, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.

For a positive finite measure $d\mu \left( \mathbf{u} \right)$ on ${{\mathbb{R}}^{d}}$ normalized to satisfy $\int{_{{{\mathbb{R}}^{d}}}d\mu \left( \mathbf{u} \right)}=1$ , the dilated average of $f\left( \mathbf{x} \right)$ is given by

$${{A}_{t}}\,f\left( \mathbf{x} \right)\,=\,\int{_{{{\mathbb{R}}^{d}}}\,f\left( \mathbf{x}\,-\,t\mathbf{u}\, \right)}d\mu \left( \mathbf{u} \right)$$

It will be shown that under some mild assumptions on $d\mu \left( \mathbf{u} \right)$ one has the equivalence

$$||{A_t}f - f|{|_B} \approx \inf \left\{ {\left( {||f - g|{|_B} + {t^2}||P\left( D \right)g|{|_B}} \right):P\left( D \right)g \in B} \right\}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\rm{for}}{\mkern 1mu} {\mkern 1mu} {\rm{t}}{\mkern 1mu} {\rm{ > }}\,{\rm{0,}}$$

where $\varphi \left( t \right)\approx \psi \left( t \right)$ means ${{c}^{-1}}\le \varphi \left( t \right)/\psi \left( t \right)\le c$ , $B$ is a Banach space of functions for which translations are continuous isometries and $P\left( D \right)$ is an elliptic differential operator induced by $\mu $. Many applications are given, notable among which is the averaging operator with $d\mu \left( \mathbf{u} \right)\,=\,\frac{1}{m\left( S \right)}{{\chi }_{S}}\left( \mathbf{u} \right)d\mathbf{u},$ where $S$ is a bounded convex set in ${{\mathbb{R}}^{d}}$ with an interior point, $m\left( S \right)$ is the Lebesgue measure of $S$, and ${{\chi }_{S}}\left( \mathbf{u} \right)$ is the characteristic function of $S$. The rate of approximation by averages on the boundary of a convex set under more restrictive conditions is also shown to be equivalent to an appropriate $K$-functional.

We consider positive linear operators of probabilistic type L1f acting on real functions f defined on the positive semi-axis. We deal with the problem of uniform convergence of L1f to f, both in the usual sup-norm and in a uniform Lp type of norm. In both cases, we obtain direct and converse inequalities in terms of a suitable weighted first modulus of smoothness of f. These results are applied to the Baskakov operator and to a gamma operator connected with real Laplace transforms, Poisson mixtures and Weyl fractional derivatives of Laplace transforms.