APPROXIMATION BY POSITIVE OPERATORS OF THE C₀–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART I

Degenerate differential equations and modified Szász-Mirakjan operators

“…By using Proposition 3.2 of [1] it is easy to obtain the following result. We omit the details for the sake of brevity.…”

“…To show the assertion we shall apply Theorem 4.1 in [1]. Note that e 1 , e 2 ∈ E m and β k e j ∈ E m−k+ j for j + k ≤ 4, because (25) implies |β k e j | ≤ |β e j+k−1 | and β e 3 ∈ E m .…”

Integral-type operators on continuous function spaces on the real line

“…In the spirit of [2] we shall modify the operators G n , in order to obtain a further positive approximation process on C w 0 (R), which satisfies an asymptotic formula generating a complete second order differential operator on the real line (see Theorem 5.2).…”

Degenerate elliptic operators, Feller semigroups and modified Bernstein‐Schnabl operators

“…In this section, by adapting an idea first developed in [3] (see also [9]), we introduce and study a modification of Bernstein-Schnabl operators. Such a modification will be shown to provide us with an efficient tool to approximate the Feller semigroup generated by (the closure of) a first-order additive perturbation of the differential operator (2.8), which in turn will enter into play when we shall investigate an asymptotic formula for our operators.…”

“…As regards the contents of the paper, we start by introducing and studying an approximation process (M n ) n ≥n 0 on C(K) obtained as a modification of the Bernstein-Schnabl operators, according to a general method introduced in [3] (see also [9]). …”