Tomasz Szostok

On Functional Equations Connected with Quadrature Rules

Abstract The functional equations of the form are considered. They are connected with quadrature rules of the approximate integration. We show that such equations characterize polynomials in the class of continuous functions. It is also shown that if the number of components is sufficiently small, then the continuity is forced by the equation itself. Unique solvability of the considered problem are established.

Some functional equations characterizing polynomials

Abstract We present a method of solving functional equations of the type where f, F: P → P are unknown functions acting on an integral domain P and parameteres are given. We prove that under some assumptions on the parameters involved, all solutions to such kind of equations are polynomials. We use this method to solve some concrete equations of this type. For example, the equation (1) for f, F: ℝ → ℝ is solved without any regularity assumptions. It is worth noting that (1) stems from a well-known quadrature rule used in numerical analysis.

Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and Levin–Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality

On the stability of the equation stemming from Lagrange MVT

\begin{aligned} f\left( \frac{x+y}{2}\right) \le \frac{1}{(y-x)^2} \int _x^y\int _x^yf\left( \frac{s+t}{2}\right) \hbox {d}s\,\hbox {d}t \le \frac{1}{y-x}\int _x^yf(t)\hbox {d}t \end{aligned}

Ohlin’s lemma and some inequalities of the Hermite–Hadamard type

fx+y2≤1(y-x)2∫xy∫xyfs+t2dsdt≤1y-x∫xyf(t)dtsatisfied by every convex function

ON A CLASS OF EQUATIONS STEMMING FROM VARIOUS QUADRATURE RULES

f{:}\,\mathbb R\rightarrow \mathbb R