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.

On spectral analysis in varieties containing the solutions of inhomogeneous linear functional equations

“…Theorem 3.3. The existence of a nonzero additive solution of (7) implies that there exist a finitely generated subfield K ⊂ C containing α i and β i (i = 1, . .…”

“…Equation ( 1) is motivated by quadrature rules of approximate integration. The problem is due to T. Szostok [5], see also [6] and [7]. To formulate the basic preliminary results and facts we need the notion of generalized polynomials.…”

“…In what follows we are going to show that spectral analysis can be applied in translation invariant closed linear subspaces (varieties) of additive functions on some finitely generated fields containing the restrictions of the solutions of functional equation (7). Note that the translation invariance is taken with respect to the multiplicative group structure.…”

On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

“…The case of p > 2 can be investigated in a similar way because of the inductive argument as follows. The first equation of system (8)…”

“…where x, y ∈ C and f, F : C → C are unknown functions. It is motivated by quadrature rules of approximate integration [6], see also [7] and [8].…”