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The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form f (x) - f (y) . [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

We examine the family of all (at most) three-point symmetric quadratures on [ - 1 , 1 ] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020). [ABSTRACT FROM AUTHOR]

Given rational vector spaces V, W a mapping $${f \colon V \to W}$$ is called a generalized polynomial of degree at most n, if there are homogeneous generalized polynomials f of degree i such that $${f = \sum_{i = 0}^n f_i}$$ . Homogeneous generalized polynomials f of degree i are mappings of the form $${f_i (x) = f_i^*(x, x, \ldots , x)}$$ with $${f_i^* \colon V^i \to W i}$$ -linear. In the literature one may find quite a lot of functional equations such that their general solution is of the form f or $${f_n + f_{n - 1}}$$ where n is a small positive integer ( ≤ 6 or ≤ 4 respectively). In this paper, given an arbitrary positive integer n and an arbitrary subset $${L \subseteq \{0, 1, \ldots, n\}}$$ such that $${n \in L}$$ , a method is described to find (many) functional equations, such that their general solution is given by $${\sum_{i \in L} f_i}$$ . For the cases $${L = \{n\}}$$ and $${L = \{n - 1, n\}}$$ additional equations are given. [ABSTRACT FROM AUTHOR]