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In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

In this paper, we utilize some known Steffensen-type identities, obtained by using Lidstone's interpolating polynomial, to prove new generalizations of Steffensen's inequality. We obtain these new generalizations by using the weighted Hermite-Hadamard inequality for (2 n + 2) - convex and (2 n + 3) - convex functions. Further, the newly obtained inequalities can be observed as an upper- and lower-bound for utilized Steffensen-type identities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the affine variety codes obtained evaluating the polynomials b y = a k x k + ... + a 1 x + a 0 , b , a i ∈ F q r , at the affine F q r -rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolute irreducibility of certain algebraic varieties. [ABSTRACT FROM AUTHOR]

In this paper we consider generalized monomial functions f , g : F → C (of possibly different degree) that also fulfill f (P (x)) = Q (g (x)) x ∈ F , where P ∈ F [ x ] and Q ∈ C [ x ] are given (classical) polynomials. [ABSTRACT FROM AUTHOR]

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]

Spectral analysis and synthesis studies translation invariant function spaces, so-called varieties over topological groups. The basic building blocks are the finite dimensional varieties. In the commutative case finite dimensional varieties are spanned by exponential polynomials. In non-commutative situations no relevant results exist. In this paper we consider finite dimensional left translation invariant linear spaces of continuous complex valued functions over the Heisenberg group. Using basic knowledge about Lie algebra we describe all left varieties of this type. In particular, it turns out that those function spaces are spanned by exponential polynomials as well. [ABSTRACT FROM AUTHOR]

In this paper we consider a generalized monomial or polynomial f : R → R that satisfies the additional equation f (x) f (y) = 0 for the pairs (x , y) ∈ D , where D ⊆ R 2 is given by some algebraic condition. In the particular cases when f is a generalized polynomial and there exist non-constant regular polynomials p and q that fulfill D = { (p (t) , q (t)) | t ∈ R } or f is a generalized monomial and there exists a positive rational m fulfilling D = { (x , y) ∈ R 2 | x 2 - m y 2 = 1 } , we prove that f (x) = 0 for all x ∈ R . [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]

Let G be an Abelian group, and let C G denote the set of complex valued functions defined on G. A map D : C G → C G is a difference operator, if there are complex numbers a i and elements b i ∈ G (i = 1 , ... , n) such that (D f) (x) = ∑ i = 1 n a i f (x + b i) for every f ∈ C G and x ∈ G . By a system of difference equations we mean a set of equations { D i f = g i : i ∈ I } , where I is an arbitrary set of indices, D i is a difference operator and g i ∈ C G is a given function for every i ∈ I , and f is the unknown function. The solvability cardinal sc (F) of a class of functions F ⊂ C G is the smallest cardinal number κ with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality < κ has a solution in F , then S itself has a solution in F . The behaviour of sc (F) is rather erratic, even for classes of functions defined on R . For example, sc (C [ x ]) = 3 , but sc (TP) = ω 1 , where TP is the set of trigonometric polynomials; sc (C R) = ω , but sc (DF) = (2 ω) + , where DF is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on R n , on normed linear spaces and, in general, on topological Abelian groups. Let P G denote the class of polynomials defined on the group G. After presenting some general estimates we prove that sc (C [ x 1 , ... , x n ]) = ω if 2 ≤ n < ∞ , and sc (P X) = ω 1 if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that sc (P G) = 3 if r 0 (G) ≤ 1 , sc (P G) = ω if 2 ≤ r 0 (G) < ∞ , and sc (P G) ≥ ω 1 if r 0 (G) is infinite, where r 0 (G) denotes the torsion free rank of G. The solvability of systems of difference equations is closely connected to the existence of projections of function classes commuting with translations (see Theorem 7.1). As an application we construct a projection from C R n onto C [ x 1 , ... , x n ] commuting with translations by vectors having rational coordinates (Theorem 7.4). [ABSTRACT FROM AUTHOR]