D'Alembert's functional equation on restricted domains

Summary In the class of functionalsf:X , whereX is an inner product space with dimX 3, we study the D'Alembert functional equationf(x + y) + f(x – y) = 2f(x)f(y) (1) on the restricted domainsX ₁ = {(x, y) X ²/x, y = 0} andX ₂ = {(x, y) X ²/x = y}. In this paper we prove that the equation (1) restricted toX ₁ is not equivalent to (1) on the whole spaceX. We also succeed in characterizing all common solutions if we add the conditionf(2x) = 2f²(x) – 1. Using this result, we prove the equivalence between (1) restricted toX ₂ and (1) on the whole spaceX. This research follows similar previous studies concerning the additive, exponential and quadratic functional equations.     

Determinantal formulas for zonal spherical functions on hyperboloids

Abstract. We construct determinantal expressions for the zonal spherical functions on the hyperboloids with p,q odd (and larger than 1). This gives rise to explicit evaluation formulas for hypergeometric series representing half-integer parameter families of Jacobi functions and (via specialization) Jacobi polynomials. Received November 18, 1999 / Published online October 30, 2000     

Pexider's equation and aggregation of allocations

An extension theorem for Pexider's equation is proved and used to generalize the results in [4] to cases with weights with more than one constraint and to more general domains in a form which can be applied to multiobjective linear programming. Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

The associativity equation revisited

Summary Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I × I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.     

On approximate solutions of the Pexider equation

Summary LetX be an abelian (topological) group andY a normed space. In this paper the following functional inequality is considered: {ie143-1} This inequality is a similar generalization of the Pexider equation as J. Tabor's generalization of the Cauchy equation (cf. [3], [4]). The solutions of our inequality have similar properties as the solutions of the Pexider equation. Continuity and related properties of the solutions are investigated as well.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On an alternative functional equation related to the Cauchy equation

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.     

A system of functional equations related to plurality functions. A method for the construction of the solutions

Summary In the first part of the present paper we prove some necessary conditions satisfied by the solutions of a system of functional equations related to Plurality Functions. In the second part we describe a geometric-combinatorial procedure for the construction of the solutions of that system. This procedure yields all possible solutions.     

On an exponential-cosine functional equation

The exponential cosine functional equationf(x + y) + (2f ²(y) – f(2y))f(x – y) = 2f(x)f(y) is studied in some detail whenf is a complex valued function defined on a Banach space. We supply conditions which ensure continuity off everywhere under the hypothesis thatf is continuous at a point. We also find solutions of the functional equation which are continuous at some point.