It is well known (see e.g. [1]) that the Cauchy functional equation

characterizes the function f:x→exx.

It was mentioned in [2D that the functions f:x→A exp(αx2m), g:x→exx/A can be characterized by the equation

(1)

but the proof was done only for m=l which was considerably simple.

Let us consider the well-known Hermite-Fejér interpolating process on the interval [—1,1] i.e. let

(1.1)

(sometimes we omit the second indices),

(1.2)

In this note we show that every (not necessarily commutative) integral domain R has a quotient ring which, although need not be a field, has the property that all of its right invariant elements are units. As an application this shows that every PRI (principal right ideal) domain can be embedded in a simple PRI domain which is, in general, not a field.

Let X1, X2, X3 … be i.i.d., Sn their nth partial sum with Sn = 0; Suppose that

This note is concerned with the implication SepII(Q)→SepI(Q) where Q is a class of subsets of some set S.

where cZ denotes S—Z.

It is well-known that in general the above implication is false (e.g. let Q be the closed subsets of the reals).

It is well known [1,3] that a commutative semigroup (S, +) can be embedded in a semigroup which is a union of groups if and only if S is separative (2a = a + b = 2b implies a = b). We extend this result to additively commutative semirings.

A semiring (S, +, ⋅) is a set S with associative addition (+) and multiplication (⋅), the latter distributing over addition from left and right. In what follows (S, +, ⋅) will denote a semiring in which the additive semigroup (S, +) is commutative. An element 0 can be adjoined, where s = s + 0, 0 = 0 ⋅ s = s ⋅ 0 for all s in S, to form S°.

Given a matrix A = (ank) (n, k = 0, 1, 2, …), let (A) denote the set of all sequences x = {xk} such that {An(x)} ∊ c where An(x) = Σk=0∞ankxk (n≥0) and c denotes the set of all convergent sequences. It is well known (see e.g. Zeller [6] or Zeller and Beekmann [7], p. 48) that given an unbounded sequence x, there exists a regular (=permanent) matrix A with ank = 0 for k > n (and indeed with ann ≠ 0) such that (A) = c ⊕x, the linear space spanned by c and x. We call A an Einfolgenverfahren. (See [7].) In [4] Rhoades considered, inconclusively, the question whether there exists a Hausdorff matrix H such that (H)= c ⊕ x (for arbitrary unbounded sequence x).