Rational approximation on the nonnegative integers

Weighted Polynomial Approximation on the Integers

We prove here some polynomial approximation theorems, somewhat related to the Szasz-Mfintz theorem, but where the domain of approximation is the integers, by dualizing a gap theorem of C. l ~ Y I for periodic entire functions. In another Paper [7], we shall prove, by similar means, a completeness theorem ibr some special sets of entire functions. I t is well known (see, for example [l]) tha t i...

Representations of Integers by Linear Forms in Nonnegative Integers*

Let Sz be the set of positive integers that are omitted values of the form f = z”= a.x. where the a, are fixed and relatively prime natural numbers *1 $1) and the xi are variable nonnegative integers. Set w = #Q and K = max 0 + 1 (the conductor). Properties of w and K are studied, such as an estimate for w (similar to one found by Brauer) and the inequality 2w > K. The so-called Gorenstein cond...

Quadratic Reciprocity for the Rational Integers and the Gaussian Integers

K-complementing Subsets of Nonnegative Integers

Let S = {S1,S2, . . .} represent a collection of nonempty sets of nonnegative integers in which each member contains the integer 0. Then S is called a complementing system of subsets for X ⊆ {0,1, . . .} if every x ∈ X can be uniquely represented as x = s1 + s2 + ··· with si ∈ Si. We will also write X = S1 ⊕ S2 ⊕ ··· and, when necessary, refer to X as the direct sum of the Si. We will denote th...

Analogues of the Ring of Rational Integers