Periodic decomposition of measurable integer valued functions

Difference functions of periodic measurable functions

We investigate some problems of the following type: For which sets H is it true that if f is in a given class F of periodic functions and the difference functions ∆hf(x) = f(x + h) − f(x) are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by H(F ,G), that is, H(F ,G) = {H ⊂ R/Z : (∃f ∈ F \ G) (∀h ∈ H) ∆hf ∈ G}, we try to charact...

Integer-valued continuous functions

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Integer-valued definable functions

We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a function f : [0,∞) → R is such that f(N) ⊆ Z, then either sup|x̄|≤r f(x̄) grows faster than exp(r), for some δ > 0,...

On Spaces of Measurable Vector - Valued Functions 399 Proof

Best approximation by integer-valued functions

Given an integer function f , the problem is to find its best uniform approximation from a set K of integervalued bounded functions. Under certain conditions on K , the best extremal (maximal or minimal) approximation is identified. Furthermore, the operator mapping f to its extremal best approximation is shown to be Lipschitzian with some constant C or optimal Lipschitzian having the smallest ...