An exponential interpolation series

Abstract

The convergence properties of the permanent exponential interpolation series

f(Z) = 1^Zf(0) + (2^Z - 1^Z)Δf(0) + (3^Z - 2.2^Z + 1^Z/2!)Δ(Δ - 1)f(0) + …

have been investigated.

Using the following notation

U_n(Z) = ∑ⁿ_k=0 (-1)^k(ⁿ_k)(n - i + 1)^Z,

Δ⁽ⁿ⁾ f(0) = Δ(Δ-1)…(Δ - n + 1)f(0),

the series can be written more compactly as

f(Z) = ∑^∞₀ U_n(Z)/n!Δ⁽ⁿ⁾ f(0).

It is shown that Δ⁽ⁿ⁾ f(0) can be represented as

Δ⁽ⁿ⁾ f(0) = M_n(f) = 1/2πi ∫_Γ (e^ω - 1)⁽ⁿ⁾ F(ω)dω,

where F(ω) is the Borel transform of f(Z) and Γ encloses the convex hull of the singularities of F(ω). It is further shown that the series

∑^∞₀ U_n(Z)/n! (e^ω - 1)⁽ⁿ⁾

forms a uniformly convergent Gregory-Newton series, convergent to e^Zω in any bounded region in the strip |I(ω)| < π/2. The Polya representation of an entire function of exponential type is then formed, and the method of kernel expansion (R. P. Boas, and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, 1964) yields the desired result. This result is summed up in the following:

Theorem

Any entire function of exponential type such that the convex hull of the set of singularities of its Borel transform lies in the strip |I(ω)| < π/2. admits the convergent exponential interpolation series expansion

f(Z) = ∑^∞_n=0 U_n(Z)/n!Δ⁽ⁿ⁾ f(0) for all Z.