An exponential interpolation series

Description

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. / M.S.

Links & Downloads

1. http://hdl.handle.net/10919/91112

Tags

LD5655.V855 1968.H66

Interpolation

Exponential functions

Additional Fields

Identifer	oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/91112
Date	January 1968
Creators	Howell, William Edward
Contributors	Mathematics
Publisher	Virginia Polytechnic Institute
Source Sets	Virginia Tech Theses and Dissertation
Language	en_US
Detected Language	English
Type	Thesis, Text
Format	iii, 30, [2] leave, application/pdf, application/pdf
Rights	In Copyright, http://rightsstatements.org/vocab/InC/1.0/
Relation	OCLC# 20183875