{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/91112"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/91112","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"An exponential interpolation series","abstract":"The convergence properties of the permanent exponential interpolation series f(Z) = 1Zf(0) + (2Z - 1Z)Δf(0) + (3Z - 2.2Z + 1Z/2!)Δ(Δ - 1)f(0) + … have been investigated. Using the following notation Un(Z) = ∑nk=0 (-1)k(nk)(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 Un(Z)/n!Δ(n) f(0). It is shown that Δ(n) f(0) can be represented as Δ(n) f(0) = Mn(f) = 1/2πi ∫Γ (eω - 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 Un(Z)/n! (eω - 1)(n) forms a uniformly convergent Gregory-Newton series, convergent to eZω 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 Un(Z)/n!Δ(n) f(0) for all Z.","abstract_html":"The convergence properties of the permanent exponential interpolation series f(Z) = 1<sup>Z</sup>f(0) + (2<sup>Z</sup> - 1<sup>Z</sup>)Δf(0) + (3<sup>Z</sup> - 2.2<sup>Z</sup> + 1<sup>Z</sup>/2!)Δ(Δ - 1)f(0) + … have been investigated. Using the following notation U<sub>n</sub>(Z) = ∑<sup>n</sup><sub>k=0</sub> (-1)<sup>k</sup>(<sup>n</sup><sub>k</sub>)(n - i + 1)<sup>Z</sup>, Δ<sup>(n)</sup> f(0) = Δ(Δ-1)…(Δ - n + 1)f(0), the series can be written more compactly as f(Z) = ∑<sup>∞</sup><sub>0</sub> U<sub>n</sub>(Z)/n!Δ<sup>(n)</sup> f(0). It is shown that Δ<sup>(n)</sup> f(0) can be represented as Δ<sup>(n)</sup> f(0) = M<sub>n</sub>(f) = 1/2πi ∫<sub>Γ</sub> (e<sup>ω</sup> - 1)<sup>(n)</sup> 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 ∑<sup>∞</sup><sub>0</sub> U<sub>n</sub>(Z)/n! (e<sup>ω</sup> - 1)<sup>(n)</sup> forms a uniformly convergent Gregory-Newton series, convergent to e<sup>Zω</sup> 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) = ∑<sup>∞</sup><sub>n=0</sub> U<sub>n</sub>(Z)/n!Δ<sup>(n)</sup> f(0) for all Z.","abstract_has_math":false,"creators":["Howell, William Edward"],"institution":"Virginia Polytechnic Institute","degree_name":"M.S.","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1968,"date_issued":"1968","date_published":"1968","updated_at":"2026-07-22T22:19:13Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/91112","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Howell, William Edward"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-07-03T20:33:48Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-07-03T20:33:48Z"]},{"key":"dc:date.issued","label":"Date","values":["1968"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/91112"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The convergence properties of the permanent exponential interpolation series f(Z) = 1Zf(0) + (2Z - 1Z)Δf(0) + (3Z - 2.2Z + 1Z/2!)Δ(Δ - 1)f(0) + … have been investigated. Using the following notation Un(Z) = ∑nk=0 (-1)k(nk)(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 Un(Z)/n!Δ(n) f(0). It is shown that Δ(n) f(0) can be represented as Δ(n) f(0) = Mn(f) = 1/2πi ∫Γ (eω - 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 Un(Z)/n! (eω - 1)(n) forms a uniformly convergent Gregory-Newton series, convergent to eZω 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 Un(Z)/n!Δ(n) f(0) for all Z."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["M.S."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["An exponential interpolation series"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Howell, William Edward"],"dc:date.accessioned":["2019-07-03T20:33:48Z"],"dc:date.available":["2019-07-03T20:33:48Z"],"dc:date.issued":["1968"],"dc:description.abstract":["The convergence properties of the permanent exponential interpolation series f(Z) = 1Zf(0) + (2Z - 1Z)Δf(0) + (3Z - 2.2Z + 1Z/2!)Δ(Δ - 1)f(0) + … have been investigated. Using the following notation Un(Z) = ∑nk=0 (-1)k(nk)(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 Un(Z)/n!Δ(n) f(0). It is shown that Δ(n) f(0) can be represented as Δ(n) f(0) = Mn(f) = 1/2πi ∫Γ (eω - 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 Un(Z)/n! (eω - 1)(n) forms a uniformly convergent Gregory-Newton series, convergent to eZω 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 Un(Z)/n!Δ(n) f(0) for all Z."],"dc:description.degree":["M.S."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/91112"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["An exponential interpolation series"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["M.S."],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:19:13Z"}