{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/87328"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/87328","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Positivity properties associated with linear differential operators","abstract":"The determination of the influence exerted on the analytic character of a real function fεC<sup>∞</sup> by the signs of its derivatives is a problem of longstanding interest in classical analysis. Most investigations of the problem have centered on extending the well known theorem of S. Bernstein which asserts that a function fεC<sup>∞</sup> with all derivatives non-negative on an interval I is necessarily real-analytic there; i.e., f is the restriction to I of a complex function analytic in a region containing I. The scope of this dissertation is the study of analogous positivity results associated with linear differential operators of the form (Ly)(t) = a₂(t)y''(t) + a₁(t)y¹(t) + a₀(t)y(t), where a₂(t), a₁(t) and a₀(t) are real-analytic in some interval I and where a₂(t) > 0 for t ε I. We call a function f ε C<sup>∞</sup> L-positive at t₀ ε I if it satisfies the \"uniform\" positivity condition L<sup>k</sup>f(t)≥0, t ε I, k = 0, 1, 2, . . . , plus the \"pointwise\" positivity condition (L<sup>k</sup>f)' (t₀) ≥ 0, k = 0, 1, 2, . . . (L⁰f = f, Lᵏf = L (Lᵏ⁻¹f), k ≥ 1). Our principal is that L-positivity of f implies analyticity of f in a neighborhood of t₀. If Ly = y'', this reduces to Bernstein's theorem. We prove our result using a generalized Taylor Series Expansion known as the L-series. The L-series expansion about t = t₀ for a function fƐC<sup>∞</sup> is: ∞ Σ Lᵏf(t₀)Φ<sub>2k</sub>(t) + √(a₂(t₀))(Lᵏf)' (t₀)Φ<sub>2k+1</sub>(t). k=0 The \"L-basis\" functions {Φ<sub>n</sub>(t)}<sub>n=0</sub><sup>∞</sup> are defined by: LΦ₀ ≡ LΦ₁ ≡ 0, Φ₀(t₀) = 1, Φ₀' (t₀) = 0, Φ₁(t₀) = 0, √(a₂(t₀))Φ₁(t₀) = 1 and LΦ<sub>𝗇+2</sub> = Φ<sub>n</sub>, Φ<sub>n+2</sub>(t₀) = Φ' <sub>n+2</sub>(t₀) = 0, n ≥ 0. Our technique is to show that L-positivity of f implies the convergence of the above series to f(t). Then we observe that the analyticity of a₂, a₁, and a₀ implies the analyticity of the Φ’s and thus the analyticity of the sum, f(t), of the series. We shall also show that the same conditions on a₂, a₁, and a₀ allow any function f, analytic in a neighborhood of t₀, to be represented by an L-series. If a₂(t) ≡ 1, the sequence {n!Φ𝗇(t)}<sub>n=0</sub><sup>∞</sup> provides a heretofore unobserved example of a Pincherle basis. The problem of dispensing with the hypothesis (Lᵏf)' (t₀) ≥ 0 in our result, L-positivity implies analyticity, is still open and does not seem to be solvable by our methods.","abstract_html":"The determination of the influence exerted on the analytic character of a real function fεC&lt;sup&gt;∞&lt;/sup&gt; by the signs of its derivatives is a problem of longstanding interest in classical analysis. Most investigations of the problem have centered on extending the well known theorem of S. Bernstein which asserts that a function fεC&lt;sup&gt;∞&lt;/sup&gt; with all derivatives non-negative on an interval I is necessarily real-analytic there; i.e., f is the restriction to I of a complex function analytic in a region containing I. The scope of this dissertation is the study of analogous positivity results associated with linear differential operators of the form (Ly)(t) = a₂(t)y&#x27;&#x27;(t) + a₁(t)y¹(t) + a₀(t)y(t), where a₂(t), a₁(t) and a₀(t) are real-analytic in some interval I and where a₂(t) &gt; 0 for t ε I. We call a function f ε C&lt;sup&gt;∞&lt;/sup&gt; L-positive at t₀ ε I if it satisfies the &quot;uniform&quot; positivity condition L&lt;sup&gt;k&lt;/sup&gt;f(t)≥0, t ε I, k = 0, 1, 2, . . . , plus the &quot;pointwise&quot; positivity condition (L&lt;sup&gt;k&lt;/sup&gt;f)&#x27; (t₀) ≥ 0, k = 0, 1, 2, . . . (L⁰f = f, Lᵏf = L (Lᵏ⁻¹f), k ≥ 1). Our principal is that L-positivity of f implies analyticity of f in a neighborhood of t₀. If Ly = y&#x27;&#x27;, this reduces to Bernstein&#x27;s theorem. We prove our result using a generalized Taylor Series Expansion known as the L-series. The L-series expansion about t = t₀ for a function fƐC&lt;sup&gt;∞&lt;/sup&gt; is: ∞ Σ Lᵏf(t₀)Φ&lt;sub&gt;2k&lt;/sub&gt;(t) + √(a₂(t₀))(Lᵏf)&#x27; (t₀)Φ&lt;sub&gt;2k+1&lt;/sub&gt;(t). k=0 The &quot;L-basis&quot; functions {Φ&lt;sub&gt;n&lt;/sub&gt;(t)}&lt;sub&gt;n=0&lt;/sub&gt;&lt;sup&gt;∞&lt;/sup&gt; are defined by: LΦ₀ ≡ LΦ₁ ≡ 0, Φ₀(t₀) = 1, Φ₀&#x27; (t₀) = 0, Φ₁(t₀) = 0, √(a₂(t₀))Φ₁(t₀) = 1 and LΦ&lt;sub&gt;𝗇+2&lt;/sub&gt; = Φ&lt;sub&gt;n&lt;/sub&gt;, Φ&lt;sub&gt;n+2&lt;/sub&gt;(t₀) = Φ&#x27; &lt;sub&gt;n+2&lt;/sub&gt;(t₀) = 0, n ≥ 0. Our technique is to show that L-positivity of f implies the convergence of the above series to f(t). Then we observe that the analyticity of a₂, a₁, and a₀ implies the analyticity of the Φ’s and thus the analyticity of the sum, f(t), of the series. We shall also show that the same conditions on a₂, a₁, and a₀ allow any function f, analytic in a neighborhood of t₀, to be represented by an L-series. If a₂(t) ≡ 1, the sequence {n!Φ𝗇(t)}&lt;sub&gt;n=0&lt;/sub&gt;&lt;sup&gt;∞&lt;/sup&gt; provides a heretofore unobserved example of a Pincherle basis. The problem of dispensing with the hypothesis (Lᵏf)&#x27; (t₀) ≥ 0 in our result, L-positivity implies analyticity, is still open and does not seem to be solvable by our methods.","abstract_has_math":false,"creators":["Winfrey, William Randolph"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1976,"date_issued":"1976","date_published":"1976","updated_at":"2026-07-22T22:20:34Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/87328","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":["Winfrey, William Randolph"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-01-31T19:03:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-01-31T19:03:56Z"]},{"key":"dc:date.issued","label":"Date","values":["1976"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"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":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"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/10919/87328"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The determination of the influence exerted on the analytic character of a real function fεC<sup>∞</sup> by the signs of its derivatives is a problem of longstanding interest in classical analysis. Most investigations of the problem have centered on extending the well known theorem of S. Bernstein which asserts that a function fεC<sup>∞</sup> with all derivatives non-negative on an interval I is necessarily real-analytic there; i.e., f is the restriction to I of a complex function analytic in a region containing I. The scope of this dissertation is the study of analogous positivity results associated with linear differential operators of the form (Ly)(t) = a₂(t)y''(t) + a₁(t)y¹(t) + a₀(t)y(t), where a₂(t), a₁(t) and a₀(t) are real-analytic in some interval I and where a₂(t) > 0 for t ε I. We call a function f ε C<sup>∞</sup> L-positive at t₀ ε I if it satisfies the \"uniform\" positivity condition L<sup>k</sup>f(t)≥0, t ε I, k = 0, 1, 2, . . . , plus the \"pointwise\" positivity condition (L<sup>k</sup>f)' (t₀) ≥ 0, k = 0, 1, 2, . . . (L⁰f = f, Lᵏf = L (Lᵏ⁻¹f), k ≥ 1). Our principal is that L-positivity of f implies analyticity of f in a neighborhood of t₀. If Ly = y'', this reduces to Bernstein's theorem. We prove our result using a generalized Taylor Series Expansion known as the L-series. The L-series expansion about t = t₀ for a function fƐC<sup>∞</sup> is: ∞ Σ Lᵏf(t₀)Φ<sub>2k</sub>(t) + √(a₂(t₀))(Lᵏf)' (t₀)Φ<sub>2k+1</sub>(t). k=0 The \"L-basis\" functions {Φ<sub>n</sub>(t)}<sub>n=0</sub><sup>∞</sup> are defined by: LΦ₀ ≡ LΦ₁ ≡ 0, Φ₀(t₀) = 1, Φ₀' (t₀) = 0, Φ₁(t₀) = 0, √(a₂(t₀))Φ₁(t₀) = 1 and LΦ<sub>𝗇+2</sub> = Φ<sub>n</sub>, Φ<sub>n+2</sub>(t₀) = Φ' <sub>n+2</sub>(t₀) = 0, n ≥ 0. Our technique is to show that L-positivity of f implies the convergence of the above series to f(t). Then we observe that the analyticity of a₂, a₁, and a₀ implies the analyticity of the Φ’s and thus the analyticity of the sum, f(t), of the series. We shall also show that the same conditions on a₂, a₁, and a₀ allow any function f, analytic in a neighborhood of t₀, to be represented by an L-series. If a₂(t) ≡ 1, the sequence {n!Φ𝗇(t)}<sub>n=0</sub><sup>∞</sup> provides a heretofore unobserved example of a Pincherle basis. The problem of dispensing with the hypothesis (Lᵏf)' (t₀) ≥ 0 in our result, L-positivity implies analyticity, is still open and does not seem to be solvable by our methods."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Positivity properties associated with linear differential operators"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Winfrey, William Randolph"],"dc:date.accessioned":["2019-01-31T19:03:56Z"],"dc:date.available":["2019-01-31T19:03:56Z"],"dc:date.issued":["1976"],"dc:description.abstract":["The determination of the influence exerted on the analytic character of a real function fεC<sup>∞</sup> by the signs of its derivatives is a problem of longstanding interest in classical analysis. Most investigations of the problem have centered on extending the well known theorem of S. Bernstein which asserts that a function fεC<sup>∞</sup> with all derivatives non-negative on an interval I is necessarily real-analytic there; i.e., f is the restriction to I of a complex function analytic in a region containing I. The scope of this dissertation is the study of analogous positivity results associated with linear differential operators of the form (Ly)(t) = a₂(t)y''(t) + a₁(t)y¹(t) + a₀(t)y(t), where a₂(t), a₁(t) and a₀(t) are real-analytic in some interval I and where a₂(t) > 0 for t ε I. We call a function f ε C<sup>∞</sup> L-positive at t₀ ε I if it satisfies the \"uniform\" positivity condition L<sup>k</sup>f(t)≥0, t ε I, k = 0, 1, 2, . . . , plus the \"pointwise\" positivity condition (L<sup>k</sup>f)' (t₀) ≥ 0, k = 0, 1, 2, . . . (L⁰f = f, Lᵏf = L (Lᵏ⁻¹f), k ≥ 1). Our principal is that L-positivity of f implies analyticity of f in a neighborhood of t₀. If Ly = y'', this reduces to Bernstein's theorem. We prove our result using a generalized Taylor Series Expansion known as the L-series. The L-series expansion about t = t₀ for a function fƐC<sup>∞</sup> is: ∞ Σ Lᵏf(t₀)Φ<sub>2k</sub>(t) + √(a₂(t₀))(Lᵏf)' (t₀)Φ<sub>2k+1</sub>(t). k=0 The \"L-basis\" functions {Φ<sub>n</sub>(t)}<sub>n=0</sub><sup>∞</sup> are defined by: LΦ₀ ≡ LΦ₁ ≡ 0, Φ₀(t₀) = 1, Φ₀' (t₀) = 0, Φ₁(t₀) = 0, √(a₂(t₀))Φ₁(t₀) = 1 and LΦ<sub>𝗇+2</sub> = Φ<sub>n</sub>, Φ<sub>n+2</sub>(t₀) = Φ' <sub>n+2</sub>(t₀) = 0, n ≥ 0. Our technique is to show that L-positivity of f implies the convergence of the above series to f(t). Then we observe that the analyticity of a₂, a₁, and a₀ implies the analyticity of the Φ’s and thus the analyticity of the sum, f(t), of the series. We shall also show that the same conditions on a₂, a₁, and a₀ allow any function f, analytic in a neighborhood of t₀, to be represented by an L-series. If a₂(t) ≡ 1, the sequence {n!Φ𝗇(t)}<sub>n=0</sub><sup>∞</sup> provides a heretofore unobserved example of a Pincherle basis. The problem of dispensing with the hypothesis (Lᵏf)' (t₀) ≥ 0 in our result, L-positivity implies analyticity, is still open and does not seem to be solvable by our methods."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/87328"],"dc:language.iso":["en"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Positivity properties associated with linear differential operators"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:34Z"}