Positivity properties associated with linear differential operators
| dc.contributor.author | Winfrey, William Randolph | en |
| dc.contributor.department | Mathematics | en |
| dc.contributor.department | Mathematics | en |
| dc.date.accessioned | 2019-01-31T19:03:56Z | en |
| dc.date.available | 2019-01-31T19:03:56Z | en |
| dc.date.issued | 1976 | en |
| 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. | en |
| dc.description.degree | Ph. D. | en |
| dc.format.extent | iii, 68 pages, 2 unnumbered leave | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.uri | http://hdl.handle.net/10919/87328 | en |
| dc.language.iso | en | en |
| dc.publisher | Virginia Polytechnic Institute and State University | en |
| dc.relation.isformatof | OCLC# 40244507 | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject.lcc | LD5655.V856 1976.W56 | en |
| dc.title | Positivity properties associated with linear differential operators | en |
| dc.type | Dissertation | en |
| dc.type.dcmitype | Text | en |
| thesis.degree.discipline | Mathematics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | doctoral | en |
| thesis.degree.name | Ph. D. | en |
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