Positivity properties associated with linear differential operators
| Autor: | Winfrey, William Randolph |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 1976 |
| Předmět: | |
| Druh dokumentu: | Dissertation |
| Popis: | The determination of the influence exerted on the analytic character of a real function fεC∞ 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∞ 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∞ L-positive at t₀ ε I if it satisfies the "uniform" positivity condition Lkf(t)≥0, t ε I, k = 0, 1, 2, ..., plus the "pointwise" positivity condition (Lkf)' (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∞ is: ∞ Σ Lᵏf(t₀)Φ2k(t) + √(a₂(t₀))(Lᵏf)' (t₀)Φ2k+1(t). k=0 The "L-basis" functions {Φn(t)}n=0∞ are defined by: LΦ₀ ≡ LΦ₁ ≡ 0, Φ₀(t₀) = 1, Φ₀' (t₀) = 0, Φ₁(t₀) = 0, √(a₂(t₀))Φ₁(t₀) = 1 and LΦ𝗇+2 = Φn, Φn+2(t₀) = Φ' n+2(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)}n=0∞ 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. Ph. D. |
| Databáze: | Networked Digital Library of Theses & Dissertations |
| Externí odkaz: |
|