Positivity properties associated with linear differential operators

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 L^kf(t)≥0, t ε I, k = 0, 1, 2, . . . , plus the "pointwise" positivity condition (L^kf)' (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)Φ_2k(t) + √(a₂(t₀))(Lᵏf)' (t₀)Φ_2k+1(t). k=0

The "L-basis" functions {Φ_n(t)}_n=0^∞ are defined by:

LΦ₀ ≡ LΦ₁ ≡ 0, Φ₀(t₀) = 1, Φ₀' (t₀) = 0, Φ₁(t₀) = 0,

√(a₂(t₀))Φ₁(t₀) = 1

and LΦ_𝗇+2 = Φ_n, Φ_n+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.