Eigenvalues, Almost Periodic Functions, and the Derivative of an Integral

Abstract

NOTES Edited by William Adkins Eigenvalues, Almost Periodic Functions, and the Derivative of an Integral Mark Finkelstein and Robert Whitley In [4], examples are given of functions g(x), bounded and continuous on (0, l], for which the indefinite integral g(t) dt is not differentiable from the right at zero. The purpose of this note is to show that such examples arise in a natural way and that the class of such examples is quite large. First, to see how such examples can come up, consider the operator l; Hf(x) = - I x 1x f (t) dt. A result due to Hardy [S, Equation 9.9.1 ], (7, Exercise 8.14] asserts that His a bounded linear operator on the Banach space LP[O, 1] for p > I with norm less than or equal to p/(p - 1). Actually Hardy gives the result for LP[O, oo), but it is somewhat simpler to work with LP[O, I]. Considering the function f(x) = x 0 (a real), which is in U[O, 1] when a > - 1/ p, computing Hf (x) , and letting a -4 - 1/ p shows that Hardy's bound for the norm of H is sharp and also shows that H is unbounded on L 1 (0, 1 ]. The case p = oo is a limiting case in which the norm of H is one. In working with the function f(x) = x 0 of the preceding paragraph, it will not have escaped the reader's notice that this function is an eigenfunction for H with eigenvalue 1/(1 +a). A basic operator theory question is: What are the eigenvalues of H? If A. is a nonzero eigenvalue with associated eigenvector f, then 11x f(t) dt x = A.f(x) , where equality holds almost everywhere. But equation (1) shows that f is absolutely continuous on (0, I], and ( 1) can be taken to hold not just almost everywhere but for all x in (0, l]. That is, a continuous f satisfying (l) for all x in (0, 1] can be chosen from the equivalence class of functions in LP[O, 1] that satisfy (1) almost everywhere. It is legitimate to differentiate equation (1) and, by rearranging, to get A.xj'(x) +(A. - l)f(x) = 0, which is an Euler differential equation [2, chap. 4, sec. 2] on (0, 1]. A priori (2) also holds only almost everywhere, but on solving for f' we see, as earlier, that a continuous representative from the equivalence class of f' can be chosen so that (2) holds for all x in (0, l]. It bas solutions of the form f(x) = xf3, where f3 =a+ ib, with f in LP [0, L] if a > -1 / p > - 1. The associated eigenvalue is A. = 1 / (1 + {3). It is immediate from (1) that the eigenfunction f is continuous at 0 (from the right, since f has domain [0, 1]) if and only if F (x) = f (t) dt is differentiable (from the right) at zero, and this occurs exactly when a > 0. l; August-September 2005] NOTES