Approximation of an analytic function on a finite real interval by a bandlimited function and conjectures on properties of prolate spheroidal functions

摘要

We show that an arbitrary function f, analytic within a rectangle around the real interval x ∈[―1,1], can be approximated by a bandlimited function of bandwidth c with an accuracy on x ∈ [―1,1] which decreases exponentially fast with the bandwidth c. We explicitly construct the approximation f~(bl)(x; c) and show that its Fourier transform is well-behaved (i.e., belongs to the "Schwartx space" of rapidly decaying C~∞ functions). We also show that an alternative method of constructing a bandlimited approximation using prolate spheroidal functions yields a series for the Fourier transform of f which is usually divergent. We offer numerical evidence for two conjectures about properties of prolate spheroidal functions. First, the eigenvalues λ_n of the prolate integral equation, ∫_(-1)~1, exp(icxy)ψ_n(y; c) dy = λ_nψ_n(x; c), decay "supergeometrically", that is, faster than exp(―qn) for any finite q. Second, the prolate coefficients in the expansions of most functions f(x) are O(1) until n > (2/π)c, after which they fall geometrically.