We propose a new notion of variable bandwidth and develop the basic theory. Starting with a strictly positive function p on R (a bandwidth parametrization), a function is of variable bandwidth Ω, if it is contained in the spectral subspace of the elliptic operator A_p f = - d/dx(p(x)d/dx)f with spectrum [0, Ω]. We derive (i) (nonuniform) sampling theorems and corresponding algorithms, and (ii) necessary density conditions in the style of Landau. The main results say that, in a neighborhood of x ∈ R, a function of variable bandwidth behaves like a bandlimited function with bandwidth (Ω/p(x))~(1/2).
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