Some Asymptotic Expressions for Prolate Spheroidal Functions and for the Eigenvalues of Differential and Integral Equations of Which They Are Solutions

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Abstract
The asymptotic behavior of the prolate spheroidal functions of order zero fn(x, c), where n is the number of zeros of the function in the interval − 1 ≤ x ≤ 1, is studied for large values of the parameter c and all values of n. The method used involves solving the differential equation which defines the functions by using a classical approximation. The corresponding eigenvalues χn are given by an implicit equation and the norm of the functions is calculated. The functions fn(x, c) are also solutions of an integral equation and associated with eigenvalues λn(c). Asymptotic expressions of [1 − λn(c)] are derived by using the values obtained for the norm of fn(x, c). All these results generalize and interpolate partial results obtained by Slepian and others in two special cases, namely, n finite and n ≃ c.

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