Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

For xed c, Prolate Spheroidal Wave Functions (PSWFs), denoted by ψn,c, form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith c. They have been largely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the ψn,c in n and c, are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at 1 in terms of the Legendre complete elliptic integral of the rst kind. Also, we give an explicit formula for the accurate approximation the eigenvalues of the Sturm-Liouville operator associated with the PSWFs. 2010 Mathematics Subject Classi cation. Primary 42C10, 65L70. Secondary 41A60, 65L15.