Uniform bounds of prolate spheroidal wave functions and eigenvalues decay

Abstract The prolate spheroidal wave functions (PSWFs) form a set of special functions with remarkable properties. They are defined on [ − 1 , 1 ] as the bounded eigenfunctions ψ n , c of a Sturm–Liouville differential operator L c as well as the eigenfunctions of the linear integral operator Q c with kernel sin ( c ( x − y ) ) π ( x − y ) . We give new bounds for the values ψ n , c ( 0 ) , ψ n , c ′ ( 0 ) and ψ n , c ( 1 ) , which allow us to obtain estimates for the L p norms of the PSWFs and for eigenvalues of L c and Q c . We get in particular an almost sharp exponential lower decay rate of the eigenvalues of Q c .