Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences

The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal in ℓ2(Z) which strictly bandlimited to frequency band [−W,W] and maximally concentrated time interval {0,…,N−1}. timelimited DPSSs (sometimes referred as the Slepian basis) an vectors CN whose Fourier transform (DTFT) is [−W,W]. Due these properties, have wide variety signal processing applications. eigensequences timelimit-then-bandlimit operator basis eigenvectors so-called matrix. eigenvalues both cases same, they exhibit particular clustering behavior – slightly fewer than 2NW very close 1, N−2NW 0, few not near 1 or 0. This eigenvalue critical many applications used. There asymptotic characterizations number 0 1. In contrast, there non-asymptotic results, don't fully characterize DPSS eigenvalues. this work, we establish two novel bounds on between ϵ 1−ϵ. Also, obtain detailing how first ≈2NW last ≈N−2NW Furthermore, extend results wave functions (PSWFs), continuous-time version DPSSs. Finally, present numerical experiments demonstrating quality our