FREQUENCY-DOMAIN BOUNDS FOR NONNEGATIVE BAND-LIMITED FUNCTIONS

In this paper we present pointwise and integral frequency-domain bounds for non-negative, band-limited functions of one continuous or discrete variable, and for radially symmetric functions of two continuous variables. We use the Akhiezer-Krein theorem to prove pointwise and integral bounds for the case of functions of one continuous variable, and to sharpen these bounds for the case of radially symmetric functions of two continuous variables. We use the Riesz-Fejer theorem to prove a particular type of pointwise bound for the case of functions of one continuous variable. We use the Bochner theorem to prove pointwise bounds for the case of functions of one discrete variable. The bounds we present are of particular interest to Fourier optics (Lukosz-type bounds). Many of the results amount to finding the largest eigenvalue of certain truncated convolution operators and Toeplitz matrices. Our explicit results on pointwise bounds are also of interest for the characterization of the feasibility region of the partial autocorrelation problem of finite-length, discrete-time signals considered recently by Steinhardt and Makhoul, and by Delsarte, Genin and Kamp.