Real orthogonal polynomials in frequency analysis

Real orthogonal polynomials in frequency analysis

We study the use of para-orthogonal polynomials in solving the frequency analysis problem. Through a transformation of Delsarte and Genin, we present an approach for the frequency analysis by using the zeros and Christoffel numbers of polynomials orthogonal on the real line. This leads to a simple and fast algorithm for the estimation of frequencies. We also provide a new method, faster than th...

Para-orthogonal polynomials in frequency analysis

and we assume α−j = αj, and ω−j = −ωj ∈ (0, π) for j = 1, 2, . . . , I. The constants αj represent amplitudes, the quantities ωj are frequencies, and m is discrete time. The frequency analysis problem is to determine the numbers {αj, ωj : j = 1, 2, . . . , I}, and n0 = 2I when values {x(m) : m = 0, 1, . . . , N − 1} (observations) are known. The Wiener-Levinson method, formulated in terms of Sz...

Complexity analysis of hypergeometric orthogonal polynomials

The complexity measures of the Crámer-Rao, Fisher-Shannon and LMC (López-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density ρn(x) = ω(x)p 2 n(x) of the polynomials pn(x) orthogonal with respect to the weight function ω(x), x ∈ (a, b), are used to quantify various two-fold facets of the spreading of the Hermite, Laguerre and Jacobi systems all over their corresponding orthogona...

Orthogonal Polynomials

The theory of orthogonal polynomials plays an important role in many branches of mathematics, such as approximation theory (best approximation, interpolation, quadrature), special functions, continued fractions, differential and integral equations. The notion of orthogonality originated from the theory of continued fractions, but later became an independent (and possibly more important) discipl...

Orthogonal Polynomials