A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures

A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures

Gaussian quadratures for a broad class of functions. While some of the components of this algorithm have been previously published, we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works. The performance of the resulting procedure is illustrated...

On Generalized Gaussian Quadratures for Bandlimited Exponentials

We review the methods in [4] and [24] for constructing quadratures for bandlimited exponentials and introduce a new algorithm for the same purpose. As in [4], our approach also yields generalized Gaussian quadratures for exponentials integrated against a non-sign-definite weight function. In addition, we compute quadrature weights via l and l∞ minimization and compare the corresponding quadratu...

On Generalized Gaussian Quadratures\ for Exponentials and Their Applications

We introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions. We use a generalization of a representation theorem due to Carathéodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix con...

Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators

Gaussian quadratures for oscillatory integrands

We consider a Gaussian type quadrature rule for some classes of integrands involving highly oscillatory functions of the form f (x) = f 1 (x) sin ζ x + f 2 (x) cos ζ x, where f 1 (x) and f 2 (x) are smooth, ζ ∈ R. We find weights σ ν and nodes x ν , ν = 1, 2,. .. , n, in a quadrature formula of the form 1 −1 f (x) dx ≈ n ν=1 σ ν f (x ν) such that it is exact for all polynomials f 1 (x) and f 2 ...