David W. Kammler

Approximation with sums of exponentials in Lp[0, ∞)

Abstract We consider the problem of approximating a given f from L p [0, ∞) by means of the family V n ( S ) of exponential sums; V n ( S ) denotes the set of all possible solutions of all possible n th order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomials all lie in the set S . We establish the existence of best approximations, show that the distance from a given f to V n ( S ) decreases to zero as n becomes infinite, and characterize such best approximations with a first-order necessary condition. In so doing we extend previously known results that apply in L p [0, 1].

Existence of best approximations by sums of exponentials

Abstract In this paper we shall show that each ƒϵ L p [0,1] (1 ⩽ p ⩽ ∞) has a best L p approximation from the set of exponential sums, V n ( S ), provided S is closed. Here V n ( S ) denotes the set of all solutions of all n -th order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomial all lie in S . We thus extend the previously known existence theorems which apply only in the special cases where S is compact or where S = R .

Characterization of best approximations by sums of exponentials

Abstract An exponential sum y can be specified by giving the coefficients b, c of the corresponding initial value problem (Dn + c1Dn − 1 + ··· + cn)y = 0, Dj − 1y(0) = bj, j = 1, 2,…, n. We discuss some of the topological properties of this parametric form, noting that the associated tangential manifold does not suffer a “loss of dimension” when the exponential parameters are allowed to coalesce. Using this representation, we formulate a first order necessary condition which may often be used to characterize a local best Lp-approximation to a given ƒ ϵ L p [0, 1], 1 ⩽ p ⩽ ∞ , from the set Vn(S). A sufficient condition is also given, and the use and limitations of the theorems are illustrated by means of several carefully chosen examples.

A perturbation analysis of the intrinsic conditioning of an approximate null vector computed with a SVD

Let A be an m ×n real matrix with singular values ?1 ? ··· ? ?n?1 ? ?n ? 0. In cases where ?n ? 0, the corresponding right singular vector ?n is a natural choice to use for an approximate null vector ofA. Using an elementary perturbation analysis, we show that ? = ?1/(?n?1 ? ?n) provides a quantitative measure of the intrinsic conditioning of the computation of ?n from A.

An alternation characterization of best uniform approximations on noncompact intervals

Abstract We present a characterization for a best uniform approximation to a given bounded continuous function f defined on the real but not necessarily compact interval T from an n -dimensional subspace S of the bounded continuous functions on T . When S is a Haar subspace and each element of S satisfies an additional endpoint regularity condition, such a best approximation may be characterized by an appropriate generalization of the familiar alternation criterion which holds for compact T . One such best approximation that has an alternating error curve may be obtained as the uniform limit of a sequence whose v th term is the unique best uniform approximation to f on the v th member of a suitably chosen expanding sequence of compact subintervals of T . The results apply in the special case where T = [0, + ∞) and S is a family of exponential sums with real exponents.

The fast Fourier transform method and ill-conditioned matrices

We study the problem of finding numerical solutions of the linear algebraic equation, a*x=b, where a denotes an NxN ill-conditioned coefficient matrix. It is well-known that Gaussian elimination methods coupled with pivoting strategies are ineffective in this setting due to round-off error. We propose a new and simple application of the fast Fourier transform (FFT) method. Other viable methods, such as the QR method (QRM) or the singular value decomposition method (SVDM), have been proposed in the literature. The goal of this paper is to investigate the performance of the proposed method and compare it to other popular methods. The comparison is illustrated by computer simulation results using MATLAB.

Existence of best approximations by exponential sums in several independent variables

Abstract In this paper we establish the existence of a best Lp approximation, 1 ⩽ p ⩽ ∞, to a given function f∈Lp( D , where D ⊂ Rm is a bounded domain, from the family Vn(S) of all nth order exponential sums in m independent variables for which the corresponding exponential parameters lie in the closed set S ⊆ C. In so doing we extend the previously known existence theorem which corresponds to the special case where m = 1 and D is a finite interval.

Existence of good and best approximations on unbounded domains by exponential sums in several independent variables

Abstract In this paper we consider the problem of using exponential sums to approximate a given complex-valued function f defined on the possibly unbounded domain D in R m. We establish the existence of a best approximation from the set of exponential sums having order at most n and formulate a Weierstrass-type density theorem. In so doing we extend previously known results which apply only in the special cases where D is bounded or where m = 1.