Darrell Schmidt

EIGENVALUES OF TRIDIAGONAL PSEUDO-TOEPLITZ MATRICES

Abstract In this article we determine the eigenvalues of sequences of tridiagonal matrices that contain a Toeplitz matrix in the upper left block.

On Poreda's problem on the strong unicity constants

Let c‘(I) denote the space of continuous. real~valued functions on the inter\,al I ICI. 6) endowed with the uniform norm /I 11. Let U,, denote the set of all algebraic polynomials of degree II or less. and let II denote the set of all algebraic polynomials. For a given SF C(I). with best uniform approximation r,,(.f) from n,,. Newman and Shapiro 1 IO] showed that there is a constant ;’ > 0 such that

Uniform strong unicity for rational approximation

Abstract Let R n m denote the class of rational functions defined on a closed interval I with numerators in the class of polynomials of degree at most n and positive valued denominators in the class of polynomials of degree at most m . If f ϵ C ( I ) is normal, the well-known strong unicity theorem asserts that there is a smallest positive constant γ n , m ( f ) such that ∥ f − R ∥ ⩾ ∥ f − R f ∥ + γ n , m ( f )∥ R − R f ∥ for all R ϵ R n m , where R f is the best uniform approximation to f from R n m . In this paper, the dependence of γ n , m ( f ) on f is investigated. Sufficient conditions are given to insure that inf fϵΓ γ n , m ( f ) > 0, where Γ is a subset of C( I ). Necessity of these conditions is investigated and examples are given to show that known results for R n 0 do not directly extend to R n m for m > 0.

On stability of systems of delay differential equations

Abstract In this paper we give necessary and sufficient conditions for the asymptotic stability of the zero solution of the system of linear delay differential equations of the form x′(t)=αAx(t)+(1−α)Ax(t−τ), where A is an n×n matrix, τ>0 is constant, and 0⩽α⩽1. We reduce this to systems of first- and second-order problems. Our stability results are given in terms of the eigenvalues of A. The proof of our results are carried out by an application of Pontryagins criterion for quasi-polynomials to the characteristic functions of subsystems of the delay differential equations. We also provide four algorithmic stability tests and include several examples.

Lipschitz conditions, strong uniqueness, and almost Chebyshev subspaces of C(X)

Abstract For a finite dimensional subspace M of C ( X ), X a compact metric space, it is well known that the (set valued) metric projection P M is (Hausdorff) continuous at any f ϵ C ( X ) having a unique best approximation from M and is point Lipschitz continuous at any f ϵ C ( X ) having a strongly unique best approximation from M . The converses of these classical results are studied. It is shown that if f has a unique best approximation and P M is point Lipschitzian at f , then f has a strongly unique best approximation. If M is an almost Chebyshev subspace of C ( X ), then the converses of both statements above are shown to hold. Using a theorem of Garkavi, the validity of these converses actually characterizes the almost Chebyshev subspaces of C(X) .

Stability criteria for certain delay integral equations of Volterra type

Abstract In this paper we study the asymptotic stability of the solution of the following delay integral equation of Volterra type: α ∫ o x (a 0 + a 1 (x − s))y(s)ds + (1 − alpha; ∫ o x (a 0 + a 1 (x − s))y(s − τ) ds , y ( x ) = ψ ( x ), − τ ⩽ x τ > 0 is constant and 0 ⩽ α ⩽ 1. Stability criteria are provided for certain αs and the parameters a 0 , a 1 and τ. The aim of this study is to understand the effect of the delay on the asymptotic stability of the solution of Volterra integral equations. As such the parameters α and 1 − α appear with the same kernel in both integrals of the equation. We also provide four algorithmic stability tests and include several examples and stability regions for certain values of the parameters α, a 0 , a 1 and τ.

On A -spaces and their relation to the Hobby-Rice theorem

Abstract If U is an n -dimensional subspace of C [ a , b ] and w is a positive, continuous function on [ a , b ], the Hobby-Rice theorem asserts that there exist points a = x 0 x 1 x m x m + 1 = b such that m ⩽ n and ∑ j = 1 m + 1 (−1) j ∫ x j − 1 x j uw dμ = 0 for all u ϵ U . It is shown that for the canonical set { x j } j = 1 m to contain n points and be unique for all admissible w , it is necessary and sufficient that U satisfy the WT -property and the splitting property (i.e., if u ϵ U and u ≡ 0 on [ c , d ] where a c d b , then uχ [ a , d ] ϵ U ). A new proof is given for the previously known result that for the canonical set to contain n points, be unique, and have full rank relative to U for all admissible w , it is necessary and sufficient that U be an A -space. For a WT -space U with the splitting property, the canonical sets for WT -extensions of U are shown to interlace with the canonical set for U , and a formula for the rank of canonical sets for U is given. In addition, it is shown that every A -space on an interval has an A -space extension.

Some properties of A-spaces and their relationship toL1-approximation

Two new characterizations of A-spaces on an interval are obtained establishing a connection between the A-property and the Hobby-Rice theorem. A complete characterization of tensor product A-spaces on a rectangle is also given.

Error bounds for polynomial product approximation

Abstract This paper establishes bounds on the uniform error in the approximation of a continuous function defined on a rectangle by polynomial product approximations. The dependence of product approximations on the basis functions used for the associated polynomial spaces is investigated.

Almost Chebyshev properties for L 1 -approximation of continuous functions

have unique best 11. I/ ,-approximations from U. We say that U is C~~~~~~~~ in C,(K) if 4&,(U) = C(K).