A generalized moment problem for self-adjoint operators

Let the sequence {λ _i } (i≧0) satisfy condition (1.1) and let {A _n} (n≧0) be a sequence of bounded self-adjoint operators over a complex Hilbert spaceH. We give a necessary and sufficient condition in order that {A _n} (n≧0) should possess the representation (1.2).     

Focusing and beam shifting properties of Fabry-Perot type interferometers

The basic formulations of a plane-wave interaction with planar, double and triple-screened interferometers are reviewed and extended to encompass slowly varying, non-planar structures. The feasibility of attaining desired phase control (focusing or beam shifting) while simultaneously preserving the local resonance conditions is demonstrated analytically and simulated numerically.     

On the Stability and Accuracy of Least Squares Approximations

We consider the problem of reconstructing an unknown function f on a domain X from samples of f at n randomly chosen points with respect to a given measure ρ _X . Given a sequence of linear spaces (V _m ) _m>0 with dim(V _m )=m≤n, we study the least squares approximations from the spaces V _m . It is well known that such approximations can be inaccurate when m is too close to n, even when the samples are noiseless. Our main result provides a criterion on m that describes the needed amount of regularization to ensure that the least squares method is stable and that its accuracy, measured in L ²(X,ρ _X ), is comparable to the best approximation error of f by elements from V _m . We illustrate this criterion for various approximation schemes, such as trigonometric polynomials, with ρ _X being the uniform measure, and algebraic polynomials, with ρ _X being either the uniform or Chebyshev measure. For such examples we also prove similar stability results using deterministic samples that are equispaced with respect to these measures.     

Shape-Preserving Widths of Weighted Sobolev-Type Classes of Positive, Monotone, and Convex Functions on a Finite Interval

   Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ ^s ₊ L _q the subset of all functions y∈ L _q such that the s -difference Δ ^s _τ y(t) is nonnegative on I ,

Positive Results and Counterexamples in Comonotone Approximation

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [?1,1], that change monotonicity s??1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)??n ^??? , n??1. We ask whether the degree of comonotone approximation is necessarily ??c(??,s)n ^??? , n??1, and if not, what can be said. It turns out that for each s??1, there is an exceptional set A _s of ????s for which the above estimate cannot be achieved.     

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, , and 1p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x^(r)_Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in L_q for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n^-2.     

Are the degrees of best (co)convex and unconstrained polynomial approximation the same?

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ² the set of convex functions f ∈ ℂ[−,1]. Also, let E _n (f) and E _n ⁽²⁾ (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ² by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E _n ⁽²⁾ (f), and Lorentz and Zeller proved that the inverse inequality E _n ⁽²⁾ (f) ≦ cE _n (f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ². In this paper we prove, for every α > 0 and function f ∈ Δ², that

ChemInform Abstract: Structural and Electrochemical Study od a New Crystalline Hydrate Iron(III) Phosphate FePO4·H2O Obtained from LiFePO4(OH) by Ion Exchange.

FePO₄·H₂O is prepared by Li⁺/H⁺ exchange from an aqueous HNO₃ suspension of LiFePO₄(OH) at about 60 °C.