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111.
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).  相似文献   
112.
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.  相似文献   
113.
On the Stability and Accuracy of Least Squares Approximations   总被引:1,自引:0,他引:1  
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 )=mn, 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.  相似文献   
114.
   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 ,
τ>0 . Further, denote by
, 0≤α<∞ , the classes of functions x on I with the seminorm ||x (r) ρ α ||_ L p ≤ 1 , such that Δ s τ x≥ 0 , τ>0 . For s=0,1,2 , we obtain two-sided estimates of the shape-preserving widths
where M n is the set of all linear manifolds M n in L q , such that dim M n ≤ n , and satisfying
.  相似文献   
115.
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.  相似文献   
116.
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 Lq 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.  相似文献   
117.
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
where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases s≦ 1 and s ≧ 2. Dedicated to Jóska Szabados on his 70th birthday  相似文献   
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