Lower bounds for the approximation with variation-diminishing splines

Lower bounds for the approximation with variation-diminishing splines

We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval [0, 1] in terms of classical moduli of smoothness depending on the degree of the spline basis. For this purpose we use a functional analysis framework in order to characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates.

Best Approximation and Cyclic Variation Diminishing Kernels

where the kernel K(x, y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from ...

Improved Lower Bounds on the Total Variation Distance for the Poisson Approximation

New lower bounds on the total variation distance between the distribution of a sum of independent Bernoulli random variables and the Poisson random variable (with the same mean) are derived via the Chen-Stein method. The new bounds rely on a non-trivial modification of the analysis by Barbour and Hall (1984) which surprisingly gives a significant improvement. A use of the new lower bounds is ad...

Lower Bounds for Approximation by Nonlinear Manifolds

which is the precision to which F approximates K. An instance of classical interest is that in which L is C([0, 1]), K is A"—i.e. those functions bounded by one, and whose modulus of continuity is dominated by the modulus of continuity function o>—and F is the family Pn of polynomials of degree n 1. For this case it is known that there are positive constants A and B, independent of to and n, su...

Lower Bounds and Approximation Guarantees