Strong unicity constants for spline functions

In the last years there has been a great deal of interest in strong unicity constants for the space of polynomials of degree at most n. We consider analogous questions for the space of spline functions of degree at most n with k fixed knots. In this case we give formulas and bounds for strong unicity constants and investigate strong unicity for generalized convex functions of order n, i.e. functions whose (n+1)–st derivative is positive.     

Strong unicity in normed linear spaces

We consider best approximations in a real or complex normed linear space by elements of a finite dimensional subspace. It is the purpose of this paper to characterize, when a best approximation to a given element is strongly unique.     

Strong unicity and alternation for linear optimization

We consider linear semi-infinite optimization problems and prove characterizations for strong unicity. One of these is a weak alternation property. This result suggests the introduction of the property of regular strong unicity, which is equivalent to a stronger alternation property. The theorems are used in order to prove a Haar-type theorem for linear optimization problems. An application to best Chebyshev approximation is given.Communicated by A. V. Fiacco     

On the unicity of solutions of problems of best approximation

Summary J. C. Mairhuber has given a topological characterization of those compact spaces in which Haar's problem of best approximation may have a unique solution. Under certain restrictive conditions a similar characterization is given here for the case of the complex field investigated by Kolmogroff. To Enrico Bompiani on his scientific Jubilee. The paper was prepared while the second named author received support from the Office of Ordnance Research.     

Location of best simultaneous approximations

Amir and Ziegler have proved that a real normed space E of dimension ≥ 3 is an inner product space if and only if, for any x,y ε E and any two dimensional subspace M of E, the expression max(‖x — w‖,‖y — w‖) attains its minimum in some point w of each segment [u,v], with u and v, respectively, best approximations to x and y from M. We extend this result to expressions π(‖x — w‖,‖y — w‖), where π denotes a monotonic norm in R²     

Strong approximations of the Q-Q process

We obtain strong approximation results for the empirical Q-Q process. Moreover, some Glivenko-Cantelli-type results are also obtained. These results are applicable to construct confidence bands for Q-Q plots.     

Strong approximations of three-dimensional Wiener sausages

We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L ²-norm estimates between the Wiener sausage and the Brownian intersection local times. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886, T 043037 and K 61052.     

Functions with prescribed best linear approximations

A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the Inverse Best Approximation Property (IBAP) if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to investigate problems in harmonic analysis, integral equations, signal theory, and wavelet frames.