Approximation in the mean and quasianalyticity

Let E be a compact subset of the complex plane with positive dxdy measure. For each p, 1 ? p < ∞, let R^p(Edxdy) denote the closure in L^p(E, dxdy) of the rational functions having no poles on E. We give an example where E has empty interior and the functions in R^p(E, dxdy), p ? 2, are uniquely determined by their values on any set of positive dxdy measure in E. Earlier, Sinanjan (Sibirsk Mat. ?. 6 (1965), 1365–1381) obtained a somewhat weaker result along these lines and he also proved that this phenomenon can never occur when 1 ? p < 2.     

Approximation in the mean by convex functions

We give elementary proofs for the existence and uniqueness of the best L₁-approximation to a continuous function from the class of convex functions on a closed interval, and describe thebest approximation in terms of certain piecewise linear functions.     

Approximation in the mean by rational functions

In this paper we proved that for a large class of compact subsetsK in the complex plane,R(K) is dense inL ^q () if and only if the set of analytic bounded point evaluations forR ^q (K, ) is empty. As a consequence, we showed that this result is true for allK ifR(K) is replaced byA(K). Our main result includes the corresponding result of James Thomson for polynomials approximation as such a special case thatK is a disk.     

Approximation of weighted local mean operators

Recently, weighted local mean operators are widely used in image processing, compressive sensing and other areas. A weighted local mean operator changes its characteristics depending on a function content within a local area in order to preserve the function features. The directional diffusion filter and Yaroslavsky neighbourhood filter (also called the sigma filter) are discrete versions of such operators. Although these operators are not convolution ones, due to their sparsity, the corresponding numerical algorithms have simple structure and fast performance. In this paper, we study the approximate properties of the weighted local mean operators, particularly focus on their asymptotic expansions, which are related to non-linear diffusion equations.     

Approximation in the mean by bounded analytic functions

Let 1<p< and . LetC _q denote the Bessel capacity in the plane. Let be the set of homomorphisms ofH (G) such that (z)= and letNP denote the set of points in G for which is not a peak set forH (G). In this note, we show that ifC _q (NP)=0, thenH (G) is dense inL _a ^p (G), the Bergman space overG.Partially supported by NSF DMS-9401234     

On a generalized asymptotic mean value property

In this paper we study the spaces of continuous functionsf on ² satisfying

Approximation to the mean curve in the LCS problem

The problem of sequence comparison via optimal alignments occurs naturally in many areas of applications. The simplest such technique is based on evaluating a score given by the length of a longest common subsequence divided by the average length of the original sequences. In this paper we investigate the expected value of this score when the input sequences are random and their length tends to infinity. The corresponding limit exists but is not known precisely. We derive a theoretical large deviation, convex analysis and Monte Carlo based method to compute a consistent sequence of upper bounds on the unknown limit. An empirical practical version of our method produces promising numerical results.     

Approximation of periodic functions by trigonometric polynomials in the mean

For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 15–26, July, 1974     

Approximation by entire functions in the mean on the real line

Approximations in a locally integral norm of functions representable as a convolution on the whole line by means of Fourier operators are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 121–125, January, 1991.