We are interested in the approximation of functions f ~ Lp(I), 0 < p < 1, I = [ 1 , 1], by algebraic polynomials. Such approximation has previously been studied by other authors, most notably, Storozhenko, Krotov, and Oswald [-S-K-O] and Khodak [K]. Our main departure from these previous works is that we shall prove direct estimates for the error of polynomial approximation in terms of the Ditz...

We consider the following problem: Given a graph with edge lengths satisfying the triangle inequality, partition its node set into p subsets, minimizing the total length of edges whose two ends are in the same subset. For this problem we present an approximation algorithm which comes to at most twice the optimal value. For clustering into two equal-sized sets, the exact bound on the maximum pos...

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p € (1, oo), /, and g , find u such that -V • (\Vu\"-2Vu) = f iniîcR2, u = g on a«. The finite element approximation is defined over Í2* , a union of regular triangles, yielding a polygonal approximation to Q. For sufficiently regular solutions u , achievable for a subclass of ...

Abstract. Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which give partial answers to this problem, but there are still questions which remain unknown. The Duffin-Schaeffer Conjecture is an attempt to answer all of these questions in full, and it has withstood more than...

Approximation lattices occur in a natural way in the study of rational approximations to p-adic numbers. Periodicity of a sequence of approximation lattices is shown to occur for rational and quadratic p-adic numbers. and for those only, thus establishing a p-adic analogue of Lagrange’s theorem on periodic continued fractions. Using approximation lattices we derive upper and lower bounds for th...

We present a reduction that allows us to establish completeness results for several approximation classes mainly beyond APX. Using it, we extend one of the basic results of S. Khanna, R. Motwani, M. Sudan, and U. Vazirani (On syntactic versus computational views of approximability, SIAM J. Comput., 28:164–191, 1998) by proving the existence of complete problems for the whole Log-APX, the class ...

In the present article, we introduce Chlodowsky variant of $(p,q)$-Bernstein operators and compute the moments for these operators which are used in proving our main results. Further, we study some approximation properties of these new operators, which include the rate of convergence using usual modulus of continuity and also the rate of convergence when the function $f$ belongs to the class Li...