Search

We deal with the functional equation motivated by quadrature rules of approximate integration. In previous results the solutions of this equation were found only in some particular cases. For example the coefficients λ were supposed to be rational or the equation in question was solved only for n=2. In the current paper we do not assume any particular form of coefficients occurring in this equation and we allow n to be any positive integer. Moreover, we obtain a solution of our equation without any regularity assumptions concerning the functions f and F. [ABSTRACT FROM AUTHOR]

We consider a model of recommendation systems, where each member from a given set of players has a binary preference to each element in a given set of objects: intuitively, each player either likes or dislikes each object. However, the players do not know their preferences. To find his preference of an object, a player may probe it, but each probe incurs unit cost. The goal of the players is to learn their complete preference vector (approximately) while incurring minimal cost. This is possible if many players have similar preference vectors: such a set of players with similar “taste” may split the cost of probing all objects among them, and share the results of their probes by posting them on a public billboard. The problem is that players do not know a priori whose taste is close to theirs. In this paper we present a distributed randomized peer-to-peer algorithm in which each player outputs a vector which is close to the best possible approximation of the player’s real preference vector after a polylogarithmic number of rounds. The algorithm works under adversarial preferences. Previous algorithms either made severely limiting assumptions on the structure of the preference vectors, or had polynomial overhead. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the “approximate”, or “almost” zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the “approximate decoupling polynomials”. The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of “approximate decoupling polynomials”. [Copyright &y& Elsevier]

Abstract: Let be a closed subset of the unit circle and let . We investigate the problem of uniform approximation of on by polynomials which are uniformly bounded on the unit disk . In a particular case when is a closed arc of , the problem was solved by L. Zalcman in 1982, who has also pointed out the possibility of considering more general approximation sets instead of an arc. The present paper gives a necessary and sufficient solution of the above problem. In fact we show that the (simple) description of given by Zalcman for the case of an arc also holds in the general case. [Copyright &y& Elsevier]

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( f)≤ n, n≥1. We ask whether the degree of comonotone approximation is necessarily ≤ c( α, 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 of α's for which the above estimate cannot be achieved. [ABSTRACT FROM AUTHOR]

We obtain asymptotically sharp estimates for the best (α, β) -approximations of the classes $ W_{1;\upgamma, \updelta }^r $ with natural r by algebraic polynomials in the mean. [ABSTRACT FROM AUTHOR]