In practical engineering, small-scale data sets are usually sparse and contaminated by noise. In this paper, we propose a new sequence of orthogonal polynomials varying with their coefficient, unified Chebyshev polynomials (UCP), which has two important properties, namely, orthogonality and adaptivity. Based on these new polynomials, a new kernel function, the unified Chebyshev kernel (UCK), is...

We present novel Markov-type and Nikolskii-type inequalities for multivariate polynomials associated with arbitrary downward closed multi-index sets in any dimension. Moreover, we show how the constant of these inequalities changes, when the polynomial is expanded in series of tensorized Legendre or Chebyshev or Gegenbauer or Jacobi orthogonal polynomials indexed by a downward closed multi-inde...

this work presents a hybrid method for motif discovery in dna sequences. the proposed method called spso-lk, borrows the concept of chebyshev polynomials and uses the stochastic local search to improve the performance of the basic pso algorithm as a motif finder. the chebyshev polynomial concept encourages us to use a linear combination of previously discovered velocities beyond that proposed b...

The main purpose of this paper is to consider the t-best co-approximation and t-best simultaneous co-approximation in fuzzy normed spaces. We develop the theory of t-best co-approximation and t-best simultaneous co-approximation in quotient spaces. This new concept is employed us to improve various characterisations of t-co-proximinal and t-co-Chebyshev sets.

Recently we introduced a new framework to describe some point sets used for multivariate integration and approximation [3], which we called Chebyshev lattices. The associated integration rules are equal weight rules, with corrections for the points on the boundary. In this text we detail the development of exhaustive search algorithms for good Chebyshev lattices where the cost of the rules, i.e...

While the verification of the performance of coordinate measuring machines (CMMs) is self-evidently important, the point coordinates they provide are usually the input to form and tolerance assessment software that calculate associated geometric features, such as a best fit cylinder to the data. For the measurement result to be reliable, it is also necessary to ensure that the calculations perf...

The main purpose of this paper is to consider the t-best co-approximation and t-best simultaneous co-approximation in fuzzy normed spaces. We develop the theory of t-best co-approximation and t-best simultaneous co-approximation in quotient spaces. This new concept is employed us to improve various characterisations of t-co-proximinal and t-co-Chebyshev sets.

We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function is specialized to the energy, one obtains classical notions from Approximation Theory and Convex Analys...

A result on the existence and uniqueness of metric projection for certain sets is proved, by means of a saddle point theorem. A conjecture, based on such a result and aiming for the construction of a nonconvex Chebyshev set in a Hilbert space, is presented.