We derive bounds and asymptotics for the maximum Riesz polarization quantity M n(A) := max x1,x2,... ,xn∈A min x∈A n ∑ j=1 1 |x− xj |p (which is n times the Chebyshev constant ) for quite general sets A ⊂ R with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete vers...

Several authors have examined connections between permutations which avoid 132, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of some of these results for permutations which avoid 1243 and 2143. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid 1243, 2143, and certain a...

We are concerned with the problem of minimizing the supremum norm on [0, 1] of a nonzero polynomial of degree at most n with integer coefficients. We use the structure of such polynomials to derive an efficient algorithm for computing them. We give a table of these polynomials for degree up to 75 and use a value from this table to answer an open problem due to P. Borwein and T. Erdélyi and impr...

A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey–Wilson to Wilson polynomials and from q-Racah to Racah polynomials ar...

In a recent article, the first and second kinds of multivariate Chebyshev polynomials fractional degree, relevant integral repesentations, have been studied. this we introduce pseudo-Lucas functions show possible applications these new functions. For kind, compute Newton sum rules any orthogonal polynomial set starting from entries Jacobi matrix. representation formulas for powers r×r matrix, a...

The mathematical theory of closed form functions for calculating LSFs on the basis of generating functions is presented. Exploiting recurrence relationships in the series expansion of Chebyshev polynomials of the first kind makes it possible to bootstrap iterative LSF-search from a set of characteristic polynomial zeros. The theoretical analysis is based on decomposition of sequences into symme...

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre poly...

It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. ...

A technique is presented for determining the roots of a polynomial p(x) that is expressed in terms of an expansion in orthogonal polynomials. The roots are expressed as the eigenvalues of a nonstandard companion matrix Bn whose coefficients depend on the recurrence formula for the orthogonal polynomials, and on the coefficients of the orthogonal expansion. Some questions on the numerical stabil...

BP decoding algorithm is a high-performance low-density parity-check (LDPC) code decoding algorithm, but because of its high complexity, it can’t be applied to the high-speed communications systems. So in order to further reduce the implementation complexity with the minimum affection to the system performance, , we proposed a BP algorithm based on Chebyshev polynomial fitting for its good appr...