Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and 1223 (there is no occurrence πi < πj < πj+1 such that 1 ≤ i ≤ j − 2) and provide a combinatorial interpretation for such permutations in terms of lattice paths. ...

The problem of semi-cardinal spline interpolation was solved by Schoenberg exploiting the piecewise polynomial form of the splines. In the present paper, we propose a new construction for the Lagrange functions of semi-cardinal spline interpolation , based on a radial basis and Fourier transform approach. This approach suggests a way of extending semi-cardinal interpolation to polyharmonic spli...

E. Helly’s theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is o...

We study the orthogonal polynomial expansion on sparse grids for a function of d variables in a weighted L space. A fast algorithm is developed to compute the orthogonal polynomial expansion by combining the fast cosine transform, a fast transform from the Chebyshev orthogonal polynomial basis to the orthogonal polynomial basis for the weighted L space, and a fast algorithm of computing hierarc...

A new class of time-integrators is presented for strongly nonlinear dynamical systems. These algorithms are far superior to the currently common time integrators in computational efficiency and accuracy. These three algorithms are based on a local variational iteration method applied over a finite interval of time. By using Chebyshev polynomials as trial functions andDirac–Delta functions as th...