In this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet-Neumann map for Laplace’s equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of ...

In the paper [8], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [−1, 1], and derived a compact form of the corresponding Lagrange interpolation formula. In [1] we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows ...

A new family of AN -type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin H...

Suppose that S ⊆ Fp, where p is a prime number. Let λ1, ..., λp be the Fourier coefficients of S arranged as follows |Ŝ(0)| = |λ1| ≥ |λ2| ≥ · · · ≥ |λp|. Then, as is well known, the smaller |λ2| is, relative to |λ1|, the larger the sumset S +S must be; and, one can work out as a function of ε and the density θ = |S|/p, an upper bound for the ratio |λ2|/|λ1| needed in order to guarantee that S +...

Let gα be a one-parameter family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, a superstable periodic orbit is known to exist. An example of such a family is the well-known logistic map. In this paper we deal with the effect of a quasi-periodic perturbation (with only one frequency) on this cascade. Let us call ε the perturbing param...

A d-space X = (X, %, μ) is a compact set X with respect to a quasi-metric % and a Borel measure μ such that the measure of a ball of radius r is equivalent to r, where d > 0. The paper deals with spaces B p(X;H) of Besov type where 1 < p < ∞ and s ∈ R. Here H is a bi-Lipschitzian map of the snowflaked version (X, %, μ) of X for some 0 < ε < 1, onto a fractal d/ε-set Γ = HX in some R, reducing t...

Gradient iterations for the Rayleigh quotient are elemental methods for computing the smallest eigenvalues of a pair of symmetric and positive definite matrices. A considerable convergence acceleration can be achieved by preconditioning and by computing Rayleigh-Ritz approximations from subspaces of increasing dimensions. An example of the resulting Krylov subspace eigensolvers is the generaliz...

A polynomial filtered Davidson-type algorithm is proposed for symmetric eigenproblems, in which the correction-equation of the Davidson approach is replaced by a polynomial filtering step. The new approach has better global convergence and robustness properties when compared with standard Davidson-type methods. The typical filter used in this paper is based on Chebyshev polynomials. The goal of...

In this paper we analyze a class of projection operators with values in a subspace of polynomials. These projection operators are related to the Hubert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthogonality relat...

In [MR], the second author and H. P. Rosenthal constructed the first examples of weakly null normalized sequences which do not have any unconditionally basic subsequences. Much more recently, the second author and W. T. Gowers [GM] constructed infinite dimensional Banach spaces which do not contain any unconditionally basic sequences. These later examples are of a different character than the e...