Let ψK be the Chebyshev function of a number field K. Let ψ K (x) := ∫ x 0 ψK(t) dt and ψ (2) K (x) := 2 ∫ x 0 ψ (1) K (t) dt. We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences |ψ K (x) − x 2 | and |ψ K (x) − x 3 |. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals....

Abstract. The discrete Fourier analysis on the 300–600–900 triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G2, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm–Liouville eigenvalue problem that contains two parameters, whose solutions are ...

We evaluate the matrix elements 〈Orp〉, where O = {1, β, iαnβ} are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of generalized hypergeometric functions 3F2 (1) for all suitable powers. Their connections with the Chebyshev and Hahn polynomials of a discrete variable are emphasized. As a result, we...

A Chebyshev knot C(a, b, c, φ) is a knot which has a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + φ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ ∈ R. We show that any two-bridge knot is a Chebyshev knot with a = 3 and also with a = 4. For every a, b, c integers (a = 3, 4 and a, b coprime), we describe an algorithm that gives all Cheb...

The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax = b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for res...

Generalized matrix functions (GMFs) extend the concept of a matrix function to rectangular matrices via the singular value decomposition. Several applications involving directed graphs, Hamiltonian dynamical systems, and optimization problems with low-rank constraints require the action of a GMF of a large, sparse matrix on a vector. We present a new method for applying GMFs to vectors based on...

The problem of allpass lter design for phase approximation and equalization in the Chebyshev sense is solved by using a generalized Remez algorithm. Convergence to the unique optimum is guaranteed and is achieved rapidly in the actual implementation. The well-known numerical problems for higher degree lters are analyzed and solved by a simple approach. Possible applications are: design of lters...

Ladder networks have been studied extensively using Fibonacci numbers, Chebyshev polynomials, Morgan-Voyce polynomials, Jacobsthal polynomials, etc. ([10], [11], [2], [14], [9]. [5], [3], and [4]). All these polynomials are, in fact, particular cases of the generalized polynomials defined by U„(x,y) = xU„_l(x,y)+yU„_2(x,y), (»>2) (la) with U0(x,y) = 0, Ul(x,y) = l, (lb) and V„(x, y) = xV^ix, y)...

In one of the most important methods in Density Functional Theory – the FullPotential Linearized Augmented Plane Wave (FLAPW) method – dense generalized eigenproblems are organized in long sequences. Moreover each eigenproblem is strongly correlated to the next one in the sequence. We propose a novel approach which exploits such correlation through the use of an eigensolver based on subspace it...

In this work, we discuss the problem of approximating a multivariate function by polynomials via `1 minimization method, using a random chosen sub-grid of the corresponding tensor grid of Gaussian points. The independent variables of the function are assumed to be random variables, and thus, the framework provides a non-intrusive way to construct the generalized polynomial chaos expansions, ste...