Constrained orthogonal polynomials have been recently introduced in the study of the HohenbergKohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean scalar product, to a given i-dimensional subspace Ei of polyno...

Constrained orthogonal polynomials have been recently introduced in the study of the HohenbergKohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace Ei of polynom...

R |xα| dμ(x) < ∞ (α ∈ (Z≥0)) and the support of μ has nonempty interior. Let Pn consist of all polynomials p of degree ≤ n such that ∫ Rd pq dμ = 0 for all polynomials q of degree < n. Then Pn has the same dimension ( n+d−1 n ) as the space of homogeneous polynomials of degree n in d variables. Furthermore, the spaces Pn (n = 0, 1, 2, . . .) are mutually orthogonal in L(μ). We call {Pn}n=0 a sy...

In this paper two types of multiple orthogonal polynomials on the semicircle with respect to a set of r different weight functions are defined. Such polynomials are generalizations of polynomials orthogonal on the semicircle with respect to a complex-valued inner product 1⁄2f ; g 1⁄4 R p 0 f e ih g eih w eih dh. The existence and uniqueness of introduced multiple orthogonal polynomials for cert...

The theory of orthogonal polynomials plays an important role in many branches of mathematics, such as approximation theory (best approximation, interpolation, quadrature), special functions, continued fractions, differential and integral equations. The notion of orthogonality originated from the theory of continued fractions, but later became an independent (and possibly more important) discipl...

Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and the...

In this paper, we introduce hybrid of block-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. Then, we utilize them to solve delay differential equations and time-delay system. The method is based upon e...

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.

For a measure μ on R, the situation is more subtle. One can always orthogonalize the subspaces of polynomials of different total degree (so that one gets a family of pseudo-orthogonal polynomials). The most common approach is to work directly with these subspaces, without producing individual orthogonal polynomials; see, for example [DX01]. One can also further orthogonalize the polynomials of ...