Variation diminishing transformations and orthogonal polynomials

Distributional transformations, orthogonal polynomials, and Stein characterizations

A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function’s higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates dis...

Symmetric Orthogonal Polynomials and the Associated Orthogonal L-polynomials

We show how symmetric orthogonal polynomials can be linked to polynomials associated with certain orthogonal L-polynomials. We provide some examples to illustrate the results obtained. Finally as an application, we derive information regarding the orthogonal polynomials associated with the weight function (1 + kx2)(l x2)~1/2, k>0.

Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

n=0 |eiφlβ n+1 − β n | <∞ for some φl ∈ R. This includes discrete Schrödinger operators on a half-line or line with finite linear combinations of Wigner–von Neumann potentials cos(nφ+α)/n , where γ > 0. For the real line, our results state that in the Lebesgue decomposition dμ = fdm+ dμs of such measures, supp(dμs) ∩ (−2, 2) is contained in an explicite finite set S (thus, dμ has no singular co...

Orthogonal Polynomials

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...

Orthogonal Polynomials