Two-dimensional analogy of the Korous inequality (original) (raw)

Korous Type Inequalities for Orthogonal Polynomials in two Variables

Tatra Mountains Mathematical Publications, 2014

J. Korous reached an important result for general orthogonal polynomials in one variable. He dealt with the boundedness and uniform boundedness of polynomials { Pn(x)}∞ n=0 orthonormal with the weight function h(x) = δ(x) ̃h(x), where ̃h(x) is the weight function of another system of polynomials { ̃Pn(x) }∞ n=0 orthonormal in the same interval and δ(x) ≥ δ0 > 0 is a certain function. We generalize this result for orthogonal polynomials in two variables multiplying their weight function h(x, y) by a polynomial, dividing h(x, y) by a polynomial, and multiplying h(x, y) with separated variables by a certain function δ(x, y).

ON A CLASS OF ORTHOGONAL POLYNOMIALS

2007

In this note we study a system of polynomials {b Pk} orthogonal with respect to the modified measure db (t) = t d t c w(t)dt, t 2 (0,1) where d,c < 0 and w is a weight function, using orthogonal polynomials {Pk} with respect to the measure dw(t).

Properties of Some of Two-Variable Orthogonal Polynomials

Bulletin of the Malaysian Mathematical Sciences Society, 2019

The present paper deals with various recurrence relations, generating functions and series expansion formulas for two families of orthogonal polynomials in two variables, given Laguerre-Laguerre Koornwinder polynomials and Laguerre-Jacobi Koornwinder polynomials in the limit cases. Several families of bilinear and bilateral generating functions are derived. Furthermore, some special cases of the results presented in this study are indicated.

A few remarks on orthogonal polynomials

2014

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials { p_n} _n≥ 0 that are orthogonal with respect to this distribution, coefficients of expansion of x^n in the series of p_j, j≤ n, two sequences of coefficients of the 3-term recurrence of the family of { p_n} _n≥ 0, the so called "linearization coefficients" i.e. coefficients of expansion of p_j, j≤ m+n. Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials { p_n} _n≥ 0, we express with their help: coefficients of the power series expansion of p_n, coefficients of expansion of x^n in the series of p_j, j≤ n, moments of the distribution that makes polynomials { p_n} _n≥ 0 orthogonal. Further having two different families of orthogonal polynomials { p_n} _n≥ 0 and { q_n} _n≥ 0 and knowing for each of them sequences of the 3-term recurren...

A weighted polynomial inequality

1984

In the theory of orthogonal polynomials for weights with noncompact support, much use is made of inequalities relating weighted integrals of polynomials over infinite and finite ranges. Using a short new method of proof, we show such inequalities hold for very general weights in L" and certain Orlicz spaces.

Estimates for jacobi—sobolev type orthogonal polynomials

Applicable Analysis, 1997

Let the Sobolev-type inner product f, g = R f gdµ 0 + R f g dµ 1 with µ 0 = w + M δ c , µ 1 = N δ c where w is the Jacobi weight, c is either 1 or −1 and M, N ≥ 0. We obtain estimates and asymptotic properties on [−1, 1] for the polynomials orthonormal with respect to .,. and their kernels. We also compare these polynomials with Jacobi orthonormal polynomials; as a consequence, a result about the convergence acceleration to c of the zeros is given.

The weighted Lp-norms of orthonormal polynomials for Erdös weights

Computers & Mathematics with Applications, 1997

polynomial growth at infinity. For example, we consider Q(x) = exp k/(]xla), a > 1, where expk = exp(exp(.., exp(...))) denotes the k th iterated exponential. Weights of the form W 2 for such W are often called ErdSs weights. We compute the growth of the Lp-norms (0 < p < oo) of the weighted orthonormal polynomials Pn (W 2, x)W(x) for a large class of Erd6s weights, based on recent work of the author with Levin and Mthembu on the Loo-norm of pn(W 2, x)W(x). As an auxiliary result, we obtain bounds on the fundamental polynomials of Lagrange interpolation at the zeros of pn(W2,x), and as a corollary, we deduce finer spacing for the zeros of pn(W 2, .). The growth of the Lp-norms of orthonormal polynomials is a key factor in investigating convergence of orthogonal expansions and Lagrange interpolation in Lp-norms.