Results 1  10
of
12
Biorthogonal Polynomials on the Unit Circle, regular semiclassical Weights and Integrable Systems
 Construct. Approx
"... Abstract. The theory of biorthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functionaldifference equations of certain coefficient functions appea ..."
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Cited by 4 (3 self)
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Abstract. The theory of biorthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functionaldifference equations of certain coefficient functions appearing in the theory. A natural formulation of the RiemannHilbert problem is presented which has as its solution the above system of biorthogonal polynomials and associated functions. In particular for the case of regular semiclassical weights on the unit circle w(z) = ∏ m j=1 (z − zj(t)) ρ j, consisting of m ∈ Z>0 finite singularities, difference equations with respect to the biorthogonal polynomial degree n (LaguerreFreud equations or discrete analogs of the Schlesinger equations) and differential equations with respect to the deformation variables zj(t) (Schlesinger equations) are derived completely characterising the system. 1.
Subexponentially localized kernels and frames induced by orthogonal expansions
"... Abstract. The aim of this paper is to construct supexponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions. 1. ..."
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Cited by 3 (2 self)
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Abstract. The aim of this paper is to construct supexponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions. 1.
DECOMPOSITION OF SPACES OF DISTRIBUTIONS INDUCED BY TENSOR PRODUCT BASES
, 902
"... Abstract. Rapidly decaying kernels and frames (needlets) in the context of tensor product Jacobi polynomials are developed based on several constructions of multivariate C ∞ cutoff functions. These tools are further employed to the development of the theory of weighted TriebelLizorkin and Besov spa ..."
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Cited by 1 (1 self)
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Abstract. Rapidly decaying kernels and frames (needlets) in the context of tensor product Jacobi polynomials are developed based on several constructions of multivariate C ∞ cutoff functions. These tools are further employed to the development of the theory of weighted TriebelLizorkin and Besov spaces on [−1,1] d. It is also shown how kernels induced by cross product bases can be constructed and utilized for the development of weighted spaces of distributions on products of multidimensional ball, cube, sphere or other domains. 1.
MARKOVNIKOLSKII TYPE INEQUALITY FOR ABSOLUTELY MONOTONE POLYNOMIALS OF ORDER k
"... Abstract. A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q ′ (x) ≥ 0,..., Q (k) (x) ≥ 0, for all x ∈ I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in Lp[−1, 1], p> 0, is esta ..."
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Abstract. A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q ′ (x) ≥ 0,..., Q (k) (x) ≥ 0, for all x ∈ I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in Lp[−1, 1], p> 0, is established. One may guess that the right Markov factor is cn 2 /k and, indeed, this turns out to be the case. Moreover, similarly sharp results hold in the case of higher derivatives and MarkovNikolskii type inequalities. There is a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm, and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of this paper. 1.
GEORGE LORENTZ AND INEQUALITIES IN APPROXIMATION
"... Abstract. George Lorentz influenced the author’s research on inequalities in approximation in many ways. This is the connecting thread of this survey paper. The themes of the survey are listed at the very beginning of the Introduction. 1. Bernsteintype inequalities for exponential sums. 2. Remezty ..."
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Abstract. George Lorentz influenced the author’s research on inequalities in approximation in many ways. This is the connecting thread of this survey paper. The themes of the survey are listed at the very beginning of the Introduction. 1. Bernsteintype inequalities for exponential sums. 2. Remeztype inequalities for exponential sums.
The Scattering Problem for a Noncommutative Nonlinear Schrödinger Equation ⋆
"... Abstract. We investigate scattering properties of a Moyal deformed version of the nonlinear Schrödinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and a ..."
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Abstract. We investigate scattering properties of a Moyal deformed version of the nonlinear Schrödinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has solitary wave solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.
unknown title
, 2000
"... www.elsevier.com/locate/compchemeng Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods ..."
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www.elsevier.com/locate/compchemeng Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods
, the orthonormal Jacobi polynomials P(α,β)
"... k (x) = O max 1, (α 2 + β 2) 1/4}) max x∈[−1,1] (1 − x)α+ 1 2 (1 + x) β+1 2 [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602614]. Here we will confirm this conjecture in the ultraspherical case α = β ≥ 1+√2, even in a str ..."
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k (x) = O max 1, (α 2 + β 2) 1/4}) max x∈[−1,1] (1 − x)α+ 1 2 (1 + x) β+1 2 [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602614]. Here we will confirm this conjecture in the ultraspherical case α = β ≥ 1+√2, even in a stronger 4 form by giving very explicit upper bounds. We also show that δ2 − x2 2 α (1 − x) ( P (α,α)
unknown title
, 2002
"... Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight ..."
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Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight
NEW BOUNDS ON THE HERMITE POLYNOMIALS
, 2004
"... Abstract. We shall establish twoside explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of Hk(x)e −x2 /2, on the real axis, where Hk are the Hermite polynomials. 1. ..."
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Abstract. We shall establish twoside explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of Hk(x)e −x2 /2, on the real axis, where Hk are the Hermite polynomials. 1.