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Lance L. Littlejohn

Other affiliations: KAIST, University of Texas at San Antonio, Utah State University  ...read more
Bio: Lance L. Littlejohn is an academic researcher from Baylor University. The author has contributed to research in topics: Orthogonal polynomials & Classical orthogonal polynomials. The author has an hindex of 23, co-authored 113 publications receiving 1480 citations. Previous affiliations of Lance L. Littlejohn include KAIST & University of Texas at San Antonio.


Papers
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Journal ArticleDOI
TL;DR: In this paper, it was shown that there is anorthogonalizing weight distribution for polynomials of degree ϕ(x) that simultaneously satisfies n distributional differential equations of orders 1, 3, 5,... (2n−1).
Abstract: Suppose ϕm(x) is a polynomial of degree in that satisfies the differential equation $$\sum\limits_{i = 1}^{2n} {b_i } (x)y^{(i)} (x) = \lambda _m y(x), m = 0, 1, 2, ...$$ (1) where n is some fixed integer ≧1. We show that, under certain conditions, there exists anorthogonalizing weight distribution for {ϕm,(x)} that simultaneously satis]ies n distributionaldifferential equations of orders 1, 3, 5, ... (2n−1). In particular, this weight ″ must satisfy $$nb_{2n} (x)\Lambda ' + (nb_{2n}^\prime (x) - b_{2n - 1} (x))\Lambda = 0$$ in the distributional sense. As a corollary to this result, we get part of H. L. Krall's 1938classification theorem which gives necessary and sufficient conditions on the existenee of anOPS of solutions to (1) in terms of the moments and the coefficients of bi(x). To illustrate the theory, we consider all of the known OPS's to (1).In particular, new light is shed upon the problem of finding a real weight distribution for the Bessel polynomials.

83 citations

Journal ArticleDOI
TL;DR: In this paper, a left-definite analysis associated with the self-adjoint Jacobi operator, generated from the classical second-order Jacobi differential expression, in the Hilbert space, where wα,β(t)=(1-t)α(1+ t)β, was developed.

74 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the set of polynomials orthogonal with respect to the inner product of the Sobolev inner product for a given integer n = 0, where n is a positive Borel measure.
Abstract: This paper discusses recurrence relations for sequences of polynomials which are orthogonal with respect to the Sobolev inner product defined on the set of polynomials $\mathcal{P}$ by \[ (p,q)w = \sum_{k = 0}^N {\int_\mathbb{R} {p^{(k)} (x)\bar q^{(k)} (x)d\mu _k (x)\quad (p,q \in \mathcal{P})} } \] for some integer $N \geq 1$, where each $\mu _k $, $0 \leq k \leq N$, is a positive Borel measure. It is proven that there exists a real-valued polynomial $h:\mathbb{R} \to \mathbb{R}$ satisfying \[ ( * )\qquad (hp,q)_W = (p,hq)_W\quad(p,q \in \mathcal{P})\] if and only if each of the measures $\mu _k $, $1 \leq k \leq N$, is purely atomic with a finite number of mass points. In addition it is proven that $R_j $, the set of real roots of ${{d^j h} / {dx^j }}$, $(1 \leq j \leq N)$, is nonempty and that $(\mu _k ) \subset \cap _{i = 1}^k R_i $. It is also shown that if h satisfies the condition $(*)$, then the polynomials orthogonal with respect to the inner product $( \cdot , \cdot )W$ will satisfy a recurrenc...

66 citations

Journal ArticleDOI
TL;DR: In this article, the classification of differential equations of the form ∑ k = 1 N a k (x)y (k) (x)=λy(x) having a sequence of polynomial eigenfunctions { p n (x )} n = 0 ∞ that are orthogonal with respect to some real bilinear form is surveyed.

63 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed the left-definite spectral theory associated with the self-adjoint operator A(k) in L2(- 1, 1), generated from the classical second-order Legendre differential equation lL,k[y](t) = -((1 - t2)y')' + ky = λy (t∈ (-1,1)), that has the Legendre polynomials {Pm(t)}m=0∞ as eigenfunctions; here, k is a fixed, nonnegative

56 citations


Cited by
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Book
19 Mar 2001
TL;DR: In this article, the authors considered the properties of orthogonal polynomials on the unit sphere, root systems and Coxeter groups, and the Summability of Orthogonal expansions.
Abstract: Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

1,026 citations

Book ChapterDOI
31 Dec 1939

811 citations

Journal ArticleDOI
TL;DR: In this article, a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type was studied.
Abstract: We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.

775 citations

Book
01 Jan 1966
TL;DR: Boundary value problems in physics and engineering were studied in this article, where Chorlton et al. considered boundary value problems with respect to physics, engineering, and computer vision.
Abstract: Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

733 citations

Journal ArticleDOI

640 citations