The subject of orthogonal polynomials is a classical one whose origin can be traced back to Legendre's work on planetary motions. In early stage of this century, a great deal of progress in this field of orthogonal polynomials has been made by the important applications to Quantum mechanics, Probability and Statistics, and other branches of mathematical analysis. Orthogonal polynomials are also strongly related to the theory of continued fractions, special functions and interpolation in approximation.
Until the late 1970's, orthogonal polynomials linking differential equations have been occasionally studied by S. Bochner, W. Hahn and H. L. Krall. Much works were developed on the interplay between the theories of orthogonal polynomials and differential equations in late 1970's, primarily by A. M. Krall and L. L. Littlejohn, which strongly motivates this work. To be precise, we are concerned with the differential equations of spectral type
◁수식 삽입▷(원문을 참조하세요)
where $ℓ_i(x)$ are polynomials, independent of n, and $λ_n$ is an eigenvalue parameter. Our primary concerns are
(1) When does the differential equation ($^*$) have an orthogonal polynomial system (OPS, in short) as eigenfunctions ?
(2) If it does, how to construct orthogonalizing weight distributions for the corresponding OPS's ?
(3) What is the general structure of such distributional weights ?
We first find necessary and sufficient conditions for the equation ($^*$) to have an OPS as solutions. Then, we find an overdetermined system of nonhomogeneous differential equations, that must be satisfied by orthogonalizing weight distributions of such OPS's. It turns out that the corresponding homogeneous system is exactly the symmetry equations of the equations ($^*$), of which any non-trivial classical solution (if it exists) must be a symmetry factor of the differential operator $L_N[ㆍ]$ in ($^*$). We then find a necessary condition for $L_N[ㆍ]$ to be symmetrizable, which will play a crucial role in the spectral analysis of such differential operators. Using this system, we can also analyze the structure of the differential operator $L_N[ㆍ]$ and distributional weights for corresponding OPS's.
Finally, we generalize these results to the so-called Sobolev orthogonal polynomial systems (SOPS, in short), which is orthogonal relative to a symmetric bilinear form
◁수식 삽입▷(원문을 참조하세요)
where $d\mu_0$ and $d\mu_1$ are signed Stieltjes measures with all moments finite. We, in particular, find necessary and sufficient conditions for the differential equation ($^*$) to have an SOPS as solutions via system of overdetermined equations, which must be satisfied by distributional representations of two measures $dμ_0$ and $dμ_1$. This result not only generalizes the classical Krall theorem for the case $dμ_1 \equiv 0$ but also gives important and much needed connections between SOPS's and differential equations of spectral type.
차수가 N이고 계수가 연속 실함수인 다음과 같은 미분방정식
◁수식 삽입▷(원문을 참조하세요)
에 대하여, 많은 사람들이 직교다항식을 고유함수 해로 갖는 이런 형태의 미분방정식을 분류하고자 하여 왔다. 여기서 직교다항식 ${P_n(x)}^ ∞_{n=0}$은 어떤 Borel측도 dμ(x)에 대해서
◁수식 삽입▷(원문을 참조하세요)
의 성질을 만족하는 다항식이다.
이런 분류 문제의 중요성은 부분적으로 Hilbert공간에서 미분작용소의 스펙트랄 이론(spectral theory)의 응용성에 기인한다. 사실, 지금까지 알려진 그런 모든 미분방정식은 대칭성(symmetrizable)이 있으며 경계치에서 어떤 조건을 만족하면 자체 대응(self-adjoint)임을 보인다.
본 연구에서는 ($^**$)과 같은 미분방정식이 직교다항식을 고유함수 해로 갖기위한 필요충분 조건을 제시하며 흥미로운 응용도 소개된다. 즉 ${P_n(x)}^ ∞_{n=0}$이 초함수 ω(x)에 대해서 직교성을 보이고 ($^**$)과 같은 미분방정식을 만족하면, 우리는 ω(x)가 어떤 동질(homogeneous)의 방정식을 만족해야함을 보이는데 이는 ${P_n(x)}^ ∞_{n=0}$을 직교시키는 무게함수를 찾는 한 방법을 제시한 것이다. 그리고 또한 ω(x)의 구조를 면밀히 관찰하여 미분작용소 $L_N[ㆍ]$이 대칭성을 규명하였다.
끝으로 고전적 직교성 $^***$})을 일반화된 Sobolev 직교성으로 확장하여 미분방정식이 Sobolev 직교다항식을 해로 갖기 위한, Krall의 정리를 확장한, 동치 조건을 찾고 몇 개의 예들을 나열하였다.