Sobolev Orthogonal Polynomials

• Sobolev orthogonal polynomials are defined via Sobolev-type inner products that couple function values and derivatives, offering a framework for analyzing differential equations.
• They are constructed using Casorati determinants derived from classical Jacobi polynomials, leading to exact polynomial sequences with finite-order recurrence relations.
• These polynomials serve as eigenfunctions of higher-order differential operators, playing a crucial role in spectral theory and approximation methods.

Sobolev orthogonal polynomials are orthogonal polynomial systems with respect to a Sobolev-type inner product—one in which derivatives of the polynomials (possibly of various orders) and function values at discrete points are coupled, sometimes involving matrix-valued measures. In the fully discrete Jacobi–Sobolev setting, these polynomials may satisfy finite-order differential equations and arise as eigenfunctions of higher-order self-adjoint operators. Their construction, recurrence structure, and spectral properties are central both in approximation theory and in the spectral analysis of differential equations.

1. Discrete Jacobi–Sobolev Bilinear Forms and Defining Orthogonality

The discrete Jacobi–Sobolev bilinear form central to the study in "Differential equations for discrete Jacobi-Sobolev orthogonal polynomials" (Durán et al., 2015) is defined for parameters α, β ∈ ℝ, m₁, m₂ ∈ ℕ₀ (m = m₁ + m₂ ≥ 1), and positive semi-definite real matrices M, N. Let the generalized Jacobi weight be: $$w_{α−m₂,β−m₁}(x) = (1 - x)^{α-m_2}(1 + x)^{β-m_1},\quad x \in (-1,1).$$ For polynomials $p, q$, the Sobolev bilinear form is: $$(p, q) = \int_{-1}^{1} p(x)\,q(x)\,w_{α-m_2, β-m_1}(x)\,dx + T_{m_1}(p)\,M\,T_{m_1}(q)^T + T_{m_2}(p)\,N\,T_{m_2}(q)^T$$ where $$T_{m_1}(p) = (p(-1), p'(-1), \dots, p^{(m_1-1)}(-1))$$, and $T_{m_2}(p)$ similar at $x=+1$.

Orthogonal polynomials with respect to this form are sequences $(Q_n)$ with

$(Q_n, x^k) = 0, \quad k=0,\dots,n-1,$

and unique (up to scalars) of exact degree $n$.

2. Construction via Casorati Determinants

Discrete Jacobi–Sobolev orthogonal polynomials are constructed using Casorati (quasi-determinant) structures:

• Jacobi base: Use classical Jacobi polynomials $J_n^{(α,β)}$ normalized so $D_{α,β} J_n^{(α,β)} = n(n+α+β+1) J_n^{(α,β)}$ for the hypergeometric-type operator

$$D_{α,β} = (1-x^2) \frac{d^2}{dx^2} + [(β-α) - (α+β+2)x] \frac{d}{dx}.$$

• Auxiliary sequences: From the matrices $M, N$, define $m=m_1+m_2$ sequences $z_\ell(n)$, each as explicit sums involving Jacobi polynomials and matrix entries.
• Casorati determinant: Let $p(n), q(n)$ be rising and falling Pochhammer-type products, then assemble

$$A(n) = \det[p(n-j) q(n-j);\, z_\ell(n-j)]_{1\le \ell,j\le m}$$

and form

$$Q_n(x) = \frac{1}{A(n)}\; \det \begin{pmatrix} J_n^{(α,β)}(x) & \dots & J_{n-m}^{(α,β)}(x) \ z_1(n) & \dots & z_1(n-m) \ \vdots & & \vdots \ z_m(n) & \dots & z_m(n-m) \end{pmatrix}.$$

If $A(n)\neq0$ for all $n$, then these $Q_n$ are orthogonal (of exact degree $n$) for the Sobolev bilinear form (Durán et al., 2015).

3. Differential Operators and Finite-Order Eigenproblems

A central result is that discrete Jacobi–Sobolev orthogonal polynomials are eigenfunctions of finite-order differential operators. For integer parameters α, β, the eigenoperator is explicitly constructed:

• D-operator scheme: Using first-order operators

$$\mathcal{D}_1 = (1-x)\frac{d}{dx} + \frac{β+1}{2};\qquad \mathcal{D}_2 = (1+x)\frac{d}{dx} - \frac{α+1}{2},$$

any rational function $S(x)$, and auxiliary polynomials $Y_i(\theta)$, the constructed operator

$$\mathbb{D}_{Q,S} = P_S(D_{α,β}) + \sum_{i=1}^m M_i(D_{α,β})\,\mathcal{D}_i\,Y_i(D_{α,β})$$

yields

$$\mathbb{D}_{Q,S} Q_n = \Lambda_n Q_n, \quad \Lambda_n = P_S(n(n+α+β+1)).$$

For the Sobolev case, a specific rational $S(x)$ in terms of the matrices and combinatorial factors ensures that the Casorati determinant and the operator have the required algebraic properties (Durán et al., 2015).

• Order bound: The order of the differential operator is determined via the weighted ranks of $M,N$: it does not exceed

$$\text{order}(\mathbb{D}_{Q,S}) \le 2\left[\beta-wr(M) + \alpha-wr(N) + 1\right],$$

where wr(M), wr(N) are weighted ranks defined from defect sequences arising from the structure of the Casorati determinant.

4. Algebraic Structure and Spectral Properties

Discrete Jacobi–Sobolev polynomials exhibit banded recurrence relations and generalized eigenvalue structures.

• Recurrence: In the generic case, a recurrence with bandwidth $m+2$, where $m=m_1+m_2$ is the total number of mass points, arises: $$x Q_n(x) = Q_{n+1}(x) + \sum_{k=0}^{m} \beta_{n,k} Q_{n-k}(x).$$ The precise coefficients are computable via the structure of the Casorati determinant and the matrices $M,N$.
• Spectral Theory: The polynomials are generalized eigenfunctions corresponding to a higher-order differential operator, satisfying

$\mathbb{D}_{Q,S}(Q_n) = \Lambda_n Q_n,$

where the eigenvalues $\Lambda_n$ and the operator are determined by the constructed symbol $S(x)$ (Durán et al., 2015).

• Limiting Cases: When both $M=N=0$, the polynomials reduce to classical Jacobi polynomials, yielding Bochner’s second-order self-adjoint operator. With one Dirac mass (e.g., $M>0$, $N=0$ or vice versa) and integer α or β, one recovers higher-order Jacobi–Koornwinder (“generalized Jacobi”) operators.
• Unification and Extension: The construction encompasses all previously known finite-order Sobolev-type ODEs, including those for Laguerre–Sobolev and discrete Jacobi–Sobolev polynomials, and supplies a route to extend to discrete–difference analogues (e.g., for Hahn/Meixner families) via the same D-operator methods.

5. Connection to Broader Theory and Open Problems

• Generality of Construction: The approach unifies the treatment of Sobolev orthogonal polynomials with arbitrary mass-point structures, providing explicit (and minimal, up to open conjectures in low-dimensional or symmetric settings) order computations for the associated differential operators (Durán et al., 2015).
• **Relation to Coherent Pair