Bivariate Koornwinder–Sobolev Orthogonal Polynomials

The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function ρ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (t)$$\end{document} such that ρ(t)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (t)^2$$\end{document} is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given.


Introduction
An interesting non trivial tool for generating orthogonal polynomials in two variables was introduced in 1965 by S. A. Agahanov ( [2]). Ten years later, T. H. Koornwinder used that procedure for classical Jacobi polynomials and introduced Two variable analogues of the classical orthogonal polynomials (see [6]). In fact, given two univariate weight functions ω i (t) defined on the intervals (a i , b i ) ⊂ R, for i = 1, 2, and a function ρ(t) on (a 1 , b 1 ), such that ρ(t) 2 is a polynomial of degree less than or equal to 2, orthogonal polynomials in two variables associated with the weight function defined by W (x, y) = ω 1 (x) ω 2 (y/ρ(x)) (1.1) orthogonal families of polynomials. In fact, the most usual bivariate families correspond to this scheme. For instance, eight of the nine different cases of Krall and Sheffer classical bivariate polynomials (see [8]) can be constructed in this way. Two-dimensional Krall and Sheffer polynomials are analogues of the classical orthogonal polynomials since they are eigenfunctions of second order linear partial differential operators and, moreover, they satisfy orthogonality conditions. In [5], Harnad et al. proved that all Krall-Sheffer polynomials are connected with two-dimensional superintegrable systems on spaces with constant curvature.
In [13], the authors exploited the structure of Koornwinder's weight function to obtain the coefficients of the three term relation satisfied by the bivariate polynomials when both weight functions ω 1 and ω 2 are classical. Moreover, in [11] differential properties for the weight function W (x, y) are deduced. Those properties were the extension to the Koornwinder case of the Pearson's differential equation of univariate classical weight functions.
Recently, using a similar construction, Olver and Xu ( [14]) presented explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. They also constructed orthogonal polynomials on the boundary of quadratic surfaces of revolution. In such a construction, they replaced the univariate weight function ω 2 with the classical weight function on the d-dimensional ball or with the Lebesgue measure on the sphere, respectively.
In Koornwinder's construction, a family of bivariate polynomials orthogonal with respect to (1.1) can be defined by means of where {p (m) n (t)} n≥0 is an orthogonal polynomial sequence (OPS in short) associated with the weight function ρ 2m+1 (t)ω 1 (t), m ≥ 0, and {q n (t)} n≥0 is an OPS associated with ω 2 (t). In this work, we consider an extension of Koornwinder's construction to the Sobolev realm. To this end, we modify the family of bivariate polynomials (1.2) replacing one of the univariate OPS by a univariate Sobolev OPS. First, we consider the case where {p (m) n (t)} n≥0 is a Sobolev OPS and, next, we study the case where {q n (t)} n≥0 is a Sobolev OPS. In both cases, we show that the bivariate polynomials in (1.2) are orthogonal with respect to a bivariate Sobolev inner product involving first order partial derivatives. The Sobolev inner product obtained in the second case is quite similar to the inner product studied by Kwon and Littlejohn in [9]. Whatever, our most interesting result is the existence of connection formulas relating the bivariate Sobolev and standard orthogonal polynomials. Algorithms to obtain the coefficients in those connection formulas are provided.
We refer to the survey paper [12] as the most recent presentation of the state of the art on Sobolev orthogonal polynomials both in the univariate as well as the bivariate case.
In the one-dimensional case, the natural framework of application of Sobolev orthogonal polynomials seems to be the implementation of spectral methods for boundary value problems for elliptic differential operators. For instance, in [15] the second-and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered, and generalized Jacobi spectral schemes are proposed. In fact, the authors construct an orthogonal basis of Jacobi-Sobolev polynomials which allows the diagonalization of the involved discrete systems. The corresponding error estimates and numerical results illustrate the effectiveness and the spectral accuracy of the method. In several variables, Sobolev orthogonal polynomials on the unit ball have been considered by Xu in the numerical solution of boundary value problems for elliptic partial differential operators. For details, we refer again to the survey paper [12] and the references therein. The structure of the paper is as follows: In Sect. 2, we introduce the notation and basic results on orthogonal polynomials used throughout our work. Section 3 contains the definition and properties of the first type of bivariate Sobolev orthogonal polynomials. The case where the first family in (1.2) is a classical one is studied in Sect. 4. Section 5 contains the definition and properties of the second type of bivariate Sobolev orthogonal polynomials. In Sect. 6 we have included several explicit examples. Finally, we added an appendix containing the technical proofs of our results.

Preliminaries
We need to fix the notation and recall the basic results used throughout this work, in order to be self-contained. We will need univariate and bivariate tools, and we start briefly with the univariate ones.

Univariate Basic Tools
Let u be a linear functional defined on Π, the linear space of real polynomials in one variable, by means of its moments and extended by linearity to Π. Hence, we say that u is a moment functional. We will work with polynomial sequences on Π, {p n (t)} n≥0 , such that deg p n = n, for n ≥ 0, and then {p n (t)} n≥0 is always a basis of Π. In addition, if u, p n p m = 0, n = m, and u, p 2 n = h n = 0, n ≥ 0, we say that {p n (t)} n≥0 is an orthogonal polynomial sequence (OPS) associated with u.
Following [3,16], given a moment functional u defined as above, there is not always an OPS associated with it. If an OPS associated with u exists, then u is called quasi-definite. It is well known that if u is quasi-definite, then its OPS is unique except for a constant factor. A moment functional u is positive definite if u, p 2 > 0 for all non zero polynomial p ∈ Π; positive definite moment functionals are quasi-definite, and OPS associated with u exists. In addition, u is symmetric if all odd moments are zero, that is, u, x 2n+1 = 0, n ≥ 0.

Bivariate Koornwinder Orthogonal Polynomials
Bivariate Koornwinder polynomials are constructed from two univariate families of orthogonal polynomials associated with inner products defined by means of weight functions [2,4,6], and an auxiliary function. This kind of polynomials are orthogonal with respect to a new bivariate inner product defined by a specific weight function. We use the extension of this construction to moment functionals. This extension, which was studied by the authors in [13], uses two univariate quasidefinite moment functionals in order to build a quasi-definite bivariate moment functional. We recall this construction.
Consider two univariate quasi-definite moment functionals u (x) and v (y) . The superscript corresponds to the variable that each moment functional acts upon. Additionally, we need a non-zero function ρ(x) such that ρ 2 (x) is a real polynomial of degree less than or equal to 2. We have two cases as follows: In both cases, we also impose that the functional u m . Moreover, let {q n (y)} n≥0 be an OPS associated with v (y) .
It was shown in [13] that the set {P n,m (x, y) : 0 ≤ m ≤ n, n ≥ 0} is a mutually orthogonal basis with respect to the moment functional w defined by

First Type of Bivariate Sobolev Orthogonal Polynomials
Here, we extend the construction described in Sect. 2.3 to include a class of Sobolev bivariate orthogonal polynomials generated from univariate orthogonal polynomials. Hence, let u ≡ u (x) and v ≡ v (y) be univariate quasi-definite moment functionals acting on the variables x and y, respectively. When there is no confusion, we remove the corresponding superscript in the notation. For n (x)} n≥0 be an orthogonal polynomial sequence associated with the quasi-definite moment functional u m = ρ(x) 2m+1 u, and let {q m (y)} m≥0 be an orthogonal polynomial sequence associated with v. In addition, we denote with h (m) n = 0 and h (q) m = 0, for n, m ≥ 0. Let ρ ≡ ρ(x) be a function satisfying the conditions of either Case I or Case II. Furthermore, we introduce the univariate bilinear form For m ≥ 0, let us define the Sobolev bilinear form in both Case I and Case II, or equivalently,  [12] and the references therein), and the analytic and algebraic properties of the associated orthogonal polynomials constitute an interesting topic in itself. Now, we present the announced extension to the Sobolev case as follows: where λ ∈ R and ∇ is the usual gradient operator ∇ = (∂ x , ∂ y ) t . Moreover, where h Proof. Let us define the change of variables x = s and y = t ρ(s). Thus, for any polynomial P (x, y) ∈ Π 2 , we get Then, Using this jointly with (2.2) and (3.5), we compute Since {s n } n≥0 is orthogonal with respect to (· , ·) m , the theorem is proved.
Notice that the 2 × 2 matrix in (3.6) is positive semidefinite. In Case I, the entries of this matrix are polynomials, but in Case II, this matrix has rational functions as entries since ρ (x) is of the following form: Then, multiplication by ρ(x) ς = ρ(x) 2 cancels out the denominators in the matrix of the bilinear form (3.6).

The Classical First-Type Univariate Sobolev Orthogonal Polynomials
In this section, we show that if the univariate moment functional u ≡ u (x) involved in the construction presented in Sect. 2.3 is classical and ρ is related to the polynomial coefficients of the Pearson equation (2.1) satisfied by u, then the orthogonal sequence {s (m) n (x)} n≥0 associated with the bilinear form (·, ·) m defined in (3.3) can be computed recursively.
First, we introduce some notation to be used in the sequel. We write the three-term recurrence relation satisfied by following: {p and if where h (m) n was defined in (3.1). We work with classical moment functionals u, that is, moment functionals satisfying Pearson equation (2.1). If ρ(x) and φ(x) are related, the classical character is inherited by the moment functional u m .
We have the following preliminary result: Proof. If a moment functional u is classical, then it satisfies the Pearson In Case I, since ρ(x) divides φ(x), then is a polynomial of degree equal to 1. In Case II, we have is a polynomial of degree equal to 1. As a consequence, in both cases u k is classical.
Notice that the Hermite, Laguerre, Jacobi, and Bessel are the only families of classical orthogonal polynomials as it was shown in [10], among others. In these cases, the polynomial φ(x) is usually normalized as 1, x, 1 − x 2 , and x 2 , respectively. Recall that the Hermite, Laguerre, and Jacobi polynomials are associated with a positive definite moment functional, and the Bessel polynomials are associated with a quasi-definite moment functional. By Lemma 4.1, in Case I, ρ(x) must divide φ(x) and, in Case II, ρ(x) 2 must divide φ(x). Then, non trivial cases are obtained by taking ρ(x) = a x in the Laguerre and Bessel cases, and ρ(x) = a (1 ± x) in the Jacobi case, for a ∈ R. In Case II, non trivial cases are obtained by taking ρ(x) 2 = a x in the Laguerre and Bessel cases, and ρ(x) 2 = a(1 − x 2 ) in the Jacobi case. We remark that in Case II the functional v must be symmetric.
We return to the construction presented in Sect. 2.3. Under the hypotheses of Lemma 4.1, u m = ρ(x) 2m+1 u is classical for m ≥ 0. As proved in [10], this means that {p (m) n (x)} n≥0 satisfies the so-called Second Structure Relation for some constants ξ n (x) the polynomial of degree n defined by It follows by definition that Now, we study Case I and Case II separately.

Case I
In this case, ρ(x) = r 1 x + r 0 , |r 1 | + |r 0 | > 0, and the bilinear form (·, ·) m reads (4.5) Next, we can express the polynomial π (m) n (x) in terms of the Sobolev polynomials. The proof is given at the Appendix A.
Therefore, from (4.2) and (4.6), we deduce a short relation between univariate Sobolev orthogonal polynomials and the first family of orthogonal polynomials.

Corollary 4.3.
For n ≥ 1, the following relation holds: and s The polynomials {s and, for n ≥ 1, where Proof. On the one hand, using (4.6), we get On the other hand, using (4.1), (4.3), (4.4), and the explicit expression of the Sobolev bilinear form (4.5), we obtain Moreover, using directly (4.5), we get h Now, we want to analyse how to compute explicitly the Sobolev orthogonal polynomials s  First of all, we know that 2 , and so on. In Fig. 1 we can see how the algorithm generates all the tilde constants.
Finally, using expression (3.5), relation (4.9) can be extended to the bivariate case multiplying by ρ(x) m q m y ρ(x) . Therefore, bivariate Sobolev orthogonal polynomials are related to the standard bivariate orthogonal polynomials, and we can compute them recursively.

Case II
Recall that v must be a symmetric moment functional. A similar reasoning as in Proposition 4.2 allows us to prove the next result, taking into account the explicit expression of the bilinear form in this case (4.12). The explicit expressions of the coefficients are showed in the Appendix.     Theorem 4.9. We get S 0,0 (x, y) = P 0,0 (x, y), and for n ≥ 1 and 0 ≤ m ≤ n, the following relation holds: An interesting case appears when ρ(x) 2 = φ(x). In this case, not only u m is classical for m ≥ 0 (Lemma 4.1), and {p  n (x)} n≥0 satisfies the structure relation (see, for instance, [10]) as follows: where ρ(x) 2 = 2 x 2 + 2 1 x + 0 , and ϑ In such a situation, the relation between standard and Sobolev polynomials is shorter than (4.17).

22)
and, if we denote ρ(x) = 2 x + 1 , then Proof. Observe that Using the explicit expression of the Sobolev bilinear form (4.12) and the structure relation (4.19), we deduce that f  Finally, the norms can be computed in a simpler way.
1 , and, for n ≥ 1 (Fig. 3),  where As in Case I, relation (4.20) can be translated to the bivariate case by multiplying by ρ(x) m q m y ρ(x) . In this case, we deduce an expression of a standard bivariate polynomial as a linear combination of three consecutive first-type bivariate Sobolev orthogonal polynomials. n−m,2 S n−2,m (x, y), with S 0,0 (x, y) = P 0,0 (x, y), and S i,j (x, y) = 0 for i < j.

Second Type of Bivariate Sobolev Orthogonal Polynomials
Now, we replace the univariate linear functional acting on the second variable y in the construction described in Sect. 2.3 by a univarite quasi-definite Sobolev bilinear form and study its associated sequence of orthogonal polynomials.
Again, let u ≡ u (x) and v ≡ v (y) be univariate quasi-definite moment functionals acting on the variables x and y, respectively, and let ρ(x) be a function satisfying the conditions of either Case I or Case II above.
Define the univariate Sobolev bilinear form where λ is a real number such that (·, ·) v is quasi-definite. Observe that when v is positive definite and λ ≥ 0, then (5.1) defines an inner product.
Let {s m (y)} m≥0 be the corresponding univariate orthogonal polynomial sequence, standardized in such a way the leading coefficient of s m (y) is the same as the leading coefficient of q m (y), for m ≥ 0. Therefore, s 0 (y) = q 0 (y). In addition, let m . Observe that in Case II the moment functional v is symmetric, and then, the bilinear form (·, ·) v is also symmetric. Moreover, if v is classical, then the only possibilities are Hermite and Gegenbauer moment functionals.
Therefore, we can construct bivariate polynomials as follows: constitute a mutually orthogonal basis with respect to the bivariate Sobolev bilinear form

4)
where λ ∈ R. Moreover, for m ≥ 0, and h for some constants ξ n , σ m , and τ m , with In Case II, from the symmetry, we get σ m = 0, for m ≥ 0. As in the above section, we define the polynomial Then, d dy π m+1 (y) = q m (y), m ≥ 0.

Proposition 5.2. Suppose that v is a classical linear functional. Then,
In Case II, d m,1 = 0, m ≥ 0, and Sobolev orthogonal polynomials {s m (y)} m≥0 are also symmetric polynomials.
Using (5.5) and (5.7) we can deduce a finite relation between the Sobolev orthogonal polynomials and the second family of orthogonal polynomials.
Finally, we translate relation (5.8) to the bivariate case, relating Sobolev orthogonal polynomials to standard bivariate polynomials.
Theorem 5.5. For n ≥ 1 and 0 ≤ m ≤ n, there exist real numbers such that the following relation holds: n−m+2 (x) ρ(x) m , and we study each term of the sum. First, using (5.9) twice, we obtain Suppose that ρ(x) is a polynomial of degree ≤ 1, that is, ρ(x) = r 1 x+r 0 , with |r 1 | + |r 0 | > 0. In this case, using (5.9) and (4.1), we get n−i constants depending on r 1 and r 0 , among other factors. In the same way, Next, for m ≥ 2, we compute the third term of the sum in (5.8). Observe that, in both Cases I and II, ρ(x) 2 is a polynomial of degree less than or equal to 2, and we can denote ρ(x) 2 = s 2 x 2 + s 1 x + s 0 its explicit expression, with s 2 , s 1 , s 0 ∈ R, and |s 2 | + |s 1 | + |s 0 | > 0 (in Case I we have s 2 = r 2 1 , s 1 = 2r 1 r 0 , and s 0 = r 2 0 ). Then, applying twice the three term relation (4.1) for polynomials {p as well as Finally, from the above expressions we get the desired result.

Examples
Here we present examples of bivariate Sobolev orthogonal polynomials and study the involved univariate Sobolev orthogonal polynomials. The first two examples deal with two families of Sobolev orthogonal polynomials on the unit disk obtained by using our construction and Gegenbauer polynomials. Next two examples are devoted to construct Sobolev orthogonal polynomials on the biangle and the simplex, respectively. Then, we present an example defined on an unbounded domain which is based on Laguerre and Hermite classical orthogonal polynomials. In the last two examples we analyse quasi-definite families of first and second-type Sobolev orthogonal polynomials constructed with Bessel and Gegenbauer polynomials.
We will use the standard representation and properties for classical Jacobi and Gegenbauer polynomials considered in the literature (see for instance [1,3,16]).
Here, the bilinear form (3.6) reads From Theorem 3.1, we have that the bivariate polynomials Since u (x) satisfies the Pearson equation for the Gegenbauer moment functional we can use Proposition 4.10. To this end, we need the three-term recurrence relation for Gegenbauer polynomials (4.7.17) in [16, p. 81], and the structure relation (4.19) given in (4.7.27) in [16, p. 83]). Therefore, by Proposition 4.10 and the symmetry of Gegenbauer polynomials, we deduce a relation between classical Gegenbauer polynomials and univariate Sobolev orthogonal polynomials and h (m) 0

Second-Type Sobolev Orthogonal Polynomials on the Unit Disk
Again, we consider the bivariate orthogonal polynomials on the disk {P where u μ is the Gegenbauer moment functional (6.2). Then, by Theorem 5.1, the bivariate polynomials defined by for 0 ≤ m ≤ n, where μ m = μ + m + 1/2, are mutually orthogonal with respect to the bivariate bilinear form (5.4) Since u μ is classical, the sequence of univariate Sobolev polynomials {s m (y)} m≥0 and the Gegenbauer polynomials satisfy relation (5.8). In order to find the coefficients, we need the second structure relation (5.5) for Gegenbauer polynomials that can be found in (4.7.29) in [16, p. 83], and, by the symmetry of the Gegenbauer and the Sobolev polynomials, relation (5.8) reads
n (x)} n≥0 be a sequence of univariate orthogonal polynomials with respect to the bilinear form where u α,βm = x m+1/2 u α,β . Then, by Theorem 3.1, the polynomials defined by are mutually orthogonal with respect to the bivariate bilinear form Observe that u α,β satisfies the Pearson equation Furthermore, by Proposition 4.8, the norms h

Orthogonal Polynomials on the Simplex
For α, β, γ > −1, the polynomials defined as where β m = β + γ + 2m + 1, and P (a,b) n denotes the n-th Jacobi polynomial orthogonal on the interval [0, 1], are mutually orthogonal with respect to the moment functional w defined as . This functional is constructed by taking the univariate Jacobi functionals u (x) = u α,β+γ and v (y) = u β,γ , and the function ρ(x) = 1 − x.
For λ > 0, consider the bivariate Sobolev inner product n (x)} n≥0 be the sequence of orthogonal polynomials associated with the univariate Sobolev inner product are mutually orthogonal with respect to (6.5).
Observe that u (x) satisfies (6.4) and that ρ( Furthermore, by Proposition 4.4, the norms h

Sobolev Orthogonal Polynomials on an Unbounded Domain
Let α > −1. We define the bivariate Laguerre-Hermite Sobolev polynomials as where {L For λ > 0, consider the bivariate Sobolev inner product defined as (P, Q) = w, P Q + λ (∇P ) t x 2 x y x y y 2 ∇Q , P,Q ∈ Π 2 , (6.7) and, for m ≥ 0, let {s (m) n (x)} n≥0 denote the sequence of univariate orthogonal polynomials associated with the univariate Sobolev inner product By Theorem 3.1, the bivariate polynomials defined as are mutually orthogonal with respect to (6.7). The classical Laguerre polynomials are orthogonal with respect to the moment functional which satisfies the Pearson equation in the Pearson equation, we can use Proposition 4.2 to deduce the relation between the Laguerre polynomials and the univariate Sobolev orthogonal polynomials. To this end, using the explicit expression of the Laguerre polynomials ( [16]) we get the three-term recurrence relation , and the structure relation and h (m) 0 Notice that the second type Sobolev orthogonal polynomials as discussed in Sect. 5 based on the Laguerre-Hermite polynomials (6.6) is a straightforward situation taking into account that the Hermite polynomials are orthogonal with respect to both the moment functional v, p = (0) = 1 (see [7]). Note that the Bessel polynomials are associated with the quasi-definite linear functional defined as u a,b , p = 1, p a,b , which satisfies the Pearson equation D x 2 u a,b = (a x + b)