Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations

The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of spherical harmonics, algebraic and differential properties will be deduced.


Introduction
In 1940 (see [8]), H.L. Krall classified orthogonal polynomials satisfying fourth-order differential equations. Apart from the classical orthogonal polynomials we can only find three other families, the now called Jacobi-type, Legendre-type and Laguerre-type polynomials. The corresponding orthogonality measures are some particular cases of classical measures modified by the addition of a Dirac delta at the end points of the interval of orthogonality. Further details on those families of polynomials were given later by A.M. Krall [7], L.L. Littlejohn [10] and T.H. Koornwinder [6].
Our main goal in this work is to extend the study of orthogonal polynomials satisfying fourth-order differential equations to a multidimensional context. In particular, we consider a nonclassical weight function supported on the d-dimensional unit ball and we will show that the corresponding orthogonal polynomials will satisfy a fourth-order partial differential equation.
In our study, classical orthogonal polynomials on the unit ball B d of R d play an essential role. They are orthogonal with respect to the inner product f, g µ := 1 where W µ (x) is the weight function given by W µ (x) := 1 − x 2 µ−1/2 on B d , µ > −1/2, and ω µ is a normalizing constant such that 1, 1 µ = 1. The condition µ > −1/2 is necessary for the inner product to be well defined.
In the present paper, we will consider orthogonal polynomials with respect to the inner product f, g λ µ = 1 where λ > 0, dσ denotes the surface measure on the unit sphere S d−1 , and σ d−1 denotes the area of S d−1 . The presence of the extra spherical term in (1.1) changes the importance of the values on the sphere of a given function f when approximating by means of the corresponding Fourier series. Using spherical harmonics and polar coordinates we shall construct explicitly a sequence of mutually orthogonal polynomials with respect to f, g λ µ , which depends on a family of polynomials of one variable. The latter ones can be expressed in terms of Jacobi polynomials. A similar construction was used in [2] and [11] to obtain a sequence of mutually orthogonal polynomials with respect to a Sobolev inner product. In [2], the inner product includes the outward normal derivatives over the sphere and aside from getting the connection with classical polynomials on the ball, some asymptotic properties are obtained. On the other hand, the Sobolev inner product in [11] was defined in terms of the standard gradient operator.
Then, in the case µ = 1/2, we will show that the real space of polynomials in d variables admits a mutually orthogonal base whose elements are eigenfunctions of a fourth-order linear partial differential operator where the coefficients are independent of the degree of the polynomials. The corresponding eigenvalues depend on the different indices of the polynomials. Thus, we get a non-trivial example of a family of multivariate polynomials orthogonal with respect a measure and satisfying a fourth-order differential equation.
Multivariate polynomials satisfying fourth-order differential equations were constructed previously in [4,Section 6.2]. In this work, the author uses also polar coordinates and spherical harmonics to construct a basis of polynomials which are eigenfunctions of a differential operator where the polynomial coefficients are independent of the indexes of the polynomials. The eigenvalues in [4] depend only on the total degree of the polynomials. However, the polynomials are orthogonal with respect to an inner product involving the spherical Laplacian, that is, a Sobolev product (see [4, equation (6.8)]). Our approach is different, we start from the standard inner product defined in (1.1) and we find a fourth-order partial differential operator having the orthogonal polynomials as eigenfunctions.
The paper is organized as follows. In the next section, we state some materials on orthogonal polynomials on the unit ball and spherical harmonics that we will need later. In Section 3, using spherical harmonics, a basis of mutually orthogonal polynomials associated to (1.1) is constructed. In Section 4, in the case µ = 1/2, we consider the differential properties of the polynomials in the radial parts of the polynomials in the previous section. Finally, in Section 5, we show that these polynomials are eigenfunctions of a fourth-order differential operator where the coefficients do not depend on the degree of the polynomials.

Classical orthogonal polynomials on the ball
In this section we describe background materials on orthogonal polynomials and spherical harmonics. The first subsection collects properties on the Jacobi polynomials that we shall need later. The second subsection is devoted to the Jacobi-type orthogonal polynomials, associated to a weight function obtained as a linear combination of the classical Jacobi weight function and a delta function at 1. The third subsection recalls the basic results on spherical harmonics and classical orthogonal polynomials on the unit ball.

Classical Jacobi polynomials
We collect some properties of the classical Jacobi polynomials P (α,β) n (t), all of them can be found in [12]. For α, β > −1, these polynomials are orthogonal with respect to the Jacobi inner The Jacobi polynomial P (α,β) n (t) is normalized by The derivative of a Jacobi polynomial is again a Jacobi polynomial The polynomials P (α,β) n (t) satisfy the second-order differential equation where y(t) = P (α,β) n (t).

Orthogonal polynomials on the unit ball and spherical harmonics
The number |ν| = ν 1 + · · · + ν d is called the total degree of x ν . A polynomial P in d variables is a finite linear combination of monomials. Let Π d denote the space of polynomials in d real variables. For a given nonnegative integer n, let Π d n denote the linear space of polynomials in several variables of (total) degree at most n and let P d n denote the space of homogeneous polynomials of degree n. The unit ball and the unit sphere in R d are denoted, respectively, by where x denotes as usual the Euclidean norm of x.
and consider the inner product which is normalized so that 1, 1 µ = 1. A polynomial P ∈ Π d n is called orthogonal with respect to W µ on the ball if P, Q µ = 0 for all Q ∈ Π d n−1 , that is, if it is orthogonal to all polynomials of lower degree. Let V d n (W µ ) denote the space of orthogonal polynomials of total degree n with respect to W µ .
For n ≥ 0, let {P n ν (x) : |ν| = n} denote a basis of V d n (W µ ). Notice that every element of V d n (W µ ) is orthogonal to polynomials of lower degree. If the elements of the basis are also orthogonal to each other, that is, P n ν , P n η µ = 0 whenever ν = η, we call the basis mutually orthogonal. If, in addition, P n ν , P n ν µ = 1, we call the basis orthonormal. Harmonic polynomials of degree n in d-variables are polynomials in P d n that satisfy the Laplace Let H d n denote the space of harmonic polynomials of degree n. It is well known that Spherical harmonics are the restriction of harmonic polynomials to the unit sphere. If Y ∈ H d n , then in spherical-polar coordinates x = rξ, r > 0 and ξ ∈ S d−1 , we get so that Y is uniquely determined by its restriction to the sphere. We shall also use H d n to denote the space of spherical harmonics of degree n.
Let dσ denote the surface measure and σ d−1 denote the surface area, Spherical harmonics of different degrees are orthogonal with respect to the inner product If Y (x) is a harmonic polynomial of degree n, by Euler's equation for homogeneous polynomials, we deduce In spherical-polar coordinates x = rξ, r > 0 and ξ ∈ S d−1 , the differential operators ∆ and x, ∇ can be decomposed as follows (cf. [1]):

5)
x, ∇ = r ∂ ∂r . (2.6) The operator ∆ 0 , which is the spherical part of the Laplacian, is called the Laplace-Beltrami operator and it has spherical harmonics as its eigenfunctions. More precisely, it holds (cf. [1]) In spherical-polar coordinates x = rξ, r > 0 and ξ ∈ S d−1 , a mutually orthogonal basis of V d n (W µ ) can be given in terms of the Jacobi polynomials and spherical harmonics (see, for instance, [3]).
It is known that orthogonal polynomials with respect to W µ are eigenfunctions of a secondorder differential operator D µ . More precisely, we have

An inner product on the ball with an extra spherical term
Let us define the inner product where λ > 0, dσ denotes the surface measure on the unit sphere S d−1 , σ d−1 denotes the area of S d−1 and ω µ is the normalizing constant (2.3). As a consequence of the central symmetry of the inner product, we can use a construction analogous to (2.8) to obtain a basis of mutually orthogonal polynomials with respect to f, g λ µ . This time, the radial parts constitute a family of polynomials in one variable related to Jacobi polynomials. (t) be the k-th orthogonal polynomial with respect to Then the polynomials with 1 ≤ k ≤ n/2, 1 ≤ ν ≤ a d n−2k constitute a mutually orthogonal basis of V d n (W µ , λ) the linear space of orthogonal polynomials of degree exactly n with respect to ·, · λ µ .
Proof . The proof of this theorem uses the following well known identity that arises from the spherical-polar coordinates x = rξ, ξ ∈ S d−1 .
In order to check the orthogonality, we need to compute the product Let us start with the computation of the first integral Using polar coordinates, relation (3.2), and the orthogonality of the spherical harmonics we obtain Finally, the change of variables t = 2r 2 − 1 moves the integral to the interval [−1, 1], Let us now compute the second integral in (3.3), To end the proof, we just have to put together (3.4) and (3.5) to get the value of (3.3) in terms of the inner product (3.1) as And the result follows from the orthogonality of the univariate polynomial q where Then they are orthogonal with respect to the inner product
Now consider the case j = 0. Then Next, an iterated integration by parts reduces the integral in the last expression to a beta integral which finally gives and the result follows from (3.7).
The partial differential equation (2.9) satisfied by classical orthogonal polynomials on the ball {P n k,ν (x)} can be deduced from the second-order differential equation corresponding to the Jacobi polynomials in the radial parts and the fact that spherical harmonics are the eigenfunctions of the Laplace-Beltrami operator.
In the case µ = 1/2, we are going to identify the polynomials in the radial parts of {Q n k,ν (x)} as a family of polynomials satisfying a fourth-order differential equation. Next, using polar coordinates and spherical harmonics we will obtain the explicit expression of a fourth-order partial differential operator having the multivariate orthogonal polynomials as eigenfunctions.

The case µ = 1/2
In this case, the inner product reduces to The mutually orthogonal basis is given by } an orthonormal basis of spherical harmonics. The polynomials in the radial part q (0,β k ,λ) k are orthogonal with respect to the inner product Finally, using (2.3) and (2.4) we get σ d−1 /ω 1/2 = d, and writing M = d 2λ we conclude that they are orthogonal with respect to Therefore, we can recognize q and satisfy the normalization q β,M k (1) = M . Using the differential equation for the Jacobi polynomials {P (0,β) k (t)} k 0 (2.2), relation (4.1) can be written as Therefore, both families of orthogonal polynomials are connected by means of a second-order linear differential operator whose polynomial coefficients are independent of the degree of the polynomials. If we denote by dµ β the measure defined by integrating by parts we can deduce that for arbitrary polynomials f and g we get Therefore, using the orthogonality properties of P = (M + k(k + β))(M + (k + 1)(k + β + 1))P (0,β) k (t), (4.4) where the coefficient of P (0,β) k (t) has been obtained identifying the leading coefficients in both sides of (4.4).
Finally, if we combine (4.2) and (4.4) we deduce that the polynomials q β,M k (t) are the eigenfunctions of a fourth-order differential operator with polynomials coefficients not depending on k. The quantities (M + k(k + β))(M + (k + 1)(k + β + 1)) are the corresponding eigenvalues.
Summarizing, we have got the following proposition. Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations 9 Then, the Jacobi-type polynomials q β,M k (t) satisfy