Abstract
Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.
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1 Introduction
Let \(\Omega \) be a wedge on the plane that consists of two line segments sharing a common endpoint. For a positive measure \(\mathrm{d}\mu \) defined on \(\Omega \), we study orthogonal polynomials of two variables with respect to the bilinear form
We also study orthogonal polynomials on the boundary of a parallelogram. Without loss of generality, we can assume that our wedge is of the form
and consider the bilinear form defined by
Since \(\Omega \) is a subset of the zero set of a quadratic polynomial \(l_1(x,y) l_2(x,y)\), where \(l_1\) and \(l_2\) are linear polynomials, the structure of orthogonal polynomials on \(\Omega \) is very different from that of ordinary orthogonal polynomials in two variables [4] but closer to that of spherical harmonics. The latter are defined as homogeneous polynomials that satisfy the Laplace equation \(\Delta Y = 0\) and are orthogonal on the unit circle, which is the zero set of the quadratic polynomial \(x^2 + y^2-1\). The space of spherical polynomials of degree n has dimension 2 for each \(n \ge 1\) and, furthermore, one basis of spherical harmonics when restricted on the unit circle are \(\cos n \theta \) and \(\sin n \theta \), in polar coordinates \((r,{\theta })\), and the Fourier orthogonal expansions in spherical harmonics coincide with the classical Fourier series.
In Sect. 2, we consider orthogonal polynomials on a wedge. The space of orthogonal polynomials of degree n has dimension 2 for each \(n \ge 1\), just like that of spherical harmonics, and they satisfy the equation \(\partial _1 \partial _2 Y = 0\). The main results are
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An explicit expression in terms of univariate orthogonal polynomials when \(w_1(x) = w_2(x) =w(x)\) where w is any weight function on [0, 1] (Theorem 2.2),
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Sufficient conditions for pointwise and uniform convergence (Theorem 2.4), as well as normwise convergence (Corollary 2.5),
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Explicit expression in terms of Jacobi polynomials when \(w_1(x) = w_{{\alpha },{\gamma }}(x)\) and \(w_2(x) = w_{{\beta },{\gamma }}(x)\) (Theorem 2.7),
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Sufficient conditions for normwise convergence (Theorem 2.9).
In Sect. 3 we study orthogonal polynomials on the boundary of a parallelogram, which we can assume as the square \([-1,1]^2\) without loss of generality. For a family of generalized Jacobi weight functions that are symmetric in both x and y, we are able to deduce an orthogonal basis in terms of four families of orthogonal bases on the wedge in Theorem 3.2. Furthermore, the convergence of the Fourier orthogonal expansions can also be deduced in this fashion, as shown in Theorem 3.3.
In Sect. 4 we use orthogonal polynomials on the boundary of the square to construct an orthogonal basis for the weight function \(w(\max \{|x|,|y|\})\) on the square \([-1,1]^2\). This mirrors the way in which spherical harmonics can be used to construct a basis of orthogonal polynomials for the weight function \(w(\sqrt{x^2+y^2})\) on the unit disc. However, unlike the unit disc, the orthogonal basis we constructed are no longer polynomials in x, y but are polynomials of x, y and \(s=\max \{|x|,|y|\}\).
The study is motivated by applications. In particular, we wish to investigate how the applications of univariate orthogonal polynomials can be translated to multivariate orthogonal polynomials on curves. As a motivating example, univariate orthogonal polynomials give rise to a determinantal point process that is linked to the eigenvalues of unitary ensembles of random matrix theory. In Sect. 5, we investigate the statistics of the determinantal point process generated from orthogonal polynomials on the wedge, and find experimentally that they have the same local behaviour as a Coulomb gas away from the corners/edges.
In “Appendix A”, we give the Jacobi operators associated with a special case of weights on the wedge, whose entries are rational. Finally, in “Appendix B” we show that the Stieltjes transform of our family of orthogonal polynomials satisfies a recurrence that can be built out of the Jacobi operators of the orthogonal polynomials, which can in turn be used to compute Stieltjes transforms numerically. This is a preliminary step towards using these polynomials for solving singular integral equations.
2 Orthogonal Polynomials on a Wedge
Let \({\mathcal P}_n^2\) denote the space of homogeneous polynomials of degree n in two variables; that is, \({\mathcal P}_n^2 =\mathrm {span}\, \{x^{n-k}y^k: 0 \le k \le n\}\). Let \(\Pi _n^2\) denote the space of polynomials of degree at most n in two variables.
2.1 Orthogonal Structure on a Wedge
Given three non-collinear points, we can define a wedge by fixing one point and joining it to other points by line segments. We are interested in orthogonal polynomials on the wedge. Since the three points are non-collinear, each wedge can be mapped to
by an affine transform. Since the polynomial structure and the orthogonality are preserved under the affine transform, we can work with the wedge \(\Omega \) without loss of generality. Henceforth we work only on \(\Omega \).
Let \(w_1\) and \(w_2\) be two nonnegative weight functions defined on [0, 1]. We consider the bilinear form define on \(\Omega \) by
Let I be the polynomial ideal of \({\mathbb R}[x,y]\) generated by \((1-x)(1-y)\). If \(f \in I\), then \({\langle }f, g{\rangle }_{w_1,w_2} =0\) for all g. The bilinear form defines an inner product on \(\Pi _n^2\), modulo I, or equivalently, on the quotient space \({\mathbb R}[x,y]/I\).
Proposition 2.1
Let \({\mathcal H}_n^2(w_1,w_2)\) be the space of orthogonal polynomials of degree n in \({\mathbb R}[x,y]/I\). Then
Furthermore, we can choose polynomials in \({\mathcal H}_n^2(w_1,w_2)\) to satisfy \(\partial _x \partial _y p = 0\).
Proof
Since \((1-x)(1-y) {\mathcal P}_{n-2}\) is a subset of I, the dimension of \(\dim {\mathcal H}_n^2(w_1,w_2) \le 2\). Applying the Gram–Schmidt process on \(\{1, x^k, y^k, k \ge 1\}\) shows that there are two orthogonal polynomials of degree exactly n. Both these polynomials can be written in the form of \(p(x) + q(y)\), since we can use \(xy \equiv x+y-1\) mod I to remove all mixed terms. Evidently such polynomials satisfy \(\partial _x \partial _y (p(x) + q(y))=0\). \(\square \)
In the next two subsections, we shall construct an orthogonal basis of \({\mathcal H}_n^2(w_1,w_2)\) for certain \(w_1\) and \(w_2\) and study the convergence of its Fourier orthogonal expansions. We will make use of results on orthogonal polynomials of one variable, which we briefly record here.
For w defined on [0, 1], we let \(p_n(w)\) denote an orthogonal polynomial of degree n with respect to w, and let \(h_n(w)\) denote the norm square of \(p_n(w)\),
Let \(L^2(w)\) denote the \(L^2\) space with respect to w on [0, 1]. The Fourier orthogonal expansion of \(f \in L^2(w)\) is defined by
The Parseval identity implies that
The n-th partial sum of the Fourier orthogonal expansion with respect to w can be written as an integral
where \(k_n(w)\) denotes the reproducing kernel defined by
2.2 Orthogonal Structure for \(w_1=w_2\) on a Wedge
In the case of \(w_1 = w_2 = w\), we denote the inner product (2.1) by \({\langle }\cdot ,\cdot {\rangle }_w\) and the space of orthogonal polynomials by \({\mathcal H}_n^2(w)\). In this case, an orthogonal basis for \({\mathcal H}_n^2(w)\) can be constructed explicitly.
Theorem 2.2
Let w be a weight function on [0, 1] and let \(\phi w(x): = (1-x)^2 w(x)\). Define
Then \(\{P_n, Q_n\}\) are two polynomials in \({\mathcal H}_n^2(w)\) and they are mutually orthogonal. Furthermore,
Proof
Since \(P_n(x,1) = P_n(1,x)\) and \(Q_n(x,1) = - Q_n(1,x)\), it follows that
for \(n \ge 0\) and \(m \ge 1\). Furthermore,
by the orthogonality of \(p_n(w)\). Similarly,
The proof is completed. \(\square \)
Let \(L^2(\Omega ,w)\) be the space of Lebesgue measurable functions with finite
norms. For \(f \in L^2(\Omega , w)\), its Fourier expansion is given by
where \(P_n\) and \(Q_n\) are defined in Theorem 2.2 and
The partial sum operator \(S_n f\) is defined by
which can be written in terms of an integral in terms of the reproducing kernel \(K_n(\cdot ,\cdot )\),
where
We show that this kernel can be expressed, when restricted on \(\Omega \), in terms of the reproducing kernel \(k_n(w;\cdot ,\cdot )\) defined at (2.3).
Proposition 2.3
The reproducing kernel \(K_n(\cdot ,\cdot )\) for \({\langle }\cdot ,\cdot {\rangle }_w\) satisfies
Proof
The other case is established similar, using \(Q_k(1,y) = - (1-y) p_{k-1}(\phi w; y)\). \(\square \)
It is well-known that the kernel \(k_n(w; \cdot ,\cdot )\) satisfies the Christoffel–Darboux formula, which plays an important role for the study of Fourier orthogonal expansion. Our formula allows us to write down an analogue of Christoffel–Darboux formula for \(K_n(\cdot ,\cdot )\), but we can derive convergence directly.
Theorem 2.4
Let f be a function defined on \(\Omega \). Define
Then
In particular, if \(s_n(w;f_e,x) \rightarrow f_e(x)\) and \(s_n(\phi w;f_o,x) \rightarrow f_o(x)\), pointwise or in the uniform norm as \(n\rightarrow \infty \), then \(S_n f(x)\) converges to f(x) likewise.
Proof
By our definition,
Similarly,
Moreover, since \(f_e(x) + (1-x) f_o(x) = f(x,1)\) and \(f_e(x) - (1-x) f_o(x) = f(1,x)\), it follows that
from which we see that the convergence of \(s_n(w;f_e)\) and \(s_n(\phi w; f_o)\) imply the convergence of \(S_n f\). \(\square \)
Since \(f\in L^2(\Omega ,w)\), it is evident that \(f_e \in L^2(w)\). Moreover, \(f_o \in L^2(\phi w)\) since
In particular, \(s_n (w, f_e)\) and \(s_n(\phi w; f_o)\) converge to \(f_e\) and \(f_o\) in \(L^2(w)\) and in \(L^2(\phi w)\), respectively.
Corollary 2.5
If \(f\in L^2(\Omega , w)\), then
Proof
By the displayed formulas at the end of the proof of the last theorem and
it is easy to see that
where we have used the identity \((a+b)^2 + (a-b)^2 = 2 (a^2+b^2)\). \(\square \)
2.3 Orthogonal Structure on a Wedge with Jacobi Weight Functions
For \({\alpha },{\gamma }> -1\), let \(w_{{\alpha },{\gamma }}\) be the Jacobi weight function defined by
We consider the inner product \({\langle }\cdot ,\cdot {\rangle }_{w_1,w_2}\) defined in (2.1) with \(w_1(x) = w_{{\alpha },{\gamma }}(x)\) and \(w_2(x) = w_{{\beta },{\gamma }}(x)\). More specifically, for \({\alpha },{\beta },{\gamma }> -1\) and \(\sigma > 0\), we define
where
2.3.1 Orthogonal Structure
We need to construct an explicit basis of \({\mathcal H}_n^2(w_{{\alpha },{\gamma }},w_{{\beta },{\gamma }})\). The case \({\alpha }= {\beta }\) can be regarded as a special case of Theorem 2.2. The case \({\alpha }\ne {\beta }\) is much more complicated, for which we need several properties of the Jacobi polynomials.
Let \(P_n^{({\alpha },{\beta })}\) denote the usual Jacobi polynomial of degree n defined on \([-1,1]\). Then \(P_n^{({\gamma },{\alpha })}(2x-1)\) is an orthogonal polynomial with respect to \(w_{{\alpha },{\gamma }}\) on [0, 1]. Moreover,
by [12, (4.3.3)]. Furthermore, \(P_n^{({\alpha },{\beta })}(1) = \left( {\begin{array}{c}n+{\alpha }\\ n\end{array}}\right) \) and, in particular, \(P_n^{(0,{\beta })}(1) =1\). Our construction relies on the following lemma.
Lemma 2.6
For \(m > n \ge 0\),
Proof
Since \(P_n^{({\gamma },{\alpha })}(2x-1)\) is an orthogonal polynomial of degree n with respect to \((1-x)^{\gamma }x^{{\alpha }}\) on [0, 1], \(I_{m,n}^{{\gamma },{\alpha }} = 0\) for \(n > m\) holds trivially. For \(m \ge n\), we need two identities of Jacobi polynomials. The first one is, see [12, (4.5.4)] or [9, (18.9.6)],
and the second one is the expansion, see [9, (18.18.14)],
Putting them together shows that
Substituting this expression into \(I_{m,n}^{{\gamma },{\alpha }}\) and using the orthogonality of the Jacobi polynomials and (2.10), we conclude that, for \(m-1 \ge n\),
Hence, the case \(m > n\) follows. The same argument works for the case \(n =m\). \(\square \)
What is of interest for us is the fact that the dependence of \(I_{m,n}^{{\gamma },{\alpha }}\) on n and \({\alpha }\) is separated, which is critical to prove that \(Q_n\) in the next theorem is orthogonal.
Theorem 2.7
Let \(P_0(x,y) =1\) and, for \(n =1,2,\ldots \), define
Then \(\{P_n, Q_n\}\) are two polynomials in \({\mathcal H}_n^2(w_{{\alpha },{\gamma }},w_{{\beta },{\gamma }})\) and
In particular, the two polynomials are orthogonal to each other if \({\beta }= {\alpha }\). Furthermore
Proof
Since \(P_n^{({\gamma },{\alpha })}(1) = P_n^{({\gamma },{\beta })}(1) = \left( {\begin{array}{c}n+{\gamma }\\ n\end{array}}\right) \), our definition shows that
By the identity in the previous lemma, if \(n > m\), then \( {\langle }P_n, Q_m {\rangle }_{{\alpha },{\beta },{\gamma }} =0\) since both \(I_{m,n}^{{\alpha },{\gamma }}=0\) and \(I_{m,n}^{{\beta },{\gamma }} =0\), whereas if \(n < m\), then
The case \(n = m\) follows from a simple calculation. Moreover, for \(m \ne n\),
by the orthogonality of the Jacobi polynomials, and it is equal to \(h_n^{{\gamma },{\alpha }} + h_n^{{\gamma },{\beta }}\) for \(m=n\). Similarly,
To derive the norm of \({\langle }Q_n,Q_n{\rangle }\), we need to use \(c_{{\gamma },{\alpha }} = ({\gamma }+1)_2/({\alpha }+{\gamma }+2)_2 c_{{\gamma }+2,{\alpha }}\). The proof is completed. \(\square \)
Corollary 2.8
For \(n =1,2,\ldots \), define
Then, for \({\alpha }\ne {\beta }\), \(\{P_n, R_n\}\) are two polynomials in \({\mathcal H}_n^2(w_{{\alpha },{\beta },{\gamma }})\) and they are mutually orthogonal. Moreover,
2.3.2 Fourier Orthogonal Expansions
Let \(L^2(\Omega , w_{{\alpha },{\gamma }}, w_{{\beta },{\gamma }})\) be the space of functions defined on \(\Omega \) such that f(1, 1) is finite and the norm
is finite for every f in this space. For \( f\in L^2(\Omega , w_{{\alpha },{\gamma }}, w_{{\beta },{\gamma }})\) we consider the Fourier orthogonal expansion with respect to \({\langle }\cdot ,\cdot {\rangle }_{{\alpha },{\beta },{\gamma }}\). With respect to the orthogonal basis \(\{P_n,R_n\}\) in Theorem 2.7 and Corollary 2.8, the Fourier orthogonal expansion is defined by
where
Its n-th partial sum is defined by
In this case, we do not have a closed form for the reproducing kernel with respect to \({\langle }\cdot ,\cdot {\rangle }_{{\alpha },{\beta },{\gamma }}\). Nevertheless, we can relate the convergence of the Fourier orthogonal expansions to that of the Fourier–Jacobi series. For \(w_{{\alpha },{\gamma }}\), we denote the partial sum defined in (2.2) by \(s_n^{{\alpha },{\gamma }} f\).
For f defined on \(\Omega \), we define \(f_1(x) = f(x,1)\) and \(f_2(x) = f(1,x)\), and
It is easy to see that if \(f(\cdot ,1) \in L^2(w_{{\alpha },{\gamma }},[0,1])\), then \(g_1 \in L^2(w_{{\alpha },{\gamma }+2}, [0,1])\), and if \(f(1,\cdot ) \in L^2(w_{{\beta },{\gamma }},[0,1])\), then \(g_2 \in L^2(w_{{\beta },{\gamma }+2}, [0,1])\).
Theorem 2.9
Let \({\alpha }, {\beta }, {\gamma }> -1\). Then the Fourier orthogonal expansion converges in \(f\in L^2(\Omega , w_{{\alpha },{\gamma }}, w_{{\beta },{\gamma }})\). Furthermore, for \(f(\cdot ,1) \in L^2(w_{{\alpha },{\gamma }})\) and \(f(1,\cdot ) \in L^2(w_{{\beta },{\gamma }})\),
where c is a constant that depends only on \({\alpha },{\beta },{\gamma }\).
Proof
Since polynomials are dense on \(\Omega \), by the Weierstrass theorem, the orthogonal basis \(\{P_n,R_n\}\) is complete, so that the Fourier orthogonal expansion converges in \(L^2(\Omega , w_{{\alpha },{\gamma }}, w_{{\beta },{\gamma }})\). By the Parseval identity,
Throughout this proof we use the convention \(A \sim B\) if \(c_1 B \le A \le c_2 A\), where \(c_1\) and \(c_2\) are constants that are independent of varying parameters in A and B. By (2.10) and the fact that \(\Gamma (n+{\alpha }+1)/n! \sim n^{\alpha }\), it is easy to see that \(h_n^{{\alpha },{\gamma }} \sim n^{-1}\), so that
and, consequently,
The Fourier–Jacobi coefficients of \(f_1\) and \(f_2\) are denoted by \(\widehat{f_1}_n^{{\alpha },{\gamma }}\) and \(\widehat{f_2}_n^{{\beta },{\gamma }}\), respectively. It follows readily that \(\widehat{f}_{P_n} \sim \widehat{f_1}_n^{{\alpha },{\gamma }} + \widehat{f_2}_n^{{\beta },{\gamma }}\), consequently,
We now consider the estimate for \(R_n\) part. By the definition of \(R_n\),
It is easy to see that
so that we only have to work with the term \({\langle }f, Q_k {\rangle }_{{\alpha },{\beta },{\gamma }}\). The definition of \(Q_k\) shows that \({\langle }1, Q_k{\rangle }_{{\alpha },{\beta },{\gamma }} = 0\), which leads to the identity
Consequently, it follows that
The proof is completed. \(\square \)
3 Orthogonal Polynomials on the Boundary of the Square
Using the results in the previous section, we can study orthogonal polynomials on a parallelogram. Since orthogonal structure is preserved under an affine transformation, we can assume without loss of generality that the parallelogram is the square \([-1,1]^2\).
For \({\alpha },{\gamma }> -1\), let \(\varpi _{{\alpha },{\gamma }}\) be the weight function
We consider orthogonal polynomials of two variables on the boundary of \([-1,1]^2\) with respect to the bilinear form
for \({\alpha },{\beta },{\gamma }> -1\). Since \((1-x^2)(1-y^2)\) vanishes on the boundary of the square, the bilinear form defines an inner product modulo the ideal generated by this polynomial, or in the space
Let \({\mathcal B}{\mathcal V}_n^2\) denote the space of orthogonal polynomials in \({\mathbb R}[x,y]/I\) with respect to the inner product \({\langle }\cdot ,\cdot {\rangle }\).
Proposition 3.1
For \(n \ge 0\), the dimension of \({\mathcal B}{\mathcal V}_n^2\) is given by
Recall that the inner product \({\langle }\cdot ,\cdot {\rangle }_{{\alpha },{\beta },{\gamma }}\) studied in the previous section contains a fixed parameter \(\sigma \). For fixed \({\alpha },{\beta }\) and \({\delta }_1,{\delta }_2 \in \{0,1\}\), we define \(p_{m,1}^{{\alpha }+{\delta }_1,{\beta }+{\delta }_2,{\gamma }}\) and \(p_{m,2}^{{\alpha }+{\delta }_1,{\beta }+{\delta }_2,{\gamma }}\) to be a basis of \({\mathcal H}_m^2(w_{{\alpha }+{\delta }_1,{\gamma }},w_{{\beta }+{\delta }_2,{\gamma }})\) for a particular choice of \(\sigma \) defined by
For example, \(p_{m,i}^{{\alpha },{\beta },{\gamma }}\) are defined with \(\sigma _{0,0} =1\) and \(p_{m,i}^{{\alpha }+1,{\beta },{\gamma }}\) are defined with \(\sigma _{1,1} = ({\alpha }+{\gamma }+2)/({\alpha }+1)\). For each pair of \({\alpha }+{\delta }_1, \, {\beta }+{\delta }_2\), we can choose, for example, \(p_{m,1}^{{\alpha }+{\delta }_1,{\beta }+{\delta }_2,{\gamma }} = P_m\) defined in (2.12) and take \(p_{m,2}^{{\alpha }+{\delta }_1,{\beta }+{\delta }_2,{\gamma }} = Q_m\) defined in (2.13) or \(p_{m,2}^{{\alpha }+1,{\beta }+1,{\gamma }} = R_m\) defined in (2.15).
Theorem 3.2
For \(n = 0, 1,2,\) a basis for \({\mathcal B}{\mathcal V}_n\) is denoted by \(Y_{n,i}\) and given by
For \(n \ge 3\), the four polynomials in \({\mathcal B}{\mathcal V}_n^2\) that are linearly independent modulo the ideal can be given by
for \(n =2m \ge 2\), and
for \(n=2m+1 \ge 3\). In particular, these bases satisfy the equation \(\partial _x^2 \partial _y^2 u = 0\).
Proof
The proof relies on the parity of the integrals. For example, it is easy to see that \({\langle }x f(x^2,y^2), g(x^2,y^2) {\rangle }=0\) and \({\langle }y f(x^2,y^2), g(x^2,y^2) {\rangle }=0\) for any polynomials f and g, which implies, in particular, that \({\langle }Y_{2m,i}, Y_{2n+1,j}{\rangle }=0\) for \(i,j = 1,2,3,4\). Furthermore, it is easy to see that \({\langle }x y f(x^2,y^2), g(x^2,y^2){\rangle }= 0\) for any polynomials f and g. Hence, \({\langle }Y_{2m,i},Y_{2k,j}{\rangle }=0\) for \(i =1,2\) and \(j=3,4\). Furthermore, using the relation
it is easy to see that
where in the second identity, we have adjusted the normalization constants of integrals from \(c_{{\alpha },{\gamma }}\) and \(c_{{\beta },{\gamma }}\) to \(c_{{\alpha }+1,{\gamma }}\) and \(c_{{\beta }+1,{\gamma }}\), respectively, and used our choice of \(\sigma _{1,1}\). Hence, with our choice of \(\sigma _{0,0}\) and \(\sigma _{1,1}\), we see that \(Y_{2m,i}\) is orthogonal to \(Y_{2k,j}\) for \(i,j =1,2\) and \(i,j = 3,4\), respectively. Similarly, by the same consideration, we obtain that
which shows, with our choice of \(\sigma _{0,1}\) and \(\sigma _{1,0}\), that \(Y_{2m+1,i}\) is orthogonal to \(Y_{2k+1,j}\) for \(i,j =1,2\) and \(i,j = 3,4\), respectively. Finally, since \(\partial _x \partial _y p_{n,i}^{{\alpha },{\beta }}(x,y) =0\), we see that \(Y_{n,j} = \xi (x,y) u(x) + \eta (x,y) v(x)\), where \(\xi \) and \(\eta \) are linear polynomial of x, y, so that it is evident that \(\partial _x^2 \partial _y^2 Y_{n,j}(x,y)=0\). \(\square \)
In our notation, the case \({\alpha }= -\frac{1}{2}\)\({\beta }= - \frac{1}{2}\) and \({\gamma }=0\) corresponds to the inner product in which the integrals are unweighted.
Let \(L^2([-1,1]^2, \varpi _{{\alpha },{\gamma }}, \varpi _{{\beta },{\gamma }})\) be the space of functions defined on the boundary of \([-1,1]^2\) such that \(f(\pm 1, \pm 1)\) are finite and the norm
is finite for every f. For \(f \in L^2([-1,1]^2, \varpi _{{\alpha },{\gamma }}, \varpi _{{\beta },{\gamma }})\), its Fourier orthogonal expansion is defined by
For \(n \ge 2\), let \(S_n (f)\) denotes its n-th partial sum defined by
For fixed \({\alpha },{\beta },{\gamma }\), let \({\langle }\cdot ,\cdot {\rangle }_{{\alpha }+\delta _1,{\beta }+\delta _2,{\gamma }}\) be the inner product defined in the previous section with \(\sigma = \sigma ^{{\alpha },{\beta },{\gamma }}\). For f defined on \([-1,1]^2\), we define four functions
where the subindices indicate the parity of the function. For example, \(F_{e,o}\) is even in x variable and odd in y variable. By definition,
We further define
and define \(\psi : {\mathbb R}^2 \mapsto {\mathbb R}^2\) by \(\psi : (x,y) \mapsto (\sqrt{x},\sqrt{y})\). Changing variables in integrals as in (3.3), we see that if \(f\in L^2([-1,1]^2, \varpi _{{\alpha },{\gamma }}, \varpi _{{\beta },{\gamma }})\), then \(G_{\delta _1,\delta _2}\circ \psi \in L^2({\mathcal B}, w_{{\alpha }+\delta _1,{\gamma }}, w_{{\beta }+\delta _2,{\gamma }})\) for \(\delta _i \in \{0,1\}\).
Theorem 3.3
For \(f\in L^2([-1,1]^2, \varpi _{{\alpha },{\gamma }}, \varpi _{{\beta },{\gamma }})\),
In particular, the norm of \(S_n f - f\) is bounded by those of \(S_{m}^{{\alpha }+{\delta }_1,{\beta }+{\delta }_2,{\gamma }} G_{{\delta }_1,{\delta }_2} - G_{{\delta }_1,{\delta }_2}\) as in Theorem 2.9.
Proof
Using the parity of the function, it is easy to see that
where we have used the fact that \(F_{e,e}\) is even in both variables and use the change of variables in integrals as in (3.3). The similar procedure can be used in the other three cases, as \(G_i(x,y)\) is even in both variables, and the result is
Since \(S_n^{{\alpha }+{\delta }_1, {\beta }+ {\delta }_2, {\gamma }} G_{{\delta }_1,{\delta }_2} \circ \psi (x^2, y^2) \rightarrow G_{{\delta }_1,{\delta }_2} (x,y)\) and
the last statement is evident. \(\square \)
4 Orthogonal System on the Square
Let w be a nonnegative weight function defined on [0, 1]. Define
We construct a system of orthogonal functions with respect to the inner product
by making use of the orthogonal polynomials on the boundary or the square, studied in the previous section. Our starting point is the following integral identity derived from changing variables \((x,y) \mapsto (s \xi ,s\eta )\),
where \(\int _{\mathcal B}\mathrm {d}\sigma \) denotes the integral on the boundary of the square,
Our orthogonal functions are similar in structure to orthogonal polynomials on the unit disc that are constructed using spherical harmonics. However, these function are polynomials in \((s, \xi ,\eta )\) for the \((x,y) = (s \xi , s\eta ) \in [-1,1]^2\), but not polynomials in (x, y).
Let \({\mathcal B}{\mathcal V}_n^2\) be the space of orthogonal polynomials on the boundary of \([-1,1]^2\) with respect to the inner product
which is the inner product with \({\alpha }= -\frac{1}{2}\), \({\beta }= - \frac{1}{2}\) and \({\gamma }=0\) studied in the previous section. Let \(Y_{n,i}\) be an orthogonal basis for \({\mathcal B}{\mathcal V}_n^2\). For \(n \le 2\), they are defined by, see Theorem 3.2,
whereas for \(n \ge 3\), they are constructed in Theorem 3.2. For \(n \ge k \), denote by \(P_{m, 2n-2k}\) the orthogonal polynomial of degree m with respect to \(t^{2n-2k+1} w(t)\) on [0, 1] and with \(P_{0,2n-2k}(s):=1\). For \(n \ge 0\) and \(0 \le k \le n\), we define
where \(i = 1,\ldots , \min \{n+1,4\}\).
Theorem 4.1
In the coordinates \((x,y) = s(\xi ,\eta )\), the system of functions
is a complete orthogonal basis for \(L^2(W; [-1,1]^2)\).
Proof
Changing variables \(x = s \xi \) and \(y= \eta \) shows
The second integral is zero if \(i \ne j\) and \(n-k \ne m-l\), whereas the second integral is zero when \(n-k = m-l\) and \(k \ne l\), so that \({\langle }Q_{k,i}^n, Q_{l,j}^m {\rangle }_{W} =0\) if \( i\ne j\), \(k\ne l\) and \(n \ne m\). By definition, \(s^{n-k} Y_{n-k,i}\left( \frac{\xi }{s}, \frac{\eta }{s} \right) \) is a polynomial of degree \(n-k\) in the variable s, so that \(Q_{k,i}^n\) is a polynomial of degree n. To show that the system is complete, we show that if \({\langle }f, Q_{k,i}^n {\rangle } =0\) for all k, i, n, then \(f(x,y)=0\). Indeed, by the orthogonality of polynomials on the boundary, we see that
modulo the ideal. Changing order of summation shows that
This completes the proof. \(\square \)
5 Sampling the Associated Determinantal Point Process
Associated with an orthonormal basis \(q_0(x),\ldots ,q_N(x)\) is a determinantal point process, which describes N points \(\lambda _1,\ldots ,\lambda _N\) distributed according to
where
is the reproducing kernel, see [1] for an overview of determinantal point processes.
In the particular case of univariate orthogonal polynomials with respect to a weight w(x), the associated determinantal process is equivalent to a Coulomb gas—that is, the points are distributed according to
where \(Z_N\) is the normalization constant—as well as the eigenvalues of unitary ensembles, see for example [3] for the case of an analytic weight on the real line or [8] for the case of a weight supported on \([-1,1]\) with Jacobi-like singularities.
In the case of our orthogonal polynomials on the wedge, the connection with Coulomb gases and random matrix theory is no longer obvious: the interaction of the points is not Coulomb (that is, it cannot be reduced to a Vandermonde determinant squared times a product of weights), nor is there an obvious distribution of random matrices whose eigenvalues are associated with the points.Footnote 1 We note that there are recent universality results due to Kroó and Lubinsky on the asymptotics of Christoffel functions associated with multivariate orthogonal polynomials [6, 7], but they do not apply in our setting.
Using the algorithm for sampling determinantal point processes associated with univariate orthogonal polynomials [10], which is trivially adapted to the orthogonal polynomials on the wedge, we can sample from this determinantal point process. We use this algorithm to calculate statistics of the points. In Fig. 1, we use the sampling algorithm in a Monte Carlo simulation to approximate the probability that no eigenvalue is present in a neighbourhood of three points for \(\alpha = \beta = \gamma = 0\). That is, we take 10,000 samples of a determinantal point process, and calculate the distance of the nearest point to \(z_0\), for \(z_0\) equal to (1, 1), (0, 1), (0.5, 1) and (0.7, 1). The plots are of a complementary empirical cumulative distribution function of these samples. This gives an estimation of the probability that no eigenvalue is in a neighbourhood of \(z_0\). We have scaled the distributions so that the empirical variance is one: this ensures that the distributions tend to a limit as N becomes large, which is the regime where universality is present.
In Fig. 2 we plot the same statistics but for samples from the unweighted Coulomb gas on the wedge, which has the distribution
for \(\lambda _k\) supported on the wedge. As this is a Vandermonde determinant squared, it is also a determinantal point process with the basis arising from orthogonalized complex-valued polynomials \(1, (x + i y), (x+ i y)^2, \ldots \) [2]. We approximate this orthogonal basis using the modified Gram–Schmidt algorithm with the wedge inner product calculated via Clenshaw–Curtis quadrature. Again, this fits naturally into the sampling algorithm of [10], hence we can produce samples of this point process. What we observe is that, while our determinantal point process is not a Coulomb gas, it appears to be in the same universality class as the Coulomb gas away from the edge and corner, as the statistics follow the same distribution. This universality class matches that of the Gaussian Unitary Ensemble, as seen in Fig. 3 where we compare the three for \(N = 50\).
6 Conclusion
We have introduced multivariate orthogonal polynomials on the wedge and boundary of a square for some natural choices of weights. We have also generated a complete orthogonal basis with respect to a suitable weight inside the square. We have looked at determinantal point process statistics and observed a relationship between the resulting statistics and Coulomb gases, suggesting that, away from the corner and edge, they are in the same universality class.
One of the motivations for this work is to solve singular integral equations and evaluate their solutions on contours that have corners, in other words, to generalized the approach of [11]. Preliminary work in this direction is included in “Appendix B”, which shows how the recurrence relationship that our polynomials satisfy can be used to evaluate Stieltjes transforms.
Notes
If there is such a random matrix distribution, one would expect it to be a pair of commuting random matrices, whose joint eigenvalues give points on the wedge.
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Communicated by Alan Edelman.
The second author was supported in part by NSF Grant DMS-1510296.
Appendices
Appendix A. Jacobi Operators
By necessity, multivariate orthogonal polynomials have block-tridiagonal Jacobi operators corresponding to multiplication by x and y. We include here the recurrences associated with the inner product \( {\langle }f,g{\rangle }_{{\alpha },{\alpha },{\gamma }}\) (that is, \({\beta }= {\alpha }\)) that encode the Jacobi operators as they have a particularly simple form. The following lemma gives a linear combination of our orthogonal polynomials that vanish on \(x=1\):
Proposition A.1
For \(\beta = \alpha \), we have
and for \(n=1,2,\ldots ,\)
Proposition A.2
Assume \((1-x)(1-y) = 0\). Then
and, for \(n=2,3,\ldots \),
Proof
The first equation follows from Proposition A.1, since, for \(y = 1\), we have (using [9, (18.9.5)] to increment the first parameter)
The second equation also follows from Proposition A.1, since, for \(y = 1\), we have (using [9, (18.9.6)] to decrement the first parameter)
\(\square \)
The recurrences for multiplication by \(1-y\) follow from the symmetries \(P_n(x,y) = P_n(y,x)\) and \(Q_n(x,y) = -Q_n(y,x)\).
Appendix B. Stieltjes Transform of Orthogonal Polynomials
Consider the Stieltjes transform
where \(\mathrm{d}s\) is the arc-length differential. Just as in one-dimensions, the Stieltjes transform of weighted multivariate orthogonal polynomials satisfies the same recurrence as the orthogonal polynomials themselves.
Proposition B.1
Suppose \({\mathbb P}_n\) are a family of orthogonal polynomials with respect to w(x, y). Then, for \(n = 1, 2, \ldots \),
In particular, if \({\mathbb P}_n\) satisfies the recurrence relationships
then for \(A_n^z = A_{n}^x +\mathrm{i}A_{n}^y\), \(B_n^z = B_{n}^x +\mathrm{i}B_{n}^y\) and \(C_n^z = C_{n}^x +\mathrm{i}C_{n}^y\) we have
Proof We will identify \({\mathbb R}^2\) and \({\mathbb C}\) and use the notation \(\zeta = x + i y\). Note that
The first integral is zero if f is orthogonal to 1. \(\square \)
While this holds true for all families of multivariate orthogonal polynomials, in general, satisfying a single recurrence is not sufficient to determine \(\mathcal{S}_\Omega [{\mathbb P}_n w](z)\). However, since our blocks are square, in our case it is:
Corollary B.2
If \(B_n^z = B_{n}^x +\mathrm{i}B_{n}^y\) is invertible, then
This means that we can calculate the Stieltjes transform in linear time by solving the recurrence equation, using explicit formulas for the \(n=0\) and \(n=1\) terms. Unfortunately, the results are numerically unstable for both z on and off the contour. Here we sketch an alternative approach built on (F.W.J.) Olver’s and Miller’s algorithm, see [9, Section 3.6] for references in the tridiagonal setting and [5, Section 2.3] for the equivalent application to calculating Stieltjes transforms of univariate orthogonal polynomials.
For z off the contour, we can successfully and stably calculate the Stieltjes transform using a block-wise version of Olver’s algorithm, which is equivalent to solving the \(2n +1 \times 2n+1\) block-tridiagonal system
where \({\mathbf q}_0 \in {\mathbb C}^1\) and \({\mathbf q}_k \in {\mathbb C}^2\) for \(k=1,2,\ldots ,n\). Then
Olver’s algorithm consists of performing Gaussian elimination adaptively until a convergence criteria is satisfied.
For z on or near the contour, we no longer see quick decay in the Stieltjes transform (it is no longer a minimal solution to the recurrence), hence n must be prohibitively large. Instead, we adapt Olver’s algorithm in a vein similar to Miller’s algorithm to allow for a non-decaying tail. We do so by calculating two additional solutions \({\mathbf q}^1\), and \({\mathbf q}^2\) (with the same block-sizes as before) satisfying:
These three solutions avoid picking up the exponentially growing solution that forward recurrence does. Thus we can solve a \(3 \times 3\) system for constants a, b and c satisfying
We immediately have that
While this holds true for all n, we note that in practice we need to choose n bigger than the number of coefficients in order to observe numerical stability, see Fig. 4. We also find that there are still stability issues near the corner. Resolving these issues is ongoing research.
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Olver, S., Xu, Y. Orthogonal Structure on a Wedge and on the Boundary of a Square. Found Comput Math 19, 561–589 (2019). https://doi.org/10.1007/s10208-018-9393-0
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DOI: https://doi.org/10.1007/s10208-018-9393-0