Quadratic decomposition of bivariate orthogonal polynomials

In [4, 5] the author found connections between even orthogonal polynomials on the ball and simplex polynomials in $d$ variables. Following [5, section 4], let

be a weight function defined on the unit ball on $\mathbb{R}^{d}$, and let

where $\mathbf{T}^{d}$ is the unit simplex on $\mathbb{R}^{d}$. For $n\geqslant 0$, and $\alpha\in\mathbb{N}^{d}_{0}$ a multi-index, let $S_{2n,\alpha}(x_{1},x_{2},\ldots,x_{d})$ be an orthogonal polynomial associated to the weight function $W^{\mathbf{B}}$ of even degree in each of its variables. Then Y. Xu proved that it can be written in terms of orthogonal polynomials on the simplex as

where $P_{n,\alpha}(x^{2}_{1},x^{2}_{2},\ldots,x^{2}_{d})$ is, for each $n\geqslant 0$, an orthogonal polynomial of total degree $n$ on the simplex associated to $W^{\mathbf{T}}$. In this way, there exists an important partial connection between classical ball polynomials and simplex ones.

Inspired by these relations, we try to analyze the situation for the leftover polynomials in this procedure, i.e., we want to know the properties of the polynomials with odd powers that were left in the above identification. We succeed doing so in a general framework, showing that these polynomials are related to new families of bivariate orthogonal polynomials, resulting from a Christoffel modification that we explicitly identify. Hence we have a totally answer in the case $d=2$, that generalizes the one given by T. Chihara in [2].

The paper is organized as follows. In Section 2 we state the basic tools and results that we will need along the paper. The Section 3 is devoted to describe symmetric monic orthogonal polynomial sequences, starting with the basic properties and regarding how the polynomials are. In Section 4 we analyze the quadratic decomposition process. This will be done in an equivalent procedure, i.e., given a symmetric monic orthogonal polynomial sequence, we separate it in four families of polynomials in a zip way, and we deduce the inherit properties of orthogonality for each of the four families, obtaining that they are Christoffel modifications of the quadratic transformation of the original weight function. As a converse result, we construct a symmetric monic orthogonal polynomial sequence from a given one.

In Section 5 we give relations between the matrix coefficients of the three term relations of the involved families. In addition, the matrix coefficients of the Christoffel transformations for the four families of orthogonal polynomials are given in terms of the matrix coefficients of the three term relations for the symmetric polynomials. These matrices enable us to reinterpret the block Jacobi matrix associated with the four orthogonal polynomials sequences in terms of a $L\,U$ or $UL$ representation.

Finally, in the last Section, we complete the study started in [4, 5] describing explicitly the four families of orthogonal polynomials on the simplex deduced from a symmetric polynomial system orthogonal on the ball.

For each $n\geqslant 0$, let $\Pi_{n}$ denote the linear space of bivariate polynomials of total degree not greater than $n$, and let $\Pi=\bigcup_{n\geqslant 0}\Pi_{n}$. A polynomial $p(x,y)\in\Pi_{n}$ is a linear combination of monomials, i.e.,

and it is of total degree $n$ if $\sum_{i=0}^{n}|a_{n-i,i}|>0$.

A polynomial $p(x,y)$ of total degree $n$ is called centrally symmetric if $p(-x,-y)=(-1)^{n}\,p(x,y).$ As a consequence, if the degree of the polynomial is even, then it only contains monomials of even degree, and if the degree of the polynomial is odd, then the polynomial only contains monomials of odd powers.

We will say that a polynomial of partial degree $h$ in $x$ is $x$-symmetric if

Therefore, if $h$ is even, $p(x,y)$ only contains even powers in $x$, and if $h$ is odd, it only contains odd powers in $x$. Analogously, we define the $y$-symmetric for a polynomial of degree $k$ in $y$ as

A $y$-symmetric polynomial of degree $k$ in $y$ only contains odd powers in $y$ when $k$ is an odd number, and it contains only even power in $y$ when $k$ is even.

Obviously, if a polynomial is $x$-symmetric and $y$-symmetric, then it is centrally symmetric. In this case, if the polynomial has total degree $n$, with degree $h$ in $x$ and degree $k$ in $y$, and $n=h+k$, then if $h$ is an odd number (respectively if $k$ is an odd number), then the polynomial only contains odd powers in $x$ (respectively, odd powers in $y$), and if $h$ is even (respectively, if $k$ is even), then it only contains even powers in $x$ (respectively, it contains only even powers in $y$).

For each $n\geqslant 0$, let $\mathbb{X}_{n}$ denote the $(n+1)\times 1$ column vector

where, as usual, the superscript means the transpose. Then $\{\mathbb{X}_{n}\}_{n\geqslant 0}$ is called the canonical basis of $\Pi$. As in [3], for $n\geqslant 0$, we denote by

such that  $\operatorname{L}_{n,1}\,\mathbb{X}_{n+1}=x\,\mathbb{X}_{n}$  and  $\operatorname{L}_{n,2}\,\mathbb{X}_{n+1}=y\,\mathbb{X}_{n}$.

For $i,j=0,1$, and $n\geqslant 0$, we introduce the matrices: $\operatorname{J}_{n}^{(i,j)}$ of dimension $(n+1)\times(2n+1+i+j)$ in the following way

such that $j_{h,2h+j}^{(n,i,j)}=1$, for $0\leqslant h\leqslant n$, the rest of the elements are zero. In particular,

Observe that the $\operatorname{J}$-matrices are obtained from the identity matrices by introducing columns of zeros. The objective of these matrices is to extract the odd or the even elements in a vector of adequate size. The transpose of these matrices introduce zeroes into a vector in the odd or even positions.

A simple computation allows us to prove the next result.

A weight function $W(x,y)$ defined on $\Omega\subset\mathbb{R}^{2}$ is called centrally symmetric (cf. [3, p. 76]) if satisfies

Therefore, by a natural change of variables, we get

and then $\mu_{h,k}=0$, for $h+k$ an odd integer number.

We introduce an additional definition of symmetry.

Obviously, if $W(x,y)$ is $x\,y$-symmetric then it is centrally symmetric. As a consequence, if $W(x,y)$ is $x\,y$-symmetric, then $\mu_{h,k}=0$ when, at least one, $n$ or $m$ are odd numbers.

Let $\{\mathbb{S}_{n}\}_{n\geqslant 0}$ be the MOPS associated with a $x\,y$-symmetric weight function satisfying

where $\mathbf{S}_{n}$ is a $(n+1)$ positive-definite symmetric matrix.

When a bivariate polynomial $S_{n,k}(x,y)$ is $x\,y$-symmetric then it has the same parity order in every variable, i.e., if the partial degree in the first variable $x$ is even (respectively, odd), then all powers in $x$ are even (respectively, odd), and analogously, if the partial degree in the second variable $y$ is even (respectively, odd), then all powers in $y$ are even (respectively, odd).

Therefore, the vector polynomial $\mathbb{S}_{n}(x,y)$ can be separated in a zip way, attending to the parity of the powers of $x$ and $y$ in its entries. In fact, for even, respectively odd degree, we get

These families will be called big vector polynomials associated with $\{\mathbb{S}_{n}\}_{n\geqslant 0}$. We must observe that the big families are formed by vectors of polynomials in the variables $(x^{2},y^{2})$, that contains polynomials of independent degree intercalated with zeros.

Our objective is to extract the odd entries in the vectors $\mathbb{P}^{(i,0)}_{n}(x,y)$, and the even entries in $\mathbb{P}^{(i,1)}_{n}(x,y)$, for $i=0,1$.

Taking into account Lemma 3.2, we start studying the inherit properties of orthogonality of the polynomial systems $\{\widehat{\mathbb{P}}_{n}^{(i,j)}\}_{n\geqslant 0}$, for $i,j=0,1$.