Let be the vector space of polynomials with complex coefficients. For , we denote by , the action of a linear functional u in the algebraic dual of . In particular we denote by the moments of u. A sequence of polynomials is said to be a monic orthogonal polynomial sequence (MOPS) w.r.t a linear functional if (see Ref.  ):
2) the leading coefficient of is equal to 1,
where for all polynôme P, denotes its degree.
Under these conditions, we say that u is regular. It will be said normalized if .
A sequence of monic orthogonal polynomials satisfies a three-term recurrence relation (see  ):
For the quantum approach of this relation, quantum probability theory produced a new point of view that the three-term recurrence relation can be interpreted in term of fundamental operators in appropriate Fock space. In the case when the linear functional is positive it has an integral representation
where is a probability measure having a finite moment of all orders. In this case, the relation (1.1) can be written
To explain the quantum interpretation, we briefly recall some notions.
Let be an orthonormal system in a Hilbert space . Defining the operators:
It’s known that are mutually adjoint and the linear subspace spanned by the set is invariant under the action of .
The quadruple is called the interacting Fock probability space associated with . The operators and are called the creation operator and the annihilation operators respectively. The linear operator given by
is called the number operator. More generally, with the sequence , we associate the preservation operator by the prescription
Let be the space of classes of complex valued, square integrable functions w.r.t . We assume that the sub-space spanned by the polynomial functions is dense in . So that is an Hilbertian basis of . In such case, we consider the isomorphism U from to whose its restriction on given by:
where . Then the U is unitary and we have
This means that the field operator is the -image of the position operator on providing, in this way, a new interpretation of the recursion relation driving by OP in term of CAP operators. Since the random variable with distribution can be identified, up to stochastic equivalence , with the position operator q on , the previous new formulation of the tri-diagonal Jacobi relation in term of the CAP operators is called the quantum decomposition of the classical random variable. In fact we have seen that
This shows that any classical random variable has a built in non commutative structure which is intrinsic and canonical, and not artificially put by hands that is a sum of three non commuting random variables. This result motivated the apparition of a series of papers      and  dealing in the same context and provided many applications in the theory of quantum probability.
Compared to the 1-dimensional case the literature available in the multidimensional case is definitively scarce, even if several publications (see e.g., Refs.   ) show an increasing interest to the problem in the past years, where it emerges in connection with different kinds of approximation problems. The need for an insightful theory was soon perceived by the mathematical community.
Several progresses followed, both on the analytical front concerning multidimensional extensions of Carlemans criteria, on the algebraic front, with the introduction of the matrix approach  and the early formulations of the multidimensional Favard lemma  However, even with these progresses in view, one cannot yet speak of a general theory of orthogonal polynomials in several variables and of a multi-dimensional Favard lemma. In fact the importance of Favard lemma consists in the fact that the pair condensates the minimal information gained from the knowledge of the nth moment with respect to the knowledge of all the kth moments with . The more recent multi-dimensional formulations of Favard lemma are based on two sequences of matrices, one of which rectangular, with quadratic constraints among the elements of these sequences (see  ). Since the multi-dimensional analogues of positive (resp. real) numbers are the positive definite (resp. Hermitian) matrices, one would intuitively expect that a multi-dimensional extension of the Favard lemma would replace the sequence ( ) by a sequence of positive definite matrices and the ( ) sequence by a sequence of Hermitian matrices for each coordinate function . The precise formulation of this naive conjecture is what we call the multi-dimensional Favard problem. The paper is organized as follows: The Section 2 is devoted to recall some necessary notations in the theory of orthogonal polynomials in several variables. In the Section 3, we give the matrix technic used by Xuan Xu to derive the three-term recurrence relation for O.P.S.V. While the Section 4 is reserved to main result which is the connection between the projection approach and matrix approach for the three-term recurrence relation. This can be done via the quantum decomposition process.
For the multi-index and the indeterminate , we denote . The total degree of is given by: .
The space of all polynomial in d-variables with real coefficients will be denoted by , i.e.,:
The subspace of of all polynomial in d-variables with real coefficients and with degree at most equal to n is denoted by
We denote by the number of the monomials of degree exactly equal to n, so that
For and for , the vector of size will be denoted , where the monomials are arranged according to the lexicographical order of .
Remark 1. The lexicographical order of the vectors is given as follows:
A multi-sequence is called positive definite if for every tuple of distinct multi-indices , the determinant of the matrix
With each multi-sequence , one can associate a linear functional on given by:
Note that if is positive definite then the associate linear functional u is square positive, that is for all non identically zero polynomial
The results obtained in this paper concern only the square positive functionals. The case of semi square positive functionals which requires only for all P will be discussed later.
Let be a nonnegative Borel measure with an infinite support on and having a finite absolute mixed moments, i.e.:
The mixed moments of this measure are given by
A d-sequence ( ) is called a moment sequence if it coincides with the sequence of mixed moments of a such measure . In this case, the associated linear functional u is called a moment functional, which has an integral representation
Two Borel measure measures are called equivalent if they have the same mixed moments sequence. If the equivalent class of measures having the same moments as consists of only, the measure is called determinate. If u is a moment functional of a determined measure, then the integral representation is unique. It is known in the literature, that u is a moment functional if it is positive, which means that whenever .
In one dimensional case ( ), positivity means for all polynomial. However, in multi-dimensional case they are no longer equivalent, which is, in fact, the cause of many problems in the multidimensional moment problem.
A square positive linear functional u induces an inner product on given by
In the remain of this paper we always assume that . Two polynomials P and Q are said to be orthogonal with respect to u, if . With respect to such an u we can apply the Gram-Schmidt orthogonalization process on the monomials arranged in lexicographical order to derive a sequence of orthonormal polynomials, denoted by , where the superscript n means that .
Let us introduce the vector notation that is essential in the development below:
Clearly that the sequence of polynomials is orthonormal with respect to u. In fact one has
and can be expressed in terms of monomial vectors as follows:
Notice that we use the notation to design the space of -matrices with real entries and when we use simply the notation .
The leading coefficient of is the matrix , which is invertible since u is square positive.
For each , let be the set of polynomials spanned by the components of .This implies that is a vector space of dimension which is orthogonal to all polynomials in . Moreover, one has
where the symbol denotes the direct orthogonal sum of vector spaces.
Remark 2. As well known that the sequence of orthonormal polynomial is not unique. Actually, it is easy to see that each orthogonal matrix of order gives rise to an orthonormal basis of and every orthonormal basis of is of the form . One can also work with other bases of that are not necessarily orthonormal. In particular, one basis consists of polynomials of the form
This basis is sometimes called monomial basis, in general . Although are orthogonal to all polynomials of lower degrees. It is easy to
see that the matrix is positive definite and . Because
of the relation, most of the results below can be stated in terms of the monomial basis.
3. Matrix Technic to the Three-Term Recurrence Relation
The development of a general theory of orthogonal polynomials in several variables starts from a three-term relation in a vector-matrix notation very much like in the one variable theory.
Theorem 3.1 For , there exist matrices and , such that
where we define and .
Proof. Since the components of are polynomials of degree .
It follows that is of the form;
The orthonormal property of implies that only the coefficient of and are nonzero. Then we obtain the relation (3.1). □
Remark 3. The matrices in the three-term relation are expressible as
As a consequence, the matrices are symmetric. If we are dealing with orthogonal polynomials, , which are not necessarily orthonormal, then the three-term relation takes the form:
where is related to by
Moreover, comparing the highest coefficient matrices at both sides of (3.1), it follows that
where the matrices which are denied by
Clearly, , and , where . For example, for we have
From the relation (3.6) and the fact that is invertible, it follows that the matrices satisfy Rank conditions. For , for , and
The importance of the three-term relation in the following analog of Favards theorem of one variable. We extend the notation (2.12) to an arbitrary sequence of polynomials . The following result is a second version of the Recursion formula in  .
Theorem 3.2. (  ). Let , be a sequence in . Then the following statements are equivalent:
1) There exists a linear functional which is square positive and which makes an orthonormal basis in .
2) there exist matrices and such that
a) the polynomial vectors satisfy the three-term relation (3.1),
b) the matrices in the relation satises the rank condition.
The theorem 3.2 is an analog of Favards theorem in one variable, but it is not as strong as the classical Favards theorem. It does not state, for example, when the linear functional in the theorem will have an integral representation. For now, we concentrate on the three-term relation (3.1). It is an analog of the three-term relation in one variable. The fact that its coefficients are matrices reflect the complexity of the structure for .
4. Orthogonal Projection Approach of the Three-Term Recurrence Relation and Quantum Decomposition
In this section we give the orthogonal projection approach of the three-term recurrence relation. This approach gives a new point of view to the tri-diagonal recurrence relation. In fact we will see that in analogy with the one variable case, there exist such a quantum decomposition of the multiplication operator . The orthogonal polynomials in several variables associated with can be replaced by a sequence of orthogonal projections.
To this goal let us consider a square positive linear function u, on . For , we denote
the u-orthogonal projection on the subspace , where we refer to the inner product induced by u as in (2.11). We define the linear map as follow:
where we adopt the notation .
Lemma 4.1. For all , the maps are orthogonal projections on the subspace spanned by the monomials . Moreover for any , one has
where denotes the identity of the space .
Proof. Let , then
which prove that . Conversely if , clearly that . This implies that .
Furthermore we have
which prove that is an orthogonal projection.
Now let . In the case , we get
The case , we obtain
When , we have seen that , which proves the identity (4.3). It remains to prove the identity (4.4).
Taking the limit as , we get (4.4). □
Theorem 4.1. For all , we have
Since it follows that
Since is increasing, if we get
If , then the first part of the prove implies
Summing up , can be nonzero only if and this proves (4.7). □
Now we consider the operators
Then it is not difficult to show that
Theorem 4.2. (  ) Defining the C-A-P (creation, annihilation and preservation)-operators
Then the following quantum decomposition holds
The connection between the approach given in Section 3 and the projection approach given here is not sufficiently clear. In the remain of this section, we discuss some possible bridges connecting them. First, one of the most fundamental questions is the existence of such quantum decomposition of when the linear functional is not positive. The second task is, even for the square positive functional, what is the form of the quantum decomposition of . Another question can be addressed to algebraist is can we obtain such form of recurrence relation if the linear functional is not positive. The answers to these questions open the gate to many future investments towards the development of the algebraic theory of orthogonal polynomials in several variables.
The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1994-cos-2016-1-12) during the academic year 1437 AH/2016 AD.
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