Towards an Algebraic Theory of Orthogonal Polynomials in Several Variables

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ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2019.75079 May 29, 2019 1185 Journal of Applied Mathematics and Physics

Towards an Algebraic Theory of Orthogonal

Polynomials in Several Variables

Habib Rebei

Department of Mathematics, College of Science, Qassim University, Al-Mulida, KSA

Abstract

In this paper, we review on a general theory of orthogonal polynomials in several variables (O.P.S.V) in which we present two different approaches for the three-term recurrence relation. We draw attention to the fact that it is possible to take advantage of the orthogonal projection approach of the three-term recurrence relation towards the development of the algebraic theory of O.P.S.V.

Keywords

Three-Term Recurrence Relation, Orthogonal Polynomials in Several Variables, Quantum Decomposition

1. Introduction

Let 

[ ]

X be the vector space of polynomials with complex coefficients. For

[ ]

P∈ X , we denote by u P, , the action of a linear functional u in the alge-braic dual of 

[ ]

X . In particular we denote by , n , 0

n

u = u X n≥ _the mo-ments of u. A sequence of polynomials

{ }

P_{n n}_≥₀ is said to be a monic orthogonal polynomial sequence (MOPS) w.r.t a linear functional u∈_

[ ]

X ′_{if (see Ref.}

[1]):

1) degP nn= ,

2) the leading coefficient of Pn is equal to 1, 3) u P P, n m =rn n mδ , , ,n m≥0,rn ≠0,n≥0,

where for all polynôme P, degP denotes its degree.

Under these conditions, we say that u is regular. It will be said normalized if

0 1

u = _.

A sequence of monic orthogonal polynomials satisfies a three-term recurrence relation (see [1]):

(

)

0 1; 1 1; n1 n1 n n n 1, 1,

P = P X= −α P+ = X −α + P −ω P n− ≥ (1.1) How to cite this paper: Rebei, H. (2019)

Towards an Algebraic Theory of Ortho-gonal Polynomials in Several Variables. Journal of Applied Mathematics and Phys-ics, 7, 1185-1196.

https://doi.org/10.4236/jamp.2019.75079

Received: February 12, 2019 Accepted: May 4, 2019 Published: May 29, 2019

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/jamp.2019.75079 1186 Journal of Applied Mathematics and Physics with

(

α ωn, n

)

∈ × \ 0 ,

{ }

n≥1.

For the quantum approach of this relation, quantum probability theory pro-duced a new point of view that the three-term recurrence relation can be inter-preted in term of fundamental operators in appropriate Fock space. In the case when the linear functional is positive it has an integral representation

( ) ( )

, d

u f =

_∫

f x

µ

x

 (1.2)

where µ_{is a probability measure having a finite moment of all orders. In this} case, the relation (1.1) can be written

1 1

n n n n n n

qP =P₊ +α P w P+ ₋ _(1.3) To explain the quantum interpretation, we briefly recall some notions.

Let

{

Φ_n,n≥0

}

be an orthonormal system in a Hilbert space 0. Defining

the operators:

1 1

n n n

B+ w

+ +

Φ = Φ (1.4)

0 0, n n n1, 1.

B− B− w n

−

Φ = Φ = Φ ≥ (1.5)

It’s known that B±_{are mutually adjoint and the linear subspace}

0

Γ ⊂ spanned by the set

{

( )

0, 0,1,2,

}

n

B+ _Φ n₌ _ _{is invariant under the action of} B±_.

The quadruple

{

_{Γ Φ},

( )

_{n n},B±

}

_{is called the interacting Fock probability} space associated with µ_{. The operators}B+_and _B−_{are called the creation} operator and the annihilation operators respectively. The linear operator given by

, 0,1,2,

n n

NΦ = Φn n= __(1.6) is called the number operator. More generally, with the sequence

{

α_n;n=1,2,_

}

_{, we associate the preservation operator}α_N∈L

( )

Γ _{by the}

pre-scription

, 1,2,

N n n n n

α Φ =α Φ = __(1.7)

Let _L2

( )

_µ

_{be the space of classes of complex valued, square integrable}

func-tions w.r.t µ. We assume that the sub-space _

[ ]

_x _⊂_L2

( )

_µ

_{spanned by the}

polynomial functions is dense in _L2

( )

_µ

_{. So that}

( )

n n

P is an Hilbertian basis of

( )

2

L

µ

. In such case, we consider the isomorphism U from Γ_to _L2

( )

_µ

whose its restriction on 0 given by:

{

}

* 0 0

: , n! n n, 1,2, ,

U Φ P w Φ P n∈ =  (1.8) where wn!=w w0 1wn. Then the U is unitary and we have

1

1 1

n n n n n n N

qP P α P w P U qU B− + α B−

+ −

= + + ⇔ = + + _(1.9)

This means that the field operator T:₌B+₊α_N₊B−_{is the}_U−1_{-image of}

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DOI: 10.4236/jamp.2019.75079 1187 Journal of Applied Mathematics and Physics Since the random variable with distribution µ can be identified, up to stochas-tic equivalence ≡, with the position operator q on _L2

( )

_µ _{, 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

N

q T B_{≡ =} +₊α ₊B−_(1.10)

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 [2] [3] [4] [5] [6] and [7] 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 multidimen-sional case is definitively scarce, even if several publications (see e.g., Refs. [8] [9]) 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 commu-nity.

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DOI: 10.4236/jamp.2019.75079 1188 Journal of Applied Mathematics and Physics

2. Preliminaries

For the multi-index

α

=

(

α

1, ,

α

d

)

∈d and the indeterminate

(

1, , d

)

X = X  X , we denote 1

1 dd,1

Xα ₌Xα Xα _{≤ ∈}d

 . The total degree of Xα_{is given by:}

1 d

α α= + +_ α _.

The space of all polynomial in d-variables X1, , Xd with real coefficients will be denoted by 

[

X1, , Xd

]

, i.e.,:

[

1, ,

]

( )

,

d

X X P X a X aα

α α α∈   =_ = ∈ _ 

∑

  

  (2.1)

The subspace of 

[

X1, , Xd

]

of all polynomial in d-variables X1, , Xd with real coefficients and with degree at most equal to n is denoted by

[

1, ,

]

,

n d

n

X X a X aα

α α α≤     =_ ∈ _   

∑

 

  (2.2)

We denote by dn the number of the monomials X1α1Xdαd of degree ex-actly equal to n, so that

[

1

]

1

[

1

]

1 1

dim , , dim , , d d n

n n d n d n d n d n d

d =  X  X −  − X  X =C+ −C− + =C − + (2.3) where

(

!

)

: , 0,1, ,

! !

k

n n

C k n

k n k

= =

−  (2.4)

For n∈ and for X =

(

X1, , Xd

)

∈d, the vector of size dn will be denoted n

( )

n

X Xα α =

= _{, where the monomials are arranged according to the} lexicographical order of

{

_α

_∈_d:

_α

₌_n

}

_.

Remark 1. The lexicographical order of the vectors Xα_{is given as follows:}

0 ₁

X =

[

]

T

1

1, , d

X = X  X

T

2 2 2 2 2 2

1, 1 2, , 1 d, 2, 2 3, , 2 d, 3, , d 1, d1 d, d

X = X X X  X X X X X  X X X  X − X X X− 

T

3 3 2 2 2 2

1, 1 2, , 1 d, 1 2, 1 2 3, , 1 3 d, 1 3,

X _{= }X X X _ X X X X X X X _ X X X X X __

A multi-sequence m=

( )

m_{α α∈}_d ⊂ is called positive definite if for every

tuple

(

_β

( )1, ,

_β

( )r

)

 of distinct multi-indices

_β

( )j _∈_d,1_{≤ ≤}_{j r} _{, the}

deter-minant of the matrix

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 2 1

2 1 2 2 2

1 2

r

r r r r

m m m

β β β β β β

+ + + + + + + + +                        is positive.

With each multi-sequence

( )

m_{α α∈}_d , one can associate a linear functional on

[

X1, , Xd

]

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( )

.

u Xα m

α

= _(2.5)

Note that if

( )

m_{α α∈}_d is positive definite then the associate linear functional

u is square positive, that is for all non identically zero polynomial

( )

,

P X a Xα

α α

=

∑

_(2.6)

on has

( )

2

, 0

u P m a aα β α β α β +

=

∑

> (2.7)

The results obtained in this paper concern only the square positive functionals. The case of semi square positive functionals which requires only _{u P}

( )

2 _≥₀_for

all P will be discussed later.

Let µ be a nonnegative Borel measure with an infinite support on _d_and having a finite absolute mixed moments, i.e.:

( )

d , .

d Xα

µ

X < +∞ ∀ ∈

α

d

∫

  (2.8)

The mixed moments of this measure are given by

( )

d d

( )

m m Xα X

α = α

µ

=

∫

_

µ

(2.9)

A d-sequence (mα) is called a moment sequence if it coincides with the se-quence of mixed moments of a such measure µ. In this case, the associated li-near functional u is called a moment functional, which has an integral represen-tation

( )

d

( ) ( )

d .

u f =

_∫

f x

µ

x

 (2.10) 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 µ_{consists of} µ _{only, the measure} µ_{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 posi-tive, which means that u P

( )

≥0 whenever P≥0.

In one dimensional case (d=1_{), positivity means}_{u P}

( )

2 _≥₀_{for all}

poly-nomial. 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

[

X1, , Xd

]

 given by

( )

[

1

]

, , , , , _d .

P Q =u PQ P Q∈_ X _ X _(2.11)

In the remain of this paper we always assume that u

( )

1 1= _{. Two polynomials}

P and Q are said to be orthogonal with respect to u, if P Q, =0. With respect to such an u we can apply the Gram-Schmidt orthogonalization process on the monomials

{ }

Xα _{arranged in lexicographical order to derive a sequence of} orthonormal polynomials, denoted by

{ }

n dn_{1 0}

k _{k n}

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DOI: 10.4236/jamp.2019.75079 1190 Journal of Applied Mathematics and Physics Let us introduce the vector notation that is essential in the development be-low:

( )

1 2 n n n n n d P X P X X P X       =         P

 (2.12)

( )

T

1 , 2 , , n

n n n

n X = P X P X P Xd 

P  (2.13)

Clearly that the sequence of polynomials

( )

P_{n n} is orthonormal with respect to u. In fact one has

(

T

)

, ,

1 0 0

0 1 0

0 0 1

n

n m m n d m n

u δ I δ

      = =       P P        (2.14)

and Pn can be expressed in terms of monomial vectors Xn as follows:

1 2 1

, 1 , 2 ,1 ,0

n n n

n =G Xn +Gn n− X − +Gn n− X − + +G Xn +Gn

P  (2.15)

where Gn n i, − ∈M

(

d dn, ;i 

)

.

Notice that we use the notation M_n

(

p q, ;

)

to design the space of p q×

-matrices with real entries and when p q= _{we use simply the notation}

(

;

)

n p

M  .

The leading coefficient of Pn is the matrix Gn=Gn n, , which is invertible

since u is square positive.

For each k≥0_{, let} E_k ⊂_n

[

X₁, ,_ X_d

]

be the set of polynomials spanned by the components of Pk.This implies that Ek is a vector space of dimension

k

d which is orthogonal to all polynomials in k−1

[

X1, , Xd

]

. Moreover, one has

[

1

]

0

, ,

n d k

k n

X X ⊥ E

≤ ≤

=

_⊕



 (2.16)

and

[

1

]

0

, , d k,

k

X X ⊥ E

≤ ≤∞

=

_⊕



 (2.17)

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 O of order dk gives rise to an orthonormal basis OPk of Ek and every orthonormal basis of

k

E is of the form OPn. One can also work with other bases of Ek that are not necessarily orthonormal. In particular, one basis consists of polynomials _Pk

α  _of the form

[

]

1 1

, , , , .

k k k

k d

P Xα R k R X X

α = + α−

α

= α− ∈ −

 _ _ _(2.18)

This basis is sometimes called monomial basis, in general _{u P P}

(

k k

)

0

α β ≠   _. Al-though _Pk

α

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DOI: 10.4236/jamp.2019.75079 1191 Journal of Applied Mathematics and Physics see that the matrix

(

T

)

n n n

H =u P P  is positive definite and 12

n =Hn n

P P . 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 va-riables starts from a three-term relation in a vector-matrix notation very much like in the one variable theory.

Theorem 3.1 For n≥0_{, there exist matrices}A_{n i}_, ∈M

(

d d_n, _n+₁;

)

and

(

)

, ;

n i n

B ∈M d  , such that

T

, 1 , 1, 1,1 ,

i n n i n n i n n i n

X P =A P+ +B P +A− P− ≤ ≤i d (3.1)

where we define P−1=0 and A−1,i=0.

Proof. Since the components of Xi nP are polynomials of degree

(

n+1

)

.

{

0, , 1

}

i n n

X P ∈ P  P+ (3.2) It follows that Xi nP is of the form;

, 0

n

i n n i i

i

X a

= =

∑

P P (3.3)

The orthonormal property of Pn implies that only the coefficient of P Pn+1, n and Pn−1 are nonzero. Then we obtain the relation (3.1). □

Remark 3. The matrices in the three-term relation are expressible as

(

T

)

(

T

)

, 1 ; ,

n i i n n n i i n n

A =u X P P+ B =u X P P (3.4)

As a consequence, the matrices Bn i, are symmetric. If we are dealing with

orthogonal polynomials, Pn, which are not necessarily orthonormal, then the three-term relation takes the form:

T

, 1 , , 1, 1 ,

i n n i n n i n n i n

X P =A P+ +B P +C P− ≤ ≤i d (3.5)

where Cn i, ∈M

(

d dn, n−1;

)

is related to An i, by

(

T

)

, 1 1,, where .

n i n n n i n n n

A H + =H C+ H =u P P 

Moreover, comparing the highest coefficient matrices at both sides of (3.1), it follows that

, 1 ,, 1 ,

n i n n n i

A G+ =G U ≤ ≤i d (3.6)

where the matrices Un i, ∈M

(

d dn, n+1;

)

which are denied by

1

, n n, 1 .

n i i

U X + ₌X X _{≤ ≤}i d _(3.7)

Clearly, rank U i

(

_n,

)

=d_n, and rank U

( )

n =dn+1, where

(

_T _T

)

T

,1| | ,

n n n d

U = U  U . For example, for d=2_{we have}

,1

1 0 0 0

0

0 0 1 0

n

U

 

 

=_ _

 

 

    

   

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DOI: 10.4236/jamp.2019.75079 1192 Journal of Applied Mathematics and Physics and

,2

0 1 0 0

0

0 0 0 1

n

U

 

 

=

 

 



   

    

From the relation (3.6) and the fact that Gn is invertible, it follows that the matrices An i, satisfy Rank conditions. For n≥0 , rank A

( )

n i, =dn for 1≤ ≤i d_{, and}

( )

(

_T _T

)

T

1, ,1, , , .

n n n n n d

rank A =d + A = A  A (3.8)

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

{ }

_n dn₁

k _k

P ₌ . The following result is a second version of the Recur-sion formula in [11].

Theorem 3.2. ([11]). Let

{ }

n n 0, 0 1

∞ = =

P P , be a sequence in 

[

X1, , Xd

]

. Then the following statements are equivalent:

1) There exists a linear functional which is square positive and which makes

{ }

_{n n}∞₀

=

P an orthonormal basis in 

[

X1, , Xd

]

.

2) there exist matrices An i, ∈M

(

d dn, n+1;

)

and Bn i, ∈M

(

dn;

)

such that a) the polynomial vectors Pn 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

u

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 re-flect the complexity of the structure for d≥2_.

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 re-currence 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

j

X j

M ≡X . The orthogonal polynomials in several variables associated with µ can be replaced by a sequence of orthogonal projections.

To this goal let us consider a square positive linear function u, on

[

X1, , Xd

]

 . For n∈, we denote

[

]

[

]

]: 1, , 1, ,

n d n d

(9)

] 1],

n n n

P =P −P− n∈ (4.2)

where we adopt the notation P−1]=0.

Lemma 4.1. For all n∈, the maps Pn are orthogonal projections on the

subspace En ⊂

[

X1, , Xd

]

spanned by the monomials

( )

Xα _{α =}_n. Moreover for any m n, ∈, one has

,

n m m n n

P P =δ P_(4.3)

and

0 n ,

n P ∞

= =

∑

1 (4.4)

where 1 denotes the identity of the space 

[

X1, , Xd

]

. Proof. Let Q=

∑

αa Xα α∈

[

X1, , Xd

]

, then

] 1]

1

n n n

n n

P Q P Q P Q a X a X a X E

α α α α α α α α α − ≤ ≤ − = = − = − = ∈

∑

which prove that ran

( )

Pn ⊂En. Conversely if Q E∈ n, clearly that

( )

ran

( )

n n

Q P Q= ∈ P _{. This implies that} ran

( )

P_n =E_n_. Furthermore we have

(

)

2

2 2 2

] 1] ] 1] ] ] 1] 1] ] 1] 1] 1] ] 1]

n n n n n n n n n

n n n n n n n

P P P P P P P P P P P P P P P P

− − − − − − − − = − = − − + = − − + = − = and

(

)

* * * *

] 1] ] 1] ] 1]

n n n n n n n n

P = P −P− =P −P− =P −P− =P (4.5)

which prove that Pn is an orthogonal projection. Now let m n, ∈__{. In the case} n m< , we get

(

] 1]

)(

] 1]

)

] ] ] 1] 1] ] 1] 1] ] 1] ] 1]

0

m n m m n n

m n m n m n m n

n n n n

P P P P P P

P P P P P P P P P P P P

− − − − − − − − = − − = − − + = − − + = The case n m> , we obtain

(

] 1]

)(

] 1]

)

] ] ] 1] 1] ] 1] 1] ] ] 1] 1]

0

m n m m n n

m n m n m n m n

m m m m

P P P P P P

P P P P P P P P P P P P

− − − − − − − − = − − = − − + = − − + =

When m n= , we have seen that 2

n n

P =P , which proves the identity (4.3). It remains to prove the identity (4.4).

We have

0] ] 1] ]

0 1

N N

n n n N

n n

P P P P− P

= =

= + − =

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DOI: 10.4236/jamp.2019.75079 1194 Journal of Applied Mathematics and Physics Taking the limit as N→ +∞_{, we get (4.4).}□

Theorem 4.1. For all n∈__{, we have}

1 1 .

j n n j n n j n n j n

X P =P X P P X P P X P+ + + − (4.7) Proof. Since

,

j j m j k

m k X =1 1X =

∑

P X P

Then

0

j n m j n

m

X P ∞ P X P

= =

∑

Since X Ej n⊂n+1

[

X1, , Xd

]

it follows that

1]

j n n j n

X P =P X P+ Since

( )

P_{m m}] is increasing, if m n> +1 we get

] 1] 1] 1] 1]

m n m n n

P P+ =P P− + =P+ hence

(

)

1] ] 1] 1] 0

m j n m n j n m m n j n

P X P P P X P= + = P −P− P X P+ = If m n< −1, then the first part of the prove implies

(

)

*

0

m j n n j m

P X P = P X P =

Summing up P XPm n, can be nonzero only if m∈

{

n−1, ,n n+1

}

and this proves (4.7). □

Now we consider the operators

, 1 : 1

j n n j n n n

A+ P X P E E

+ +

= →

, 1 : 1

j n n j n n n

A− P X P E E

− −

= →

, :

j n n j n n n

N =P X P E →E

Then it is not difficult to show that

( )

* _*

, , 1; , ,

j n j n j n j n

A+ A+ N N

+

= = _(4.8)

Theorem 4.2. ([12]) Defining the C-A-P (creation, annihilation and preserva-tion)-operators

, , ,

0 ; 0 ; 0

j j n j j n j j n

n n n

A+ ∞ A+ A− ∞ A− N ∞N

= = =

=

∑

=

∑

=

∑

_(4.9)

Then the following quantum decomposition holds

j j j j

X ₌A+₊N ₊A−_(4.10)

5. Conclusion

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li-DOI: 10.4236/jamp.2019.75079 1195 Journal of Applied Mathematics and Physics near functional is not positive. The second task is, even for the square positive functional, what is the form of the quantum decomposition of Xj. 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 alge-braic theory of orthogonal polynomials in several variables.

Acknowledgements

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.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this pa-per.

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DOI: 10.4236/jamp.2019.75079 1196 Journal of Applied Mathematics and Physics Communications, 14, 247-270. https://doi.org/10.1090/fic/014/13