Multivariate Orthogonal Polynomials and Modified Moment Functionals

0.48 MB·

by  Antonia M. Delgado

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 090, 25 pages
Multivariate Orthogonal Polynomials
and Modified Moment Functionals(cid:63)
Antonia M. DELGADO, Lidia FERNA´NDEZ, Teresa E. PE´REZ and Miguel A. PIN˜AR
IEMath – Math Institute and Department of Applied Mathematics, University of Granada,
18071, Granada, Spain
E-mail: [email protected], [email protected], [email protected], [email protected]
URL: http://www.ugr.es/local/goya/
6 Received January 28, 2016, in final form September 05, 2016; Published online September 10, 2016
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http://dx.doi.org/10.3842/SIGMA.2016.090
0
2
Abstract. Multivariateorthogonalpolynomialscanbeintroducedbyusingamomentfunc-
p
e tional defined on the linear space of polynomials in several variables with real coefficients.
S Westudytheso-calledUvarovandChristoffelmodificationsobtainedbyaddingtothemo-
ment functional a finite set of mass points, or by multiplying it times a polynomial of total
0
1 degree 2, respectively. Orthogonal polynomials associated with modified moment functio-
nals will be studied, as well as the impact of the modification in useful properties of the

] orthogonal polynomials. Finally, some illustrative examples will be given.
A
C Key words: multivariateorthogonalpolynomials; momentfunctionals; Christoffelmodifica-
. tion; Uvarov modification; ball polynomials
h
t 2010 Mathematics Subject Classification: 33C50; 42C10
a
m
[

1 Introduction
2
v
4
Using a moment functional approach as in [20, 21, 22], one interesting problem in the theory of
9
1 orthogonal polynomials in one and several variables is the study of the modifications of a quasi-
7 definitemomentfunctionaludefinedonΠ,thelinearspaceofpolynomialswithrealcoefficients.
0
In fact, there are many works devoted to this topic since modifications of moment functionals
.
1
are underlying some well known facts, such as quasi-orthogonality, relations between adjacent
0
6 families, quadrature and cubature formulas, higher-order (partial) differential equations, etc.
1 In the univariate case, given a quasi-definite moment functional u defined on Π, the (basic)
v: Uvarov modification is defined by means of the addition of a Dirac delta on a ∈ R,
i
X
v = u+λδ , such that (cid:104)v,p(x)(cid:105) = (cid:104)u,p(x)(cid:105)+λp(a), λ (cid:54)= 0.
r a
a
Apparently, this modification was introduced by V.B. Uvarov in 1969 [27], who studied the
casewhereafinitenumberofmasspointsisaddedtoameasure,andprovedconnectionformulas
for orthogonal polynomials with respect to the modified measure in terms of those with respect
to the original one.
InsomespecialcasesofclassicalLaguerreandJacobimeasures, iftheperturbationsaregiven
at the end points of the support of the measure, then the new polynomials are eigenfunctions
of higher-order differential operators with polynomial coefficients and they are called Krall
polynomials (see, for instance, [32] and the references therein).
In the multivariate case, the addition of Dirac masses to a multivariate measure was studied
in [11] and [13]. Moreover, Uvarov modification for disk polynomials was analysed in [10].
(cid:63)ThispaperisacontributiontotheSpecialIssueonOrthogonalPolynomials,SpecialFunctionsandApplica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 A.M. Delgado, L. Fern´andez, T.E. P´erez and M.A. Pin˜ar
Besides Uvarov modifications by means of Dirac masses at finite discrete set of points, in
the context of several variables it is possible to modify the moment functional by means of
moment functionals defined on lower dimensional manifolds such as curves, surfaces, etc. Re-
cently, a family of orthogonal polynomials with respect to such a Uvarov modification of the
classical ball measure by means of a mass uniformly distributed over the sphere was introduced
in [23]. The authors proved that, at least in the Legendre case, these multivariate orthogonal
polynomials satisfy a fourth-order partial differential equation, which constitutes a natural ex-
tension of Krall orthogonal polynomials [16] to the multivariate case. In [5], a modification of
a moment functional by adding another moment functional defined on a curve is presented, and
a Christoffel formula built up in terms of a Fredholm integral equation is discussed. As far as
we know, a general theory about Uvarov modifications by means of moment functionals defined
on lower dimensional manifolds remains as an open problem.
Following [31], the univariate (basic) Christoffel modification is given by the multiplication
of a moment functional u times a polynomial of degree 1, usually x−a, a ∈ R,
v = (x−a)u, acting as (cid:104)v,p(x)(cid:105) = (cid:104)u,(x−a)p(x)(cid:105).
This type of transformations were first considered by E.B. Christoffel in 1858 [9] within the
framework of Gaussian quadrature theory. Nowadays, Christoffel formulas are classical results
in the theory of orthogonal polynomials, and they are presented in many general references
(e.g., [8, 26]).
Christoffel modification is characterized by the linear relations that both families, modified
and non-modified orthogonal polynomials, satisfy. They are called connection formulas. It is
well known that some families of classical orthogonal polynomials can be expressed as linear
combinations of polynomials of the same family for different values of their parameters, the
so-called relations between adjacent families (e.g., see formulas in Chapter 22 in [1] for Jacobi
polynomials, or (5.1.13) in [26] for Laguerre polynomials). The study of such type of linear
combinations is also related with the concept of quasi-orthogonality introduced by M. Riesz in
1921 (see [8, p. 64]) as the basis of his analysis of the moment problem, and it is related with
quadrature formulas based on the zeros of orthogonal polynomials.
The extension of this kind of results to the multivariate case is not always possible. Gaussian
cubatureformulasofdegree2n−1werecharacterizedbyMysovskikh[24]intermsofthenumber
ofcommonzerosofthemultivariateorthogonalpolynomials. However, theseformulasonlyexist
inveryspecialcasesandthecaseofdegree2n−2becomesinteresting. Here, linearcombinations
of multivariate orthogonal polynomials play an important role, as it can be seen for instance in
[6, 25, 28, 29].
To our best knowledge, one of the first studies about Christoffel modifications in several
variables, multiplying a moment functional times a polynomial of degree 1, was done in [2],
where the modification naturally appears in the study of linear relations between two families
of multivariate polynomials. Necessary and sufficient conditions about the existence of orthog-
onality properties for one of the families was given in terms of the three term relations, by using
Favard’s theorem in several variables [12].
In [7] the authors show that modifications of univariate moment functionals are related
with the Darboux factorization of the associated Jacobi matrix. In this direction, in [3, 4, 5]
long discussions about several aspects of the theory of multivariate orthogonal polynomials
can be found. In particular, Darboux transformations for orthogonal polynomials in several
variables were presented in [3], and in [4] we can find an extension of the univariate Christoffel
determinantal formula to the multivariate context. Also, as in the univariate case, they proved
aconnectionwiththeDarbouxfactorizationoftheJacobiblockmatrixassociatedwiththethree
term recurrence relations for multivariate orthogonal polynomials. Similar considerations for
Multivariate Orthogonal Polynomials and Modified Moment Functionals 3
multivariate Geronimus and more general linear spectral transformations of moment functionals
can be found, among other topics, in [5].
In this paper, we study Uvarov and Christoffel modifications of quasi-definite moment func-
tionals. The study of orthogonal polynomials associated with moment functionals fits into
a quite general frame that includes families of orthogonal polynomials associated either with
positive-definite or non positive-definite moment functionals such as those generated using
Bessel polynomials, among others. We give necessary and sufficient conditions in order to
obtain the quasi-definiteness of the modified moment functional in both cases, Uvarov and
Christoffel modifications (see Theorems 3.1 and 4.3). We also investigate properties of the
polynomials associated with the modified functional, relations between original and modified
orthogonal polynomials as well as the impact of the modification in some useful properties of
the orthogonal polynomials.
When dealing with the Christoffel modification, and in the case where both moment func-
tionals, the original and the modified, are quasi-definite, some of the results are similar to those
obtained in [3, 4, 5] using a different technique. In particular, the necessary condition in our
Theorem 4.3 was proven there for an arbitrary degree polynomial, but the sufficient condition
was not discussed there.
The main results of this work can be divided in three parts, corresponding with Sections 3, 4
and 5. Section 2 is devoted to establish the basic concepts and tools we will need along the
paper. For that section, we recall the standard notations and basic results in the theory of
multivariate orthogonal polynomials following mainly [12].
Uvarov modification of a quasi-definite moment functional is studied in Section 3. In this
case, we modify a quasi-definite moment functional by adding several Dirac deltas at a finite
discrete set of fixed points. First, we will give a necessary and sufficient condition for the quasi-
definiteness of the moment functional associated with this Uvarov modification, and, in the
affirmativecase,wewilldeducetheconnectionbetweenbothfamiliesoforthogonalpolynomials.
Asimilarstudywasdonein[11]inthecasewhentheoriginalmomentfunctionalisdefinedfrom
a measure, and the modification is defined by means of a positive semi-definite matrix. In that
case, both moment functionals are positive-definite, and both orthogonal polynomial systems
exist. As a consequence, since it is possible to work with orthonormal polynomials, in [11] the
first family of orthogonal polynomials is considered orthonormal, and formulas are simpler than
in the general case considered here.
In Section 4 we study the Christoffel modification by means of a second degree polynomial.
This is not a trivial extension of the case when the degree of the polynomial is 1 studied in [2],
since in several variables not every polynomial of degree 2 factorizes as a product of polynomials
of degree 1. Again, we relate both families of orthogonal polynomials and also we deduce
the orthogonality by using Favard’s theorem in several variables. In fact, from a recursive
expression for the modified polynomials, we give necessary and sufficient conditions for the
existence of a three term relation, and we show that there exists a polynomial of second degree
constructed in terms of the first connection coefficients. Since three term relations can be
reformulated in terms of Jacobi block matrices, similar connection results can be found in [4, 5]
for arbitrary degree polynomials by using a block matrix formalism. Moreover, we study this
kind of Christoffel modification in the particular case when the original moment functional is
centrally symmetric, the natural extension of the concept of symmetry for univariate moment
functionals.
Finally, Section 5 is devoted to apply our results to two families of multivariate orthogonal
polynomials: the classical orthogonal polynomials on the ball in Rd, and the Bessel–Laguerre
polynomials, afamilyofbivariatepolynomialsorthogonalwithrespecttoanonpositive-definite
momentfunctional,thatcanbefoundin[19]assolutionofoneoftheclassicalKrallandSheffer’s
second-order partial differential equation [18].
4 A.M. Delgado, L. Fern´andez, T.E. P´erez and M.A. Pin˜ar
First, we modify the classical moment functional on the ball by adding a Dirac mass at
0 ∈ Rd. This example was introduced in [13], and here, we complete that study giving, as a new
result, the asymptotics of the modification. Next, we use the Christoffel modification by means
of a second degree polynomial to deduce a relation between adjacent families of classical ball
orthogonal polynomials in several variables. As far as we know, there is not a relation of this
kind for ball polynomials in the literature. This relation can be seen as an extension of (22.7.23)
in [1, p. 782] for Gegenbauer polynomials in one variable. Last example corresponds to a Uvarov
modificationforthenonpositive-definiteclassicalBessel–Laguerrebivariatepolynomialsdefined
in [19].
2 Basic tools
In this section we collect the definitions and basic tools about orthogonal polynomials in several
variables that will be used later. We follow mainly [12].
Along this paper, we will denote by M (R) the linear space of matrices of size h×k with
h×k
real entries, and the notation will simplify when h = k as M (R). As usual, we will say that
h
M ∈ M (R) is non-singular (or invertible) if detM (cid:54)= 0, and symmetric if Mt = M, where Mt
h
denotes its transpose. Moreover, I will denote the identity matrix of size h, and we will omit
h
the subscript when the size is clear from the context.
Let us consider the d-dimensional space Rd, with d (cid:62) 1. Given ν = (n ,n ,...,n ) ∈ Nd
1 2 d 0
a multi-index, a monomial in d variables is an expression of the form
xν = xn1xn2···xnd,
1 2 d
where x = (x ,...,x ) ∈ Rd. The total degree of the monomial is denoted by n = |ν| =
1 d
n +n +···+n . Then, a polynomial of total degree n in d variables with real coefficients is
1 2 d
a finite linear combination of monomials of degree at most n,
(cid:88)
p(x) = a xν, a ∈ R.
ν ν
|ν|(cid:54)n
Let us denote by Πd the linear space of polynomials in d variables of total degree less than or
n
equal to n, and by Πd the linear space of all polynomials in d variables.
For d (cid:62) 2 and n (cid:62) 0, if we denote
(cid:18) (cid:19)
n+d−1
rd = #{xν: |ν| = n} = ,
n n
then, unlike the univariate case, rd > 1 for d (cid:62) 2. Moreover,
n
n (cid:18) (cid:19)
(cid:88) n+d
rd = dimΠd = rd = . (2.1)
n n m n
m=0
When we deal with more than one variable, the first problem we have to face is that there is
not a natural way to order the monomials. As in [12], we use the graded lexicographical order,
that is, we order the monomial by their total degree, and then by reverse lexicographical order.
For instance, if d = 2, the order of the monomials is
(cid:8)1;x ,x ;x2,x x ,x2;x3,x2x ,x x2,x3;...(cid:9).
1 2 1 1 2 2 1 1 2 1 2 2
Ausefultoolinthetheoryoforthogonalpolynomialsinseveralvariablesistherepresentation
of a basis of polynomials as a polynomial system (PS).
Multivariate Orthogonal Polynomials and Modified Moment Functionals 5
Definition 2.1. A polynomial system (PS) is a sequence of column vectors of increasing size rd,
n
{Pn}n(cid:62)0, whose entries are independent polynomials of total degree n
P = P (x) = (cid:0)Pn(x)(cid:1) = (cid:0)Pn(x),Pn(x),...,Pn (x)(cid:1)t,
n n ν |ν|=n ν1 ν2 νrnd
where ν ,ν ,...,ν ∈ (cid:8)ν ∈ Nd: |ν| = n(cid:9) are different multi-indexes arranged in the reverse
1 2 rd 0
n
lexicographical order.
Observe that, for n (cid:62) 0, the entries of {P ,P ,...,P } form a basis of Πd, and, by extension,
0 1 n n
we will say that the vector polynomials {P }n is a basis of Πd.
m m=0 n
Using this representation, we can define the canonical polynomial system, as the sequence of
vectorpolynomialswhoseentriesarethebasicmonomialsarrangedinthereverselexicographical
order,
{Xn}n(cid:62)0 = (cid:8)(cid:0)xν1,xν2,...,xνrnd(cid:1)t: |νi| = n(cid:9)n(cid:62)0.
Thus, each polynomial vector P , for n (cid:62) 0, can be expressed as a unique linear combination of
n
the canonical polynomial system with matrix coefficients
P = G X +G X +···+G X ,
n n,n n n,n−1 n−1 n,0 0
where G ∈ M (R) for k = 0,1,...,n. Since both vectors P and X contain a system
n,k rd×rd n n
n k
of independent polynomials, the square matrix G is non-singular and it is called the leading
n,n
coefficient of the vector polynomial P . Usually, it will be denoted by
n
G (P ) = G ∈ M (R).
n n n,n rd
n
In the case when the leading coefficient is the identity matrix for all vector polynomial in a PS
G (P ) = I , n (cid:62) 0,
n n rd
n
then we will say that {Pn}n(cid:62)0 is a monic PS. With this notation, we will say that two vector
polynomials P and Q , for n (cid:62) 0, have the same leading coefficient if G (P ) = G (Q ), or,
n n n n n n
equivalently, if any entry of the vector P −Q is a polynomial of degree at most n−1.
n n
The shift operator in several variables, that is, the multiplication of a polynomial times
a variable x for i = 1,2,...,d, can be expressed in terms of the canonical PS as
i
x X = L X , n (cid:62) 0,
i n n,i n+1
where L , i = 1,2,...,d are rd ×rd full rank matrices such that (see [12])
n,i n n+1
L Lt = I .
n,i n,i rd
n
In particular, L is a matrix containing the columns of the identity matrix of size rd but
n,i n
including rd −rd columns of zeros eventually separating the columns of the identity.
n+1 n
Moreover, for i,j = 1,2,...,d, the matrix products L L ∈ M (R), are also full
n,i n+1,j rd×rd
n n+2
rank matrices and of the same type of L . In addition, since x x X = x x X , for n (cid:62) 0, we
n,i i j n j i n
get
L L = L L , i,j = 1,2,...,d, n (cid:62) 0. (2.2)
n,i n+1,j n,j n+1,i
For instance, if d = 2,
   
1 (cid:13) 0 0 1 (cid:13)
Ln,1 =  ... ..., Ln,2 = ... ... ,
(cid:13) 1 0 0 (cid:13) 1
6 A.M. Delgado, L. Fern´andez, T.E. P´erez and M.A. Pin˜ar
and
   
1 (cid:13) 0 0 0 0 1 (cid:13)
Ln,1Ln+1,1 =  ... ... ..., Ln,2Ln+1,2 = ... ... ... ,
(cid:13) 1 0 0 0 0 (cid:13) 1
 
0 1 (cid:13) 0
Ln,1Ln+1,2 = ... ... ... = Ln,2Ln+1,1.
0 (cid:13) 1 0
Let us now turn to deal with moment functionals and orthogonality in several variables.
Given a sequence of real numbers {µ } , the moment functional u is defined by means of
ν |ν|=n(cid:62)0
its moments
u: Πd −→ R,
xν (cid:55)−→ (cid:104)u,xν(cid:105) = µ ,
ν
and extended to polynomials by linearity, i.e., if p(x) = (cid:80) a xν, then
ν
|ν|(cid:54)n
(cid:88)
(cid:104)u,p(x)(cid:105) = a µ .
ν ν
|ν|(cid:54)n
Now, we recall some basic operations acting over a moment functional u. The action of u over
a polynomial matrix is defined by
(cid:104)u,M(cid:105) = ((cid:104)u,m (x)(cid:105))h,k ∈ M (R),
i,j i,j=1 h×k
where M = (m (x))h,k ∈ M (Πd), and the left product of a polynomial p ∈ Πd times u by
i,j i,j=1 h×k
(cid:104)pu,q(cid:105) = (cid:104)u,pq(cid:105), ∀q ∈ Πd.
Using the canonical polynomial system, for h,k (cid:62) 0, we define the rd×rd block of moments
h k
m = (cid:104)u,X Xt(cid:105) = mt .
h,k h k k,h
Wemustremarkthatm containsallofmomentsoforderh×k. Then, wecanseethemoment
h,k
matrix as a block matrix in the form
 
m m m ··· m
0,0 0,1 0,2 0,n
m1,0 m1,1 m1,2 ··· m1,n
 
Mn = m2,0 m2,1 m2,2 ··· m2,n
 .. .. .. .. 
 . . . . 
m m m ··· m
n,0 n,1 n,2 n,n
of dimension rd defined in (2.1).
n
Definition 2.2. A moment functional u is called quasi-definite or regular if and only if
∆d = detM (cid:54)= 0, n (cid:62) 0.
n n
Multivariate Orthogonal Polynomials and Modified Moment Functionals 7
Now, we are ready to introduce the orthogonality. Two polynomials p,q ∈ Πd are said to be
orthogonal with respect to u if (cid:104)u,pq(cid:105) = 0. A given polynomial p ∈ Πd of exact degree n is an
n
orthogonal polynomial if it is orthogonal to any polynomial of lower degree.
We can also introduce the orthogonality in terms of a polynomial system. We will say that
a PS {Pn}n(cid:62)0 is orthogonal (OPS) with respect to u if
(cid:104)u,X Pt(cid:105) = S , n (cid:62) 0, (cid:104)u,X Pt(cid:105) = 0, m < n,
n n n m n
where S is a non-singular matrix of size rd ×rd. As a consequence, it is clear that
n n n
(cid:104)u,P Pt(cid:105) = H , n (cid:62) 0, (cid:104)u,P Pt(cid:105) = 0, m (cid:54)= n,
n n n m n
where the matrix H is symmetric and non-singular. At this point we have to notice that, with
n
this definition, the orthogonality between the polynomials of the same total degree may not
hold. In the case when the matrix H is diagonal for all n (cid:62) 0 we say that the OPS is mutually
n
orthogonal.
A moment functional u is quasi-definite or regular if and only if there exists an OPS. If u
is quasi-definite then there exists a unique monic OPS. As usual u is said positive definite if
(cid:104)u,p2(cid:105) > 0, for all polynomial p (cid:54)= 0, and a positive definite moment functional u is quasi-
definite. In this case, there exist an orthonormal basis satisfying
(cid:104)u,P Pt(cid:105) = H = I .
n n n rd
n
For n (cid:62) 0, the kernel functions in several variables are the symmetric functions defined by
P (u;x,y) = P (x)tH−1P (y), m (cid:62) 0,
m m m m
n n
(cid:88) (cid:88)
K (u;x,y) = P (x)tH−1P (y) = P (u;x,y). (2.3)
n m m m m
m=0 m=0
Bothkernelssatisfytheusualreproducingproperty,andtheyareindependentoftheparticularly
chosen orthogonal polynomial system. For n = 0, we assume P (u;x,y) = K (u;x,y) = 0.
−1 −1
To finish this section, we need to recall the three term relations satisfied by orthogonal
polynomialsinseveralvariables. Asintheunivariatecase,theorthogonalitycanbecharacterized
by means of the three term relations [12, p. 74].
Theorem 2.3. Let {Pn}n(cid:62)0 = {Pνn(x): |ν| = n,n (cid:62) 0}, P0 = 1, be an arbitrary sequence in Πd.
Then the following statements are equivalent.
(1) There exists a linear functional u which defines a quasi-definite moment functional on Πd
and which makes {Pn}n(cid:62)0 an orthogonal basis in Πd.
(2) For n (cid:62) 0, 1 (cid:54) i (cid:54) d, there exist matrices A , B and C of respective sizes rd×rd ,
n,i n,i n,i n n+1
rd ×rd and rd ×rd , such that
n n n n−1
(a) the polynomials P satisfy the three term relations
n
x P = A P +B P +C P , 1 (cid:54) i (cid:54) d, (2.4)
i n n,i n+1 n,i n n,i n−1
with P = 0 and C = 0,
−1 −1,i
(b) for n (cid:62) 0 and 1 (cid:54) i (cid:54) d, the matrices A and C satisfy the rank conditions
n,i n+1,i
rankA = rankC = rd, (2.5)
n,i n+1,i n
8 A.M. Delgado, L. Fern´andez, T.E. P´erez and M.A. Pin˜ar
and,
rankA = rankCt = rd , (2.6)
n n+1 n+1
where A is the joint matrix of A , defined as
n n,i
A = (cid:0)At ,At ,...,At (cid:1)t ∈ M (R),
n n,1 n,2 n,d drd×rd
n n+1
and Ct is the joint matrix of Ct . Moreover,
n+1 n+1,i
A H = (cid:104)u,x P Pt (cid:105), B H = (cid:104)u,x P Pt(cid:105),
n,i n+1 i n n+1 n,i n i n n
C H = (cid:104)u,x P Pt (cid:105) = H At . (2.7)
n,i n−1 i n n−1 n n−1,i
In the case when the orthogonal polynomial system is monic, it follows that A = L ,
n,i n,i
n (cid:62) 0 for 1 (cid:54) i (cid:54) d. In this case, the rank conditions for the matrices A = L and A = L
n,i n,i n n
obviously hold.
3 Uvarov modification
Let u be a quasi-definite moment functional defined on Πd, let {ξ ,ξ ,...,ξ } be a fixed set of
1 2 N
distinct points in Rd, and let {λ ,λ ,...,λ } be a finite set of non zero real numbers. Then,
1 2 N
for p ∈ Πd, the expression
(cid:104)v,p(cid:105) = (cid:104)u,p(cid:105)+λ p(ξ )+λ p(ξ )+···+λ p(ξ ), (3.1)
1 1 2 2 N N
defines a moment functional on Πd which is known as a Uvarov modification of u.
If we define the diagonal matrix Λ = diag{λ ,...,λ }, then, for p,q ∈ Πd, we can write
1 N
 
q(ξ )
1
q(ξ2)
(cid:104)v,pq(cid:105) = (cid:104)u,pq(cid:105)+(p(ξ1),p(ξ2),...,p(ξN))Λ ..  = (cid:104)u,pq(cid:105)+p(ξ)tΛq(ξ), (3.2)
 . 
q(ξ )
N
where p(ξ) = (p(ξ ),p(ξ ),...,p(ξ ))t and q(ξ) = (q(ξ ),q(ξ ),...,q(ξ ))t, for all p,q ∈ Πd.
1 2 N 1 2 N
If we generalize equation (3.2) using a non-diagonal matrix Λ, then v is not a moment
functional but it can be seen as a bilinear form.
Ifthemomentfunctionalvisquasi-definite,ourfirstresultshowsthatorthogonalpolynomials
with respect to v can be derived in terms of those with respect to u. To simplify the proof of
this result we will make use of a vector-matrix notation which we will introduce next.
Throughout this section, we shall fix {Pn}n(cid:62)0 as an orthogonal polynomial system associated
with u. We denote by P (ξ) the matrix whose columns are P (ξ )
n n i
P (ξ) = (P (ξ )|P (ξ )|...|P (ξ )) ∈ M (R), (3.3)
n n 1 n 2 n N rd×N
n
denote by K the matrix whose entries are the kernels K (u;ξ ,ξ ) defined in (2.3),
n n i j
K = (cid:0)K (u;ξ ,ξ )(cid:1)N ∈ M (R), (3.4)
n n i j i,j=1 N×N
and, finally, denote by K (ξ,x) the vector of polynomials
n
(cid:0) (cid:1)t
K (ξ,x) = K (u;ξ ,x),K (u;ξ ,x),...,K (u;ξ ,x) . (3.5)
n n 1 n 2 n N
Multivariate Orthogonal Polynomials and Modified Moment Functionals 9
From the fact that K (u;x,y)−K (u;x,y) = P (u;x,y) = P (x)tH−1P (y), we have im-
n n−1 n n n n
mediately the following relations
Pt(ξ)H−1P (x) = K (ξ,x)−K (ξ,x), (3.6)
n n n n n−1
Pt(ξ)H−1P (ξ) = K −K , (3.7)
n n n n n−1
which will be used below.
Now we are ready to state and prove the main result in this section. In fact, we give a neces-
sary and sufficient condition in order to ensure the quasi-definiteness of the modified moment
functional in terms of the non-singularity of a matrix.
Theorem 3.1. Let u be a quasi-definite moment functional and assume that the moment func-
tional v defined in (3.1) is quasi-definite. Then the matrices
I +ΛK (3.8)
N n−1
are invertible for n = 1,2,..., and any polynomial system {Qn}n(cid:62)0 orthogonal with respect to v
can be written in the form
Q (x) = P (x),
0 0
Q (x) = P (x)−P (ξ)(I +ΛK )−1ΛK (ξ,x), n (cid:62) 1, (3.9)
n n n N n−1 n−1
where {Pn}n(cid:62)0 is a polynomial system orthogonal with respect to u. Moreover, the invertible
matrices Hˆ = (cid:104)v,Q Qt(cid:105) satisfy
n n n
Hˆ = H +P (ξ)(I +ΛK )−1ΛPt(ξ). (3.10)
n n n N n−1 n
Conversely, if the matrices defined in (3.8) and (3.10) are invertible then the polynomial
system {Qn}n(cid:62)0 defined by (3.9) constitutes an orthogonal polynomial system with respect to v,
and therefore v is quasi-definite.
Proof. Let us assume that v is a quasi-definite moment functional and let {Qn}n(cid:62)0 be an OPS
with respect to v. We can select an OPS {Pn}n(cid:62)0 with respect to u such that Qn has the same
leading coefficient as P , for n (cid:62) 0, in particular we have Q = P .
n 0 0
From this assumption, the components of Q −P are polynomials in Πd for n (cid:62) 1, then,
n n n−1
we can express them as linear combinations of orthogonal polynomials P ,P ,...,P . In
0 1 n−1
vector-matrix notation, this means that
n−1
(cid:88)
Q (x) = P (x)+ MnP (x),
n n j j
j=0
where Mn are matrices of size rd ×rd. These coefficient matrices can be determined from the
j n j
orthogonality of P and Q . Indeed, (cid:104)v,Q Pt(cid:105) = 0 for 0 (cid:54) j (cid:54) n−1, which shows, by definition
n n n j
of v and the fact that P is orthogonal,
j
Mn = (cid:104)u,Q Pt(cid:105)H−1 = −Q (ξ)ΛPt(ξ)H−1,
j n j j n j j
where P (ξ) is defined as in (3.3) and Q (ξ) = (cid:0)Q (ξ )|Q (ξ )|...|Q (ξ )(cid:1) is the analogous
j n n 1 n 2 n N
matrix with Q (ξ ) as its column vectors. Consequently, we obtain
n i
n−1
(cid:88)
Q (x) = P (x)− Q (ξ)ΛPt(ξ)H−1P (x) = P (x)−Q (ξ)ΛK (ξ,x), (3.11)
n n n j j j n n n−1
j=0
10 A.M. Delgado, L. Fern´andez, T.E. P´erez and M.A. Pin˜ar
where the second equation follows from relation (3.6), which leads to a telescopic sum that sums
up to K (ξ,x). Setting x = ξ , we obtain
n−1 i
Q (ξ ) = P (ξ )−Q (ξ)ΛK (ξ,ξ ), 1 (cid:54) i (cid:54) N,
n i n i n n−1 i
which by the definition of K at (3.4) leads to
n−1
Q (ξ) = P (ξ)−Q (ξ)ΛK ,
n n n n−1
and therefore
Q (ξ)(I +ΛK ) = P (ξ). (3.12)
n N n−1 n
Next,wearegoingtoshowthatthematricesI +ΛK areinvertible. Assumethatthereexists
N n−1
an index k such that I +ΛK is non regular, then there exists a vector C = (c ,c ,...,c )t
N k−1 1 2 N
satisfying (I +ΛK )C = 0. From (3.12) we deduce P (ξ)C = 0, which implies (I +ΛK )C
N k−1 k N k
= 0 using (3.7), and therefore we conclude
P (ξ)C = 0, n = k,k+1,.... (3.13)
n
Let us consider the discrete moment functional L = c δ + ··· + c δ , then (3.13) implies
1 ξ1 N ξN
that L vanishes on every orthogonal polynomial of total degree n (cid:62) k. Using duality we can
deduce the existence of a polynomial q(x) of total degree k−1 such that L = q(x)u. Let p(x)
be a non zero polynomial vanishing at ξ ,ξ ,...,ξ , then we get p(x)q(x)u = p(x)L = 0, which
1 2 N
contradicts the quasi-definite character of u.
Now, solving for Q (ξ) in equation (3.12) we get
n
Q (ξ) = P (ξ)(I +ΛK )−1, (3.14)
n n N n−1
and substituting this expression into (3.11) establishes (3.9).
Finally, from (3.9) and (3.14) we obtain
Hˆ = (cid:104)v,Q Qt(cid:105) = (cid:104)v,Q Pt(cid:105) = (cid:104)u,Q Pt(cid:105)+Q (ξ)ΛPt(ξ)
n n n n n n n n n
= H +P (ξ)(I +ΛK )−1ΛPt(ξ),
n n N n−1 n
which proves (3.10).
Conversely, if we define polynomials Q as in (3.9), then above proof shows that Q is
n n
orthogonal with respect to v. Since Q and P have the same leading coefficient, it is evident
n n
that {Qn}n(cid:62)0 is an OPS in Πd. (cid:4)
From now on, let us assume that v is a quasi-definite moment functional and {Qn}n(cid:62)0 is an
OPS with respect to v as given in (3.9). Then, the invertible matrix Hˆ = (cid:104)v,Q Qt(cid:105) can be
n n n
expressed in terms of matrices involving only {Pn}n(cid:62)0, as we have shown in Theorem 3.1. It
turns out that this happens also for Hˆ−1.
n
Proposition 3.2. In the conditions of Theorem 3.1, for n (cid:62) 0, the following identity holds,
Hˆ−1 = H−1−H−1P (ξ)(I +ΛK )−1ΛPt(ξ)H−1. (3.15)
n n n n N n n n
Proof. Formula (3.15) is a direct consequence of the Sherman–Morrison–Woodbury identity
for the inverse of the perturbation of a non singular matrix (see [14, p. 51]). (cid:4)