Minimal cubature rules on an unbounded domain

0.3 MB·English

by  Yuan Xu

MINIMAL CUBATURE RULES ON AN UNBOUNDED DOMAIN
YUANXU
3 Abstract. Afamilyofminimalcubaturerulesisestablishedonanunbounded
1
domain, which is the first such family known on unbounded domains. The
0
nodesofsuchcubaturerulesarecommonzerosofcertainorthogonalpolyno-
2
mialsontheunboundeddomain,whicharealsoconstructed.

n
a
J

4
1 1. Introduction

In two or more variables, few families of minimal cubature rules are known in
]
A the literature, none on unbounded domains. The purpose of this note is to record
N a family of minimal cubature rules on an unbounded domain.
Theprecisionofacubatureruleisusuallymeasuredbythedegreesofpolynomials
.
h that can be evaluated exactly. For a nonnegative integer m, we denote be Π2 the
t m
a space of polynomials of degree at most m. Let Ω be a domain in R2 and let W
m be a non-negative weight function on Ω. A cubature rule of precision 2n−1 with
[ respect to W is a finite sum that satisfies

1 (cid:90) N
(cid:88)
v (1.1) f(x,y)W(x,y)dxdy = λ f(x ,y ), ∀f ∈Π2 ,
k k k 2n−1
9 Ω k=1
5
8 and there exists at least one function f∗ in Π2 such that the equation (1.1) does
2n
2 not hold. A cubature rule in the form of (1.1) is called minimal, if its number of
.
1 nodes is the smallest among all cubature rules of the same precision for the same
0 integral.
3 It is well known that the number of nodes, N, of (1.1) satisfies (cf. [6, 9]),
1
: n(n+1)
v (1.2) N ≥dimΠ2 = .
i n−1 2
X
Acubatureruleofdegree2n−1withN =dimΠ2 iscalledGaussian. Incontrast
r n−1
a to the Gaussian quadrature of one variable, Gaussian cubature rules rarely exists
and there are two family of examples known [3, 7], both on bounded domains. It
is known that they do not exist if W is centrally symmetric, which means that
W(x) = W(−x) and −x ∈ Ω whenever x ∈ Ω. In fact, in the centrally symmetric
case, the number of nodes of (1.1) satisfies a better lower bound [4],
(cid:106)n(cid:107) n(n+1) (cid:106)n(cid:107)
(1.3) N ≥dimΠ2 + = + .
n−1 2 2 2
Date:January15,2013.
2000 Mathematics Subject Classification. 41A05,65D05,65D32.
Keywordsandphrases. Minimalcubaturerules,orthogonalpolynomials,unboundeddomain.
TheworkofthethirdauthorwassupportedinpartbyNSFGrantDMS-1106113.
1
2 YUANXU
A cubature rule whose number of nodes attains a known lower bound is nec-
essarily minimal. It turns out that a family of weight functions W defined
α,β,±1
2
by
(1.4) Wα,β,±1(x,y):=|x+y|2α+1|x−y|2β+1(1−x2)±21(1−y2)±21, α,β >−1,
2
on [−1,1]2 admits minimal cubature rules of degree 4n−1 on the square [−1,1]2,
which was established in [5] for α=β =0 (see also [1, 10]) and in [11] for (α,β)(cid:54)=
(0,0). Therearenotmanyothercasesforwhichminimalcubaturerulesareknown
toexistforallnandnoneintheliteraturethatareknownforunboundeddomains.
To establish the minimal cubature rules for W , the starting point in [11]
α,β,±12
is the Gaussian cubature rules in [7] and it amounts to a series of changing of
variables from the product Jacobi weight function on the square [−1,1]2 to the
weightfunctionW . Theprocedureworksforgeneralproductweightfunction
α,β,±12
on the square. Moreover, as we shall shown in this note, that the procedure also
works for an unbounded domain, which leads to our main results in this note. The
minimal cubature rules are known to be closely related to orthogonal polynomials,
as their nodes are necessarily zeros of certain orthogonal polynomials. We will
discussthisconnectionandconstructanexplicitorthogonalbasisonourunbounded
domain in Section 3.
2. Gaussian Cubature rules on an unbounded domain
Let W be a nonnegative weight function on a domain Ω ⊂ R2. A polynomial
P ∈Πd is an orthogonal polynomial of degree n with respect to W if
n
(cid:90)
P (x,y)q(x,y)W(x,y)dxdy =0, ∀q ∈Π2 .
k,n n−1
Ω
LetV2 bethespaceoforthogonalpolynomialsofdegreeexactlyn. ThendimV2 =
n n
n+1. The Gaussian cubature rules can be characterized in terms of the common
zeros of elements in V2 ([6, 9]).
n
Theorem 2.1. Let {P :0≤k ≤n} be a basis of V2. Then a Gaussian cubature
k,n n
rule of degree 2n−1 for the integral against W exists if and only if its nodes are
the common zeros of P , 0≤k ≤n.
k,n
We now describe a family of Gaussian cubature rules on an unbounded do-
main. Letw(x)beanonnegativeweightfunctiondefinedontheunboundeddomain
(cid:82)∞
[1,∞) and let c denote its normalization constant defined by c w(x)dx = 1.
w w 1
Let p (w;x) be the orthogonal polynomial of degree n with respect to w and let
n
x ,x ,...,x be the zeros of p (w;x). It is well known that x are real and
1,n 2,n n,n n k,n
distinct points in [1,∞). The Gaussian quadrature rule for the integral against w
is given by
(cid:90) ∞ n
(cid:88)
(2.1) c f(x)w(x)dx= λ f(x ), f ∈Π ,
w k,n k,n 2n−1
1 k=1
where Π denotes the space of polynomials of degree 2n−1 in one variable and
2n−1
the weights λ are known to be all positive.
k,n
A typical example of w is the shifted Laguerre weight function
w (x):=(x−1)αe−x+1, α>−1,
α
MINIMAL CUBATURE RULES ON UNBOUNDED DOMAIN 3
forwhichtheorthogonalpolynomialp (w ;x)istheLaguerrepolynomialLα(x−1)
n α n
with argument x−1.
The unbounded domain on which our Gaussian cubature rules live is given by
(2.2) Ω:={(u,v):0<u−1<v <u2/4},
whichisboundedbyaparabolaandalineforu≥1andv ≥2,whichistheshaded
area depicted in Figure 1.
v
4
3
2
1
1 2 3 4 5 u
(cid:45)1
Figure 1. Domain Ω
Thefunctionw(x)w(y)isevidentlyasymmetricfunctioninxandy. Forγ >−1,
we define the weight
(2.3) W (u,v):=w(x)w(y)|u2−4v|γ, γ >−1,
γ
where the variables (x,y) and (u,v) are related by
(2.4) u=x+y, v =xy.
Thefunctionw(x)w(y)isobviouslysymmetricinxandy,sothatitcanbewritten
as a function of (u,v) for (x,y) in the domain
(cid:52):={(x,y):1<x<y <∞}.
The function W (u,v) is the image of w(x)w(y)|x−y|2γ+1 under the changing of
γ √
variablesu=x+yandv =xy,whichhasaJacobian|x−y|and|x−y|= u2−4v.
Whenw =w istheshiftedLaguerreweight,wedenoteW byW ,whichisgiven
α γ α,γ
explicitly by
W (x,y):=(v−u+1)αe−u−2(u2−4v)γ.
α,γ
The changing of variables (2.4) immediately leads to the relation
(cid:90) (cid:90)
(2.5) f(u,v)W (u,v)dudv = f(x+y,xy)w(x)w(y)|x−y|2γ+1dxdy
γ
Ω (cid:52)
1(cid:90)
= f(x+y,xy)w(x)w(y)|x−y|2γ+1dxdy,
2
[1,∞)2
4 YUANXU
where the second equation follows from symmetry. Now, recall that x denote
k,n
the zeros of p (w;x). We define
n
u =x +x , v =x x , 0≤j ≤k ≤n.
k,j k,n j,n k,j k,n j,n
Theorem 2.2. For W on Ω, the Gaussian cubature rule of degree 2n−1 is
−1
2
(cid:90) n k
(cid:88)(cid:88)(cid:48)
(2.6) c2 f(u,v)W (u,v)dudv = λ λ f(u ,v ), f ∈Π2 ,
w −1 k j j,k j,k 2n−1
Ω 2 k=1 j=1
where(cid:80)(cid:48) meansthatthetermforj =k isdividedby2. ForW onΩ,theGaussian
1
2
cubature rule of degree 2n−3 is
(cid:90) n k−1
(cid:88)(cid:88)
(2.7) c2 f(u,v)W (u,v)dudv = λ f(u ,v ), f ∈Π2 ,
w 1 j,k j,k j,k 2n−3
Ω 2 k=2j=1
where λ =λ λ (x −x )2.
j,k j k j,n k,n
TheprooffollowsalmostverbatimfromtheproofofTheorem3.1in[11]. Infact,
by(2.5),thecubaturerule(2.6)isequivalenttothecubatureruleforw(x)w(y)dxdy
on [1,∞)2 for polynomials f(x + y,xy) with f ∈ Π2 , which are symmetric
2n−1
polynomials in x and y. By the Sobolev’s theorem on invariant cubature rules [8],
thiscubatureruleisequivalenttotheproductGaussiancubaturerulesforw(x)w(y)
on[1,∞)2,whichistheproductof (2.1). Thus,thecubaturerule(2.6)followsfrom
the product of Gaussian cubature rules under (2.4).
Theexistenceofthesecubaturerulesalsofollowfromcountingcommonzerosof
the orthogonal polynomials with respect to W . Indeed, a mutually orthogonal
±1
2
basis with respect to W on Ω is given by
−1
2
(−1)
(2.8) P 2 (u,v)=p (x)p (y)+p (y)p (x), 0≤k ≤n,
k,n n k n k
and a mutually orthogonal basis with respect to W is given by
1
2
(2.9) P(12)(u,v)= pn+1(x)pk(y)−pn+1(y)pk(x), 0≤k ≤n,
k,n x−y
both families are defined under the mapping (2.4). This was established in [2] for
the domain [−1,1]2, but the proof can be easily extended to our Ω. It is easy to
(−1)
see that the elements of {(u ,v ):0 ≤ j ≤ k ≤ n} are common zeros of P 2 ,
j,k j,k k,n
0≤k ≤n, and the cardinality of this set is dimΠ2 , which implies, by Theorem
n−1
2.1, that the Gaussian cubature rules for W exists. The proof for W works
−1 1
2 2
similarly.
3. Minimal cubature rules on an unbounded domain
We are looking for cubature rules of degree 2n−1 that satisfy the lower bound
(1.3), which are necessarily minimal cubature rules. Such cubature rules are char-
acterized by common zeros of a subspace of V2 ([4]).
n
Theorem 3.1. A cubature rule whose number of nodes attains the lower bound
(1.3) exists if, and only if, its nodes are common zeros of (cid:98)n+1(cid:99)+1 orthogonal
2
polynomials of degree n.
MINIMAL CUBATURE RULES ON UNBOUNDED DOMAIN 5
Let w be the weight function defined on [1,∞) and let W be the corresponding
γ
weight function in (2.3) defined on Ω in (2.2). We define a family of new weight
functions by
(3.1) W (x,y):=W (2xy,x2+y2−1)|x2−y2|, (x,y)∈G,
γ γ
where the domain G is centrally symmetric and defined by
G:=(−∞,−1]2∪[1,∞)2.
Thus, the weight function W is centrally symmetric on G.
γ
That W is well-defined on G is established in the next lemma. Recall that
γ
(cid:52)={(x,y):1<x<y <∞}.
Lemma 3.2. The mapping (x,y)(cid:55)→(2xy,x2+y2−1) is a bijection from (cid:52) onto
Ω. Furthermore,
(cid:90) (cid:90)
(3.2) f(u,v)W (u,v)dudv = f(2xy,x2+y2−1)W (x,y)dxdy.
γ γ
Ω G
Proof. Recallthatif(x,y)∈(cid:52),then(x+y,xy)∈Ωandthemappingisone-to-one.
For (x,y) ∈ [1,∞)2, let us write x = coshθ and y = coshφ, 0 ≤ θ,φ ≤ π. Then it
is easy to verify that
(3.3) 2xy =cosh(θ−φ)+cosh(θ+φ), x2+y2−1=cosh(θ−φ)cosh(θ+φ),
from which it follows readily that (2xy,x2+y2−1)∈Ω whenever (x,y)∈(cid:52). The
Jacobian of the change of variables u = 2xy and v = x2+y2−1 is 4|x2−y2|, so
that the mapping is a bijection. Since dudv =4|x2−y2|dxdy and the area of G is
fourtimesof(cid:52), theformula(3.3)followsfromthechangeofvariables, theintegral
(2.5) and the fact that f(2xy,x2+y2−1) is central symmetric on G. (cid:3)
In the case of W =W , we denote the weight function W by W , which is
γ α,γ γ α,γ
given explicitly by
(3.4) W (x,y):=4γ|x−y|2α(x2−1)γ(y2−1)γ|x2−y2|e−2xy−2.
α,γ
FortheweightfunctionW ontheunboundeddomainG,minimalcubaturerules
γ
of degree 4n−1 exist, as shown in the next theorem. To state the theorem, we
need a notation. For n ∈ N , let x be the zeros of the orthogonal polynomial
0 k,n
p (w;x), which are in the support set [1,∞) of the weight function w. We define
n
θ by
k,n
x =coshθ , 1≤k ≤n.
k,n k,n
Since x >1, it is evident that θ are well defined. We then define
k,n k,n
(3.5) s :=coshθj,n−θk,n and t :=coshθj,n+θk,n.
j,k 2 j,k 2
Theorem 3.3. For W on G, we have the minimal cubature rule of degree 4n−1
−1
2
with dimΠ2 +n nodes,
2n−1
(cid:90) 1(cid:88)n (cid:88)k (cid:48)
(3.6) c2 f(x,y)W (x,y)dxdy = λ λ [f(s ,t )+f(t ,s )
w G −12 4k=1 j=1 j,n k,n j,k j,k j,k j,k
+ f(−s ,−t )+f(−t ,−s )].
j,k j,k j,k j,k
6 YUANXU
ForW onG,wehavetheminimalcubatureruleofdegree4n−3withdimΠ2 +n
1 2n−3
2
nodes,
(cid:90) 1(cid:88)n k(cid:88)−1
(3.7) c2 f(x,y)W (x,y)dxdy = λ [f(s ,t )+f(t ,s )
w G 12 4k=2j=1 j,k j,k j,k j,k j,k
+ f(−s ,−t )+f(−t ,−s )],
j,k j,k j,k j,k
where λ =λ λ (coshθ −coshθ )2.
j,k j,n k,n j,n k,n
Proof. We prove only the case of W , the case of W is similar. Our starting
−1 1
2 2
point is the Gaussian cubature rule in (2.6), which gives, by (3.2),
(cid:90) n k
(cid:88)(cid:88)(cid:48)
c2 f(2xy,x2+y2−1)W (x,y)dxdy = λ λ f(u ,v ), f ∈Π2 .
w −1 j,n k,n j,k j,k 2n−1
G 2 k=1 j=1
By (3.3) or by direct verification,
coshθ +coshθ =2coshθj,n−θk,n coshθj,n+θk,n
j,n k,n 2 2
coshθ coshθ =cosh2 θj,n−θk,n +cosh2 θj,n+θk,n −1
j,n k,n 2 2
which implies that
u =x +x =2s t and v =x x =s2 +t2 −1.
j,k j,n k,n j,k j,k j,k j,n k,n j,k j,k
Consequently, the above cubature rule can be written as
(cid:90) n k
(cid:88)(cid:88)(cid:48)
c2 f(2xy,x2+y2−1)W (x,y)dxdy = λ λ f(2s t ,s2 +t2 −1)
w −1 k j j,k j,k j,k j,k
G 2 k=1 j=1
for all f ∈ Π2 . For f ∈ Π2 , the polynomial f(2xyx2+y2−1) is of degree
2n−1 2n−1
4n−1. Since the polynomials f(2xy,x2+y2−1) are symmetric polynomials and
allsymmetricpolynomialsinΠ2 canbewritteninthisway,wehaveestablished
4n−1
(3.6) for symmetric polynomials. By the Sobolev’s theorem on invariant cubature
rules, this establishes (3.6) for all polynomials in Π2 . (cid:3)
4n−1
The number of nodes of the cubature rule in (3.6) is precisely
(cid:22) (cid:23)
2n
N =dimΠ2 +n=dimΠ2 +
2n−1 2n−1 2
which is the lower bounded of (1.3) with n replaced by 2n. Thus, (3.6) attains the
lower bound (1.3). Similarly, (3.7) attains the lower bound (1.3) with n replaced
by 2n−1.
It turns out that a basis of orthogonal polynomials with respect to W can be
γ
givenexplicitly. WeneedtodefinethreemoreweightfunctionsassociatedwithW .
γ
W(1,1)(u,v):=(1−u+v)(1+u+v)W (u,v),
γ γ
(3.8) W(1,0)(u,v):=(1−u+v)W (u,v), (u,v)∈Ω,
γ γ
W(0,1)(u,v):=(1+u+v)W (u,v).
γ γ
Under the change of variables u = x+y and v = xy, 1−u+v = (x−1)(y−1)
and 1+u+v = (x+1)(y+1). The three weight functions in (3.8) are evidently
of the same type as W . We denote by {P(γ) : 0 ≤ k ≤ n} an orthonormal
γ k,n
MINIMAL CUBATURE RULES ON UNBOUNDED DOMAIN 7
basis of V (W ) under (cid:104)·,·(cid:105) . For 0 ≤ k ≤ n, we further denote by P(γ),1,1,
n γ Wγ k,n
P(γ),1,0,, P(γ),0,1 the orthonormal polynomials of degree n with respect to (cid:104)f,g(cid:105)
k,n k,n W
for W =W(1,1), W(1,0), W(0,1), respectively.
γ γ γ
Theorem 3.4. For n = 0,1,..., a mutually orthogonal basis of V (W ) is given
2n γ
by
Q(γ) (x,y):=P(γ)(2xy,x2+y2−1), 0≤k ≤n,
1 k,2n k,n
(3.9)
Q(γ) (x,y):=(x2−y2)P(γ),1,1(2xy,x2+y2−1), 0≤k ≤n−1,
2 k,2n k,n−1
and a mutually orthogonal basis of V (W ) is given by
2n+1 γ
Q(γ) (x,y):=(x+y)P(γ),0,1(2xy,x2+y2−1), 0≤k ≤n,
1 k,2n+1 k,n
(3.10)
Q(γ) (x,y):=(x−y)P(γ),1,0(2xy,x2+y2−1), 0≤k ≤n.
2 k,2n+1 k,n
Underthemapping(u,v)(cid:55)→(2xy,x2+y2−1), itiseasytoseethatW(1,1)(u,v)
γ
becomes(x2−y2)2W (x,y),W(1,0)(u,v)becomes(x−y)2W (x,y),andW(0,1)(u,v)
γ γ γ γ
becomes (x+y)2W (x,y). Hence, using Lemma 3.2, the proof can be deduced
γ
from the orthogonality of Pk,n(γ),i,j and symmetry of the integrals against W(i,j),
γ
similar to the proof of Theorem 3.4 in [12].
(±1)
Combining with (2.8) and (2.9), we can express orthogonal polynomials Q 2
i k,n
in terms of orthogonal polynomials in one variables. For example, in the case of
W in (3.4), we can express even degree orthogonal polynomials in terms of the
α,γ
Laguerre polynomials.
Proposition 3.5. Let α > −1. A mutually orthogonal basis of V (W ) is
2n α,−1
2
given by, for 0≤k ≤n and 0≤k ≤n−1, respectively,
Q(α,−21)(coshθ+1,coshφ+1)=L(α)(cosh(θ−φ))L(α)(cosh(θ+φ))
1 k,2n n k
+L(α)(cosh(θ−φ))L(α)(cosh(θ+φ)),
k n
Q(α,−12)(coshθ+1,coshφ+1)=(x2−y2)(cid:104)L(α+1)(cosh(θ−φ))L(α+1)(cosh(θ+φ))
2 k,2n n−1 k
(cid:105)
+L(α+1)(cosh(θ−φ))L(α+1)(cosh(θ+φ)) .
k n−1
In these formulas we used x = coshθ−1 and y = coshφ−1, so that Lα(x−
n
1) = Lα(coshθ) and Lα(y −1) = Lα(coshφ) and we can then use (3.3). Note,
n n n
however, we cannot express the odd degree ones in terms of Laguerre polynomials.
In fact, for Q , we need orthogonal polynomials of one variable with respect
2 k,2n+1
to w(x) = (x + 1)(x − 1)αe−(x−1), which is not a shift of the Laguerre weight
function.
Corollary 3.6. For the weight function W , the nodes of the minimal cubature
−1
2
(−1)
formulas (3.6) are common zeros of orthogonal polynomials { Q 2 :0≤k ≤n}.
1 k,2n
An analogue result can be stated for the minimal cubature formulas (3.7).
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Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.
E-mail address: [email protected]