“BABES¸–BOLYAI”, MATHEMATICA, VolumeLIV, Number 4, December 2009 ON APPLICATIONS OF THE REPRODUCING KERNEL METHOD FOR CONSTRUCTION OF CUBATURE FORMULAS EMIL A

STUDIA UNIV. “BABES¸–BOLYAI”, MATHEMATICA, VolumeLIV, Number 4, December 2009

ON APPLICATIONS OF THE REPRODUCING KERNEL METHOD FOR CONSTRUCTION OF CUBATURE FORMULAS

EMIL A. DANCIU

Abstract. In this paper we use the method of Reproducing Kernel and Gegenbauer polynomials for constructing cubature formulas on the unit ballB^d, and on the standard simplex. Also we study the relation between interpolation polynomials based on the zeros of quasi-orthogonalCheby- shevpolynomials and the nodes of near minimal degree cubature formulas.

1. Introduction

1) The Reproducing Kernel of a Hilbert space of functions

One calls reproducing Kernel of the Hilbert spaceH of functions defined on D, real valued (D ⊂R^d), a functionK =K(x, y) : D×D →R, which verifies the following conditions

i)K(·, y)∈H, for any fixedy∈D, ii)< f, K(·, y)>=f(y), ∀f ∈H.

It is known that in the Hilbert spaceH are stated the following results.

Theorem 1.1. If the Hilbert spaceH has a Reproducing Kernel, then this kernel is unique and symmetric with respect to its arguments.

Theorem 1.2. IfLis a linear and bounded functional defined on the Hilbert space H,which has a Reproducing Kernel, then the representation function corresponding toLisg(x) =Ly[K(x, y)].

Received by the editors: 05.01.2009.

2000Mathematics Subject Classification.41A25, 41A36, 65D32.

Key words and phrases. Reproducing Kernel, cubature formulas, Gegenbauer polynomials on simplex, on ball.

We consider now,H =P^dn the space of all polynomials of degree at mostn, andD⊂R^d.

It is known that dimP^dn(D) = ^n+d_d

, if and only if int(D)6=∅.

Letf ∈P^dn be a polynomial of degree exactn,and we denote µ=µ(d, n) =

n+d d

= (n+d)!

n!d! .

It was shown that the number of terms in the representation of the polynomial f is equal to µ(d, n) and this number represents the number of the monomials in the expression off =f(x).

LetW =W(x) :D→R⁺0, (D⊂R^d),be a weight function.

Theorem 1.3. For a given region (domain)D, D⊂R^d and a given weight function W = W(x) : D → R⁺0, exists and are unique r(d, n) = µ(d, n−1) = ^(n−1+d)!_(n−1)!d!

orthogonal polynomials of degreen, which are linearly independent.

Let now,{ei(x)}^∞_i=0, be the monomials which are ordered increasing, and for the same degree for certain terms, we use the lexicographic order.

So, the set {ei(x)}, i = 1, µ(d, n) represents all the monomials of degree at mostn.

By applying the Gram-Schmidt orthonormalization process, we can obtain anorthonormalizedset with respect to the scalar product

(f, g) =I(f·g) = Z

D

f(x)g(x)W(x)dx. (1.1)

2) The Gegenbauer (ultraspherical) orthogonal polynomials

We present now, some of the properties of Gegenbauer polynomials, which play an important role in the applications of the cubature formulas theory by using the Reproducing Kernel method.

TheGegenbauer polynomials are usually defined by the following generating function:

(1−2tz+z²)^−λ=

∞

X

n=0

C_n^(λ)(t)zⁿ, (1.2)

where |z|<1, |t| ≤1, λ >0.

The coefficientsCn^(λ)(t) are algebraic polynomials of degreenwhich are called theGegenbauerpolynomials associated withλ. One can prove that the family of poly- nomials{Cn^(λ)}^∞_n=0is a complete orthogonal system for the weighted spaceL2(I, W), I= [−1,1], W(t) =Wλ(t) := (1−t²)^λ−12, and we have

Z

[−1,1]

C_n^(λ)(t)C_m^(λ)(t)W(t)dt=







0, m6=n

γ_n,λ= ^π1/2(n+λ)n!Γ(λ)^{(2λ)nΓ(λ+1/2)}, m=n where we use (a)λ, thePockhammer symbol,

(a)₀:= 0, (a)_n:=a(a+ 1). . .(a+n−1) = Γ(a+n)/Γ(a).

Also we have,

C_n^(λ)(−t) = (−1)ⁿC_n^(λ)(t), C_n^(λ)(1) = (2λ)n

n! and C₀^(λ)= 1. (1.3) The Gegenbauer polynomials can also be defined by the well known Ro- drigues’s formula (see [7]Szeg¨o)

C_n^(λ)(t) = (−1)ⁿαn,λ(1−t²)^−λ+12 dⁿ

dtⁿ[(1−t²)^n+λ−12] where,

αn,λ= (2λ)n

n!2ⁿ(λ+¹₂)n

.

It is known that there exists the following identity which relatesGegenbauer polynomials with different weights

d^k

dt^kC_n^(λ)(t) = 2^k(λ)kC_n−k^(λ+k), k= 1,2, . . . n. (1.4) Forλ= 1/2, we can obtain theLegendre polynomial

Pn(t) =(−1)ⁿ 2ⁿn!

dⁿ

dtⁿ[(1−t²)ⁿ] =C_n^(1/2)(t) and forλ= 1 we obtain theChebyshev polynomial of second kind Un,

U_n =sin[(n+ 1)arccost]

√1−t² =C_n⁽¹⁾(t).

Also, we can obtain theChebyshev polynomial of the first kind T_n(t) :=cos(narccost) =C_n⁽⁰⁾,

by consideringCn⁽⁰⁾ associated with the weight functionW0(t) = (1−t²)^−1/2. We can also consider theGegenbauer polynomials Cn^(λ), forλ <0, λ∈Z⁻ namely,

C_n^(λ)(t) :=α(1−t²)^−λ+12 dⁿ dtⁿ

(1−t²)^n+λ−12 , λ <0

whereαis an constant independent oft and we can write the identity d^k

dt^kC_n^(λ)(t) =cC_n−k^(λ+k)(t), k= 1,2. . . , n, wherecis independent oft.

3) The relation between Cubature Formulas and the Reproducing Kernels The Reproducing Kernel method was first used byI.P M ysovskikh([3]) and later studied byMoller¨ ([2]).

Let a given weight functionW =W(x) be defined on a subsetD⊂R^d. Then, a cubature formula is a linear combination of function values on some points, that approximatesR

Df(x)W(x)dx.

Let I^d[f] = R

Df(x)W(x)dx, f ∈ C(D), D ⊂ R^d for which the moments I^d[x^α], α∈N^d exists and W=W(x) is nonnegative.

We say that the cubature formula has the degree of exactnessm, if it yields the exact value of the integrals for any functionf ∈P^dm, which is a polynomials of degree at mostm.

We denote the space of polynomials of degree at mostnbyP^dn. Let

{P_kⁿ: 1≤k≤r(d, n)}, 0≤n <∞,

(wherer(d, n) =µ(d, n−1) = ^d+n−1_d

), denote a sequence of orthonormal polynomi- als of degreenwith respect to the inner product (1.1), which are linearly independent, where the superscriptnmeans thatP_kⁿ∈P^dnand let denote byPn= (P₁ⁿ, . . . , P_r(d,n)ⁿ ), the vector of all these polynomials.

The n−th Reproducing Kernel K_n(x, y) of the Hilbert space H = P^dn is defined by:

Kn(x, y) =

n

X

k=0

P^T_k(x)Pk(y) =

n

X

k=0 r(d,k)

X

j=1

P_j^k(x)P_j^k(y), ∀x, y∈R^d. (1.5) The method of Reproducing Kernel requires to choose d points: a⁽¹⁾, . . . , a^(d)∈R^d,such that the hypersurfacesH1, . . . Hd, where Hi is the surface defined by Hi ={x∈R^d:Kn(x, a⁽ⁱ⁾) = 0}, intersect atn^d points. The pointsa⁽¹⁾, . . . , a^(d) are chosen as follows.

For a⁽¹⁾ we choose any point that is not a common zero of the polynomial setPn. If the pointsa⁽¹⁾, . . . , a^(r−1)have been chosen, then we choosea^(r)∈

r−1

\

k=1

Hk,

anda^(r) may be any point of this set, which is not a common zero ofPn. We assume that the infinity is not a common point ofH1, . . . , Hd. We present now the following results.

a) The Method of Reproducing Kernel

If H1, . . . , Hd defined by a⁽¹⁾, . . . , a^(d), intersect at n^d distinct points:

{x⁽ⁱ⁾, i= 1, n^d}, then there is a cubature formula of degree 2n, Qn(f) =

d

X

i=1

λif(a⁽ⁱ⁾) +

n^d

X

j=1

µjf(x^(j)), ∀f ∈P^d2n, (1.6) whereλi= 1/Kn(a⁽ⁱ⁾, a⁽ⁱ⁾).

If the weight functionW =W(x) is centrally symmetric, that is,W =W(x) and its support set D satisfy ∀x∈D ⇒ −x∈D, W(−x) =W(x), then there is a modified method of Reproducing Kernel due toMoller¨ ([2]).

LetKf_n denote:

Kfn(x, y) =

n

X

k=0 0r(d,k)

X

j=0 0

P_j^k(x)P_j^k(y), ∀x, y∈R^d, (1.7) whereP0

means that the summation is taken over thosej so that the corresponding P_j^k has the same parity as n. We choose the points a⁽ⁱ⁾ as before except that we replace H_i by the hypersurface Hf_i defined by Hf_i={x∈R^d:Kf_n(x, a⁽ⁱ⁾) = 0} and

we suppose that the infinity is not a common point ofHf₁. . . . ,Hf_d. Then we have, if W =W(x) is centrally symmetric onD⊂R^d.

b) The Modified method of Reproducing Kernel

If Hf1, . . . ,Hfd defined by a⁽¹⁾, . . . , a^(d) intersect at n^d distinct points: {x⁽ⁱ⁾, i= 1, n^d}, then there is a cubature formula of degree 2n+ 1,

Qn(f) =

d

X

i=1

λi[f(a⁽ⁱ⁾) +f(−a⁽ⁱ⁾)]/2 +

n^d

X

j=1

µjf(x^(j)), ∀f ∈P^d2n+1, (1.8)

whereλi= 1/Kfn(a⁽ⁱ⁾, a⁽ⁱ⁾).

Ifd= 2, then the method requires to choose two points a⁽¹⁾ anda⁽²⁾ so that the polynomial surfaceKfn(x, a⁽¹⁾) andKfn(x, a⁽²⁾) have n²common zeros.

In the paper [12] Y. Xu was presented a compact formula of the Reproducing Kernel for the Jacobi type weight functions on the unit ball and on the standard simplex.

The method of Reproducing Kernel yields cubature formulas of degree 2n+ 1 or 2n with n^d+dnor n^d+dn−1 nodes, which is greater than the theoretic lower bound for the number of nodes.

2. Cubature formulas on the unit ball using the reproducing kernel method Letx, y∈R^d and we use the following notations:

< x, y >=x₁y₁+· · ·+x_dy_d, the usual Euclidian inner product,

|x|²=kxk² =< x, x >, the Euclidian norm.

We consider cubature formulas on the unit ballB^d ={x∈R^d: kxk≤1}, with respect to the normalized weight function

Wµ(x) =wµ(1− kxk²)^µ−12, µ≥0, x∈B^d, (2.1) wherewµ is a constant chosen so that the integralR

B^dWµ(x)dx= 1, and we have wµ= 2

ω_d−1

Γ(µ+^d+1₂ )

Γ(µ+¹₂)Γ(^d₂) = Γ(µ+^d+1₂ ) π^d/2Γ(µ+¹₂),

whereω_d−1= 2π^d/2/Γ(d/2) is the surface area of the unit sphere inR^d.

LetK_n(., .) be then−thReproducing Kernel with respect to weight function W_µ. In [12] is presented the following compact formula for the representation of this kernel.

K_n(W_µ;x, y) =c_µ Z 1

−1

C^(µ+

d+1 2 )

n (< x, y >+p

1− kxk²p

1− kyk² t)+ (2.2)

+C^(µ+

d+1 2 )

n−1 (< x, y >+p

1− kxk²p

1− kyk²t)

(1−t²)^µ−1dt, where cµ = 1/R1

−1(1−t²)^µ−1dt andCn^(λ) is the Gegenbauer polynomial of degree n defined by the generating function (1.2), which have the property

C_n^(λ)(−t) = (−1)ⁿC_n^(λ)(t).

If we take in consideration the expressions: Kn(Wµ;x, y)±Kn(Wµ;x,−y) fornbeing even and odd, respectively then it follows from the formula (1.5) and (1.7) that the modified Reproducing Kernel functionKf_n(W_µ;...) is given by the formula

Kfn(Wµ;x, y) =cµ

Z 1

−1

C^(µ+

d+1 2 )

n (< x, y >+p

1− kxk²p

1− kyk² t)(1−t²)^µ−1dt.

(2.3) Forµ→0, in (2.2) and (2.3), one can use the limit

µ→0limcµ

Z 1

−1

f(t)(1−t²)^µ−1dt= f(1) +f(−1)

2 . (2.4)

In the caseµ= ¹₂, we have: W_1/2(x) =d/ω_d−1.

Ifµ= 0 we have: W0(x) =w0(1− kxk)^−1/2and we obtain:

Kf_n(W₀;x, y) = 1 2

C_n^(3/2)(< x, y >+p

1− kxk²p

1− kyk²)+ (2.5) +C_n^(3/2)(< x, y >−p

1− kxk²p

1− kyk²)

.

If we considerkak= 1,we have

Kf_n(W_µ;x, a) =C(µ+(d+1)/2)

n (< x, a >). (2.6)

In this case, ifk ak= 1 thenais not a common zero of the polynomial set Pn, because that Pn has no common zeros if n is even, and it has only origin as common zero ifnis odd.

2.1 The construction of a family of cubature formulas onB^d by using the Gegenbauer polynomials

One can use the properties of the Gegenbauer polynomial Cn^(λ)(t), λ=µ+ (d+ 1)/2, that all its zeros are inside (−1,1), and we denote these zeros by:

−1< t_1,n< t_2,n<· · ·< t_n,n <1, where λ=µ+ (d+ 1)/2.

It is known that these zeros are symmetric with respect to the origin, that is, they satisfy the relation ti,n = −t_{n−(i−1),n}. So, in [12] was given the following strategy to choose the pointsa⁽¹⁾, . . . , a^(d)as follows.

Letnbe fixed and lett∗,n be a fixed zero ofC(µ+(d+1)/2)

n (t), and let define:

a⁽¹⁾ = (1,0, . . . ,0), a^(k)= (b1, . . . , b_k−1,q

1−b²₁− · · · −b²_k−1,0, . . . ,0), 0 ≤ k ≤ d, where the components b1, . . . , bd−1 are determined inductively by the conditions: < a^(k), a^(k+1)>=t_∗,n, which is equivalent with

b²₁+· · ·+b²_k−1+ q

1−b²₁− · · · −b²_k−1 bk =t_∗,n, k= 1, d−1, from which are obtained:

b1=t_∗,n, b2= (t_∗,n−b²₁)/

q

1−b²₁, . . . , and we havebk≤q

1−b²₁− · · · −b²_k−1, becauset_∗,n<1, hencea^(k+1) is well defined. It follows that

k

\

i=1

H_k={x∈R^d:< x, a⁽¹⁾>=t_i1,n, . . . , < x, a^(k)>=t_ik,n, 1≤i₁, . . . , i_k ≤n}

fork= 2, d.If we assume thata^(k)∈H1T

· · ·T

Hk−1,and we require thata^(k+1)∈ Tk

i=1Hi,one observe thata⁽²⁾= (t∗,n,q

1−t²_∗,n,0, . . . ,0)∈H1. Inductively, if we assume thata^(k)∈Tk−1

i=1 Hi,that is

< a⁽ⁱ⁾, a^(k)>=t_∗,n, 1≤i≤k−1.

Sincea^(k+1) satisfies< a^(k), a^(k+1)>=t_∗,n it follows that:

< a⁽ⁱ⁾, a^(k+1)>=t_∗,n, i= 1, k, that isa^(k+1)∈H₁T

· · ·TH_k.

One can observe that H₁T· · ·TH_d containsn^d distinct points, which are given by the relations:

< x, a⁽¹⁾>= ti₁,n, . . . , < x, a^(d)>= ti_d,n, 1≤i1, . . . , id≤n. (2.7) Theorem 2.1. Let a⁽¹⁾, . . . , a^(d) be defined as above and let Hk be the surface defined byHk ={x∈R^d :Kfn(Wµ;x, a^(k)) = 0}. Then the modified method of the Reproducing Kernel yields, a cubature formula of degree 2n+1, based ona⁽¹⁾, . . . , a^(d) and then^d distinct points determined by (2.7) and have the form

Q_n(f) =

d

X

i=1

λ_i[f(a⁽ⁱ⁾) +f(−a⁽ⁱ⁾)]/2 +

n^d

X

j=1

µ_jf(x^(j)), ∀f ∈P^d2n+1, (2.8)

whereλi= 1/Kfn(a⁽ⁱ⁾, a⁽ⁱ⁾).

We obtain by using (2.6) that

λ_i= 1/Kf_n(W_µ;a⁽ⁱ⁾, a⁽ⁱ⁾) = 1/C(µ+(d+1)/2)

n (1) = 1/

n+ 2µ+d n

. (2.9) For fixeddandn, the others weightsµj in (2.8) can be determined by solving a linear system of equations.

From the fact that in definition ofa^(k), if we use the condition

< a^(k−1), a^(k)>=t∗,n,

we remark that one can chooset_∗,n to be any zero of the polynomialC(µ+(d+1)/2)

n (t)

and we can get many different formulas from this method.

Remark 2.1. Whenn is an odd integer, then C(µ+(d+1)/2)

n is an odd polynomial, and it follows thatt= 0 is a zero of this polynomial.

If we take t_∗,n = 0 in the definition of a^(k) in the above construction, then we obtain: a⁽¹⁾ =e1, . . . , a^(d)=ed, where {ei, i= 1, d} is the standard basis of R^d, that is,e1= (1,0, . . . ,0), e2= (0,1,0, . . . ,0), . . . , ed= (0, . . . ,0,1).

But from (2.6) it follows that

Kf_n(W_µ;x, e_k) =C(µ+(d+1)/2)

n (x_k), k= 1, d

and we observe that the n^d intersection points ofH₁T· · ·TH_d, namely{x⁽ⁱ⁾, i= 1, n^d}, are the tensor product of the zeros obtained from (2.7).

Letnbe an odd integer and lett1,n, . . . , tn,n be the zeros ofC(µ+(d+1)/2)

n (t).

Then there is a cubature of degree 2n+ 1 onB^d of the form:

Z

B^d

f(x)W_µ(x)dx= 1





n+ 2µ+d n





n

X

k=1

f(e_k) +f(−ek)

/2+ (2.10)

+

n

X

k₁=1

· · ·

n

X

k_d=1

µk₁,...,k_df(tk₁,n, . . . , tk_d,n), ∀f ∈P^d2n+1.

The weights µj in the formula (2.10) can be computed by solving a linear system equations for a givennandd.

In the cased= 2, we can consider the polynomialsl_k,ndefined by:

l_k,n=

n

Y

i=1,i6=k

x−ti,n

t_k,n−t_i,n = C(µ+(d+1)/2)

n (t)

(2µ+d+ 1)C_n−1^µ+(d+3)/2)(t_k,n)(x−t_k,n) ,

which are the fundamental interpolation polynomials based on the zeros of C(µ+(d+1)/2)

n (t) which satisfies the interpolation conditions: lk,n(tj,n) =δk,j,by using (1.4).

One observe that the polynomial lk₁,n(x1)lk₂,n(x2)(1−x²₁−x²₂) is of degree 2(n−1) + 2 = 2n, then it will be integrated exactly by the cubature formula (2.10), and from the interpolation property oflk,n we will obtain the values of the weights are

µ_k1,k2 = Z

B²

l_k1,n(x₁)l_k2,n(x₂)(1−x²₁−x²₂)W_µ(x₁, x₂)dx₁dx₂.

The formula (2.10) uses the tensor product of nodes of an one variable quad- rature rule. The points{t1,n, . . . , tn,n}are nodes of a Gaussian quadrature formula of degree 2n−1 on [−1,1] for the measure: W(x) = (1−x²)^µ+d/2dxon [−1,1].Moreover, {−1, t1,n, . . . , tn,n,1} form the nodes of aGauss−Lobattotype quadrature formula of degree 2n+ 1,

Z 1

−1

f(x)(1−x²)^µ+d/2dx=Af(−1) +

n

X

k=1

λkf(tk,n) +Af(1), ∀f ∈P¹2n+1. (2.11)

The tensor product of {−1, t_1,n, ..., t_n,n,1}, can be used as nodes in the fol- lowing product formula of degree 2n+ 1 for the product weight function:

W(x) =

d

Y

k=1

(1−xk)^µ+d/2on [−1,1]^d,

Z

[−1,1]^d

f(x)

d

Y

k=1

(1−x_k)^µ+d/2dx=

n+1

X

k₁=0

· · ·

n+1

X

k_d=0

λ_k1. . . λ_kdf(t_k1,n, . . . , t_kd,n), (2.12) for∀f ∈P^d2n+1, wheret0,n=−1, tn+1,n= 1 and λ0=λn+1=A.

It was showed that some nodes of the cubature formulas constructed above can lie outside of the unit ball B^d. But we can choose different valuesa^(k)in order to construct formulas with all nodes inside ofB^d.

2.2 Samples of cubature formulas of lower degree with nodes insideB^d We use the modified method of the Reproducing Kernel to construct cubature formulas of lower degree with nodes insideB^d.

a. Formulas of degree 5

We choosea⁽¹⁾= (0,0, . . . ,0) the origin ofR^d and we definea^(k+1), 1≤k≤ d−1 by

a^(k+1)= (

r 1

2µ+d+ 3, . . . ,

r 1 2µ+d+ 3,

s

d+ 3−k

2µ+d+ 3,0. . . ,0) (2.13) which hasd−kzero components.

From the properties of theGegenbauer polynomials [7], we have:

C₂^(λ)(t) =λ[2(λ+ 1)t²−1], forn= 2, whereλ=µ+ (d+ 1)/2, and follows that

K˜2(Wµ;x, y) =λ

(2µ+d+ 3)< x, y >²+(2µ+d+ 3)(1− |x|²)(1− |y|²)/(2µ+ 1)−1

. (2.14) If we take,a⁽¹⁾= (0, . . . ,0), it follows from the formula of ˜K2(Wµ;x, y) that H1 ={x:K2(x, a⁽¹⁾) = 0} ={x:|x|² = (d+ 2)/(2µ+d+ 3)} and we require that the chosen pointa^(k+1)from (2.13), belongs toHk and we obtain:

K˜₂(W_µ;x, a^k+1) = (µ+d+ 1 2 )

x²₁+· · ·+x²_k−1+ (d+ 3−k)x²_k− kxk²

,

from which we obtaina^(k+1)∈

k

\

i=1

Hi and

d

\

i=1

Hi={(±

r 1

2µ+d+ 3, . . . ,±

r 1

2µ+d+ 3,±

r 3

2µ+d+ 3)} (2.15)

Thus,

d

\

i=1

H_i has the 2^d intersection points obtained from (2.15).

Now we can apply the modified Reproducing Kernel method, from which one results that the nodes of the cubature formula are {a⁽ⁱ⁾, i = 1, d} from (2.13) and (x^(j), j= 1,2^d) from (2.15) and these nodes generates a cubature formula of degree 5 onB^d of the form (2.8). Using the formula of ˜K₂(W_µ;x, y) one get the coefficients of the formula for this choice of the nodesa^(k+1),if we considern= 2

λ1= 1/K˜2(Wµ; 0,0) = 2(2µ+ 1)

(2µ+d+ 1)(d+ 2), (2.16) λk+1= 1/K˜2(Wµ;ak+1, ak+1) = 2(2µ+d+ 3)

(2µ+d+ 1)(d+ 2−k)(d+ 3−k), k= 2, d.

Then there exists the weightsµξ such that the following cubature formula is of degree 5 forWµ onB^d [12].

Z

B^d

f(x)Wµ(x)dx= 2(2µ+ 1)

(2µ+d+ 1)(d+ 2)f(0) +2µ+d+ 3

2µ+d+ 1

d−1

X

k=1

f(a^(k+1)) +f(−a^(k+1)) (d+ 2−k)(d+ 3−k)

+ X

ξ∈{−1,1}^d

µ_ξf

ξ₁

r 1

2µ+d+ 3, . . . , ξ_d−1

r 1

2µ+d+ 3, ξ_d

r 3 2µ+d+ 3

. (2.17) In this formula the weightsµξ, ξ= (ξ1, . . . , ξd)∈ {−1,1}^dcan be determined by the condition that the formula must be exact for polynomials of degree 5.

In the case ofd= 2, we have the explicit formula Z

B²

f(x)Wµ(x)dx= 2(2µ+ 1)

4(2µ+ 3)f(0) + 2µ+ 5 12(2µ+ 3)

f(2/p

2µ+ 5,0) (2.18)

+f(−2/p

2µ+ 5,0)

+ 2µ+ 5 12(2µ+ 3)

Xf(±1/p

2µ+ 5,±√ 3/p

2µ+ 5).

The formula on B^d uses N = 2^d+ 2d−1 nodes. According with M¨oller’s lower bound [2], the cubature formula of degree 5 must have at leastN^∗≥d(d+ 1) + 1 nodes, then the formula (2.18) which haveN = 2²+ 2·2−1 = 7 is minimal.

Ford= 3, the cubature formula onB³, which was constructed by using (2.10) in [12], haveN = 13 nodes and is minimal; ford= 5, N= 2⁵+ 2·5−1 = 41 nodes which is more that the lower bound ofN^∗= 5(5 + 1) + 1 = 31.

Finally we obtain the formula (2.17). To determine the other coefficients, one can require that the formula be exact for the polynomials of degree at most 5.

Ford= 3, we can choosef(x) to be the test functions x₁, x₁x₂, x²₁, x₁x₂x₃. For the case of d > 3, it is useful the following formula for the nonzero moments of the weight functionW_µ=W_µ(x) ([12])

Z

B^d

x^2k₁ ¹. . . x^2k_d ^dWµ(x)dx=Γ(µ+ (d+ 1)/2) Γ(k1+ 1/2). . . Γ(kd+ 1/2) π^d/2Γ(µ+ (d+ 1)/2 +k₁+· · ·+k_d) . 3. Cubature formulas on the triangle using the reproducing kernel method

We consider now, cubature formulas on the triangle using the compact for- mula in [12], for a family of weight functions on ad-dimensional simplex.We use the fol- lowing notations:x∈R^d, |x|₁=|x₁|+· · ·+|x_d|, thel¹norm of x,|α|₁=α₁+· · ·+α_d, the length of multiindexα∈N^d and the standard simplex:

T^d={x∈R^d: x₁≥0, . . . , x_d≥0, 1− |x|1≥0}.

We remark that, for d = 2 we have T² which is the triangle with vertices (0,0),(1,0) and (0,1).

In [12] was found the compact formula for the Reproducing Kernel with respect to the weight function:

Wα(x) =wαx^α₁^1−1/2. . . x^α_d^d−1/2(1− |x|1)^αd+1−1/2, αi≥0, (3.1) wherewαis the normalization constant such thatR

T^dWα(x)dx= 1 , namely, w_α= Γ(|α|₁+ (d+ 1)/2)

Γ(α1+ 1/2). . .Γ(αd+1+ 1/2).

Then the reproducing KernelK_n(W_α) given in terms ofGegenbauer polyno- mials, has the expression [12]:

K_n(W_α;x, y) = Z

[−1,1]^d+1

C_2n^{(|α|1+(d+1)/2)}(√

x₁y₁t₁+· · ·+√

x_d+1y_d+1 t_d+1). (3.2)

.

d+1

Y

i=1

cα_i(1−t²_i)^αi−1dt, where

x, y∈T^d, xd+1 = 1− |x|1, yd+1= 1− |y|1, and we use limit (2.4) in the case when have oneα_i= 0.

If we takey=e_i= (0, . . .0,1,0, . . .0), thei-th element of the standard basis, with the i-th component =1, ofR^d, 1 ≤i ≤ d, then we have the following explicit formula:

Kn(Wα;x, ei) =Aα,iP_n^{(|α|1+d/2−αi,αi−1/2)}(2xi−1) where

Aα,i=C_2n^{(|α|1+(d+1)/2)}(0)/P_n^{(|α|1+d/2−αi,αi−1/2)}(−1) (see [12]).

This formula was derived in [14] from (3.2) using a product formula forJacobi polynomials.

We observe that, ei is not a common zero of Pn. This follows from the expression ofP^T_n(x)Pn(y) =P

kP_kⁿ(x)P_kⁿ(y).

Letd= 2 and α₁ =α₂=α₃ = 1/2. Then the weight functionW_α becomes a multiple of unit weight function, denoted byW_1/2, and we have: W_1/2(x) = 2.

In this case, the Reproducing Kernel takes the form:

Kn(W1/2;x, y) = 1 π³

Z

[−1,1]³

C_2n⁽³⁾(√

x1y1t1+√

x2y2t2+√ x3y3t3)

3

Y

i=1

(1−t²_i)^−1/2dt Forα= 0, we have W0(x) = (x1x2x3)^−1/2/2π.

In [11] was shown that any cubature formula for W0 with all nodes inside T²corresponds to a cubature formula on a sphereS². In this case, the Reproducing

Kernel can be represented in the following simple form:

K_n(W₀;x, y) = 1 4

XC_2n^(3/2)(√

x₁y₁±√

x₂y₂±√ x₃y₃),

where the sum is over all possible sign changes, and this formula follows from (3.2) by taking limits (2.4).

Samples of cubature formulas on the triangle

Forn= 2, we have the following explicit formula forKn(W_1/2;x, y) K2(W_1/2;x, y) = 6(1−10(x1y1+x2y2+x3y3) + 60(x1x2y1y2+x1x3y1y3+

+x2x3y2y3) + 15(x²₁y₁²+x²₂y₂²+x²₃y²₃)).

If we takea⁽¹⁾= (1,0),one obtain that K2(W_1/2, x,(1,0)) has two zeros, z₁= (5−√

10)/15, z₂= (5 +√ 10)/15.

From this fact, it follows thatK₂(W_1/2, x,(1,0)) andK₂(W_1/2, x,(z₁,0)) have 4 dis- tinct common zeros:

(5−√

10)/15,(70−7√ 10±

q

10(233−62√ 10)/90

(5 +√

10)/15,(30−3√ 10±

q

3(110−20√ 10)/90

.

4. The construction of cubature formulas by using the Chebyshev orthogonal polynomials and the reproducing kernel method

Let us consider, theChebyshev polynomial of degree n, T_n^∗(x) =cosnθ, x=cosθ,

that is

T_n^∗(x) =cos(narcosx).

The zeros ofT_n^∗ are xk = ^(2k−1)π_2n , k= 1, n,andT_n^∗ are orthogonal with respect to theChebyshev weight functionw₁(x) = (1−x²)^−1/2 on [−1,1].

The zeros of T_n^∗ can be selected as the nodes of the Gaussian quadrature formula with respect to w(x) and these zeros can be used to construct a compact interpolation formula. Let we denote the classicalChebyshev weight of the first kind

w1(x) = 1 π

√ 1

1−x², x∈(−1,1) Then the orthonormal polynomials with respect tow₁ are

T0(x) = 1, Tk(x) =√

2coskθ, k≥1, x=cosθ and Z 1

−1

w1(x)dx= 1.

Next, we can consider the productChebyshev weight function on [−1,1]² defined by W⁽²⁾(x, y) =w1(x)w1(y) = 1

π²

√ 1 1−x²

1

p1−y², (x, y)∈[−1,1]². (4.1) One can verify that the polynomials defined by

P_kⁿ(x, y) =Tn−k(x)Tk(y), k= 0, n, n∈N0, (4.2) whereP_kⁿ is of degree exactlynare orthogonal with respect toW⁽²⁾(x, y).

In [10] was established the following relations. If we denote Pn = (P₀ⁿ, ..., P_nⁿ)^T, n ∈ N0, the vector of the polynomials of degree exactly n in (4.2) and the matrices,

A_n,1=1 2

















1 0 0 . . . 0

0 1 0 . . . 0

... ... . .. ...

0 0 . . . √

2 0

















, A_n,2= 1 2

















0 √

2 0 . . . 0

0 0 1 . . . 0

... ... . .. ...

0 0 . . . 1















 ,

it can be verified that productChebyshev polynomials satisfy the three-term relation x_iP_n(x) =A_n,iP_n+1(x) +A^T_n−1,iP_n−1(x), i= 1,2, x= (x₁, x₂) orx= (x, y) (4.3) For x, y ∈ R², the Reproducing Kernel of the product Chebyshev polynomials is defined by

Kn(x, y) =

n−1

X

k=0 k

X

j=0

P_j^k(x)P_j^k(y) =

n−1

X

k=0

P^T_k(x)Pk(y) andP^T_n(x)P_n(y) =K_n(x, y)−K_n−1(x, y).

If one consider x = (cosθ₁, cosθ₂), y = (cosϕ₁, cosϕ₂), then we have the compact formula [10].

Kn(x, y) =Dn(θ1+ϕ1, θ2+ϕ2) +Dn(θ1+ϕ1, θ2−ϕ2) +Dn(θ1−ϕ1, θ2+ϕ2) +Dn(θ1−ϕ1, θ2−ϕ2),

where the functionD_n has the form Dn(θ1, θ2) = 1

2

cos(n−¹₂)θ1cos^θ₂¹ −cos(n−¹₂)θ2cos^θ₂²

cosθ₁−cosθ₂ .

One can use these formulas in order to obtain a compact formula for the Lagrange interpolation, which will be used to construct a cubature formula of degree 2n−1 with respect toW⁽²⁾(x, y) of the form

In(f) = Z

[−1,1]²

f(x, y)W⁽²⁾(x, y)dxdy'Qn(f), (4.4)

whereQn(f) =

N

X

k=0

λkf(xk), λk>0, xk ∈R²,so that we have

I_n(P) =Q_n(P), ∀P ∈P²2n−1.

According to a general result ofM¨oller for centrally symmetric weight func- tions, for example one can considerW⁽²⁾(x, y) =w1(x)w1(y), the number of nodes in the cubature formula satisfies

N ≥dimP²n−1+ [n/2] = n+ 1

2

+ [n/2].

Let considerzk be the pointszk =zk,n=cos^kπ_n , k= 0, n.

In [10] was stated, based on the three-term recurrence relation (4.3), that a cubature formula exists when the following matrix equations in the variable V are solvable

A_n−1,1(V V^T −I)A^T_n−1,2=A_n−1,2(V V^T −I)A^T_n−1,1 (4.5) and V^TA^T_n−1,1An−1,2V =V^TA^T_n−1,2An−1,1V,

whereV is a matrix of size (n+ 1)×σ, σ= [n/2] orσ= [n/2] + 1.

Ifn= 2min [10] was showed that a solution of (4.5) is

T_n−(k−1)(x)T_k−1(y)−T_k−1(x)T_n−(k−1)(y),1≤k≤n/2 + 1, (4.6) which corresponds to (A), and ifn= 2m−1,a solution of (4.5) is

T_n−(k−1)(x)T_k−1(y)−T_k−1(x)T_n−(k−1)(y), 1≤k≤(n+ 1)/2, (4.7) corresponds to (B).

If a cubature formula exists, we can consider the Lagrange interpolation prob- lem based on the nodes of the cubature formula which consists in construction of a unique polynomial which is the solution of the problem to determiningP =P(x) so thatP(xk) =f(xk), k= 1, N .

In [8], was proved that one can consider the subspace V_n²=P²n−1

[span{V⁺Pn},

whereV⁺ is the uniqueMoore-Penrose generalized inverse ofV, and in our case we haveV with full rank and we haveV⁺ = (V^TV)⁻¹V^T.

For (x, y)∈R², was used the following expression of the Reproducing Kernel K_n^∗(x, y) =K_n(x, y) + [V⁺P_n(x)]^TV⁺P_n(y). (4.8) Using a modifiedChristoffel-Darboux formula, was showed in [10] thatK_n^∗(x_k, x_j) = 0 fork6=j andK_n^∗(xk, xk)6= 0.

Finally, it follows that (Lnf)(x) =

N

X

k=1

K_n^∗(x, xk)

K_n^∗(xk, xk)f(xk) (4.9) and we have

Z

[−1,1]²

(Lnf)(x)W0(x)dx=

N

X

k=1

λkf(xk) =In(f).

From the condition on P_j^k and the definition of K_n^∗(·,·) it follows that the coefficients in the cubature formula are given by the expressionλk= 1/K_n^∗(xk, xk)

Ifn= 2m, the interpolation nodes are

x_2i,2j+1= (z_2i, z_2j+1), i= 0, m, j= 0, m−1 (4.10)

x_2i+1,2j= (z_2i+1, z_2j), i= 0, m−1, j= 0, m.

From (4.7) and the expression ofK_n^∗(x, y) one can obtain K_n^∗(x, xk,l) =1

2[Kn(x, xk,l) +K_n−1(x, xk,l)]−1

2(−1)^k[Tn(x)−Tn(y)].

Finally, one can obtain

K_n^∗(x0,2j+1, x0,2j+1) =n², K_n^∗(x2i,2j+1, x2i,2j+1) =n²/2, K_n^∗(x2i+1,0, x2i+1,0) =n², K_n^∗(x2i+1,2j, x2i+1,2j) =n²/2, i >0, j >0.

Ifn= 2m−1, the interpolation nodes are

x_2i,2j= (z_2i, z_2j), i, j= 0, m−1 x2i+1,2j+1= (z2i+1, z2j+1), i, j= 0, m−1, from which, was derived

K_n^∗(x, xk,l) =1

2[Kn(x, xk,l) +Kn−1(x, xk,l)]−1

2(−1)^k[Tn(x) +Tn(y)], from which was obtained

K_n^∗(x2i,2j, x2i,2j) =















n²/2, if 0< i, j≤m−1

n², ifi= 0 orj= 0, i+j >0 2n², ifi=j= 0,

K_n^∗(x2i+1,2j+1, x2i+1,2j+1) =















n²/2, if 0≤i, j < m−1

n², ifi=m−1 orj =m−1, i+j <2m−2 2n², ifi=j =m−1.

In [14] was proved the mean convergence of Lagrange interpolation formula corre- sponding to the weight functionW⁽²⁾(x, y) and by integrating this formula one can arrive to the following cubature formulas

Based on the nodes (x_i, x_j),we obtain the cubature formulas:

A) Forn= 2m, (A) 1

π² Z 1

−1

Z 1

−1

f(x, y) dxdy

√1−x²p

1−y² = 2 n²

n 2

X

i=0 00n

2−1

X

j=0

f(z_2i, z_2j+1)+

+2 n²

n 2−1

X

i=0

n 2

X

j=0 00

f(z2i+1, z2j),∀f ∈P²2n−1

B) Forn= 2m−1, (B) 1

π² Z 1

−1

Z 1

−1

f(x, y) dxdy

√1−x²p

1−y² = 2 n²

n−1 2

X

i=0

n−1 2

X

j=0

f(z2i, z2j)+

+ 2 n²

n−1 2

X

i=0

n−1 2

X

j=0

f(z_n−2i, z_n−2j),∀f ∈P²2n−1, where Σ⁰ means that the first term in summation is halved.

References

[1] Cools, R., Mysovskikh, I.P., Schmid, H.J.,Cubature Formulae and Orthogonal Polyno- mials, J.Comput.& Appl. Math,127(2001), 121-152.

[2] M¨oller, H.M.,Polynomideale und Kubaturformeln, Thesis, Univ. Dortmund, 1973.

[3] Mysovskikh, I.P.,Interpolatory Cubature formulas, Nauka, Moscow, (1981), (Russian).

[4] Mysovskikh, I.P.,A representation of the reproducing Kernel of a sphere, Comp. Math.

Math. Phys.,36(1996), 303-308.

[5] Stancu, D.D.,Generalizarea unor formule de interpolare pentru funct¸iile de mai multe variabile ¸si unele considerat¸ii asupra formulelor de integrare numeric˘a a lui Gauss, Ed.

Acad. R.P.R., Bul.S¸t. Sec. St. Mat. ¸si Fiz.,2(1957).

[6] Stroud, A.H.,Approximate calculation of multiple integrals, Prentice Hall, Englewood Cliffs, NJ, (1971).

[7] Szeg¨o, H., Orthogonal polynomials, 4th ed., Amer. Math. Soc. Collaq. Publ. vol. 23, Providence, RI, (1975).

[8] Xu, Y.,Common zeros of polynomials in several variables and Higher dimensional quad- rature, Pitman Research Notes in Mathematics series, Longman, Essex, (1994).

[9] Xu, Y., A Class of Bivariate orthogonal polynomials and cubature formulas, Nu- mer.Math.,69(1994), 233-241.

[10] Xu, Y.,Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory, 87(1996), 220-238.

[11] Xu, Y.,Orthogonal polynomials and cubature formulae on spheres and on balls, SIAM J. Math. Anal.,29(1998), 779-793.

[12] Xu, Y., Constructing cubature formulae by the method of reproducing kernel, Numer.

Math.,85(2000), 155-173.

[13] Xu, Y., Li - Sumability of the product Jacobi series, J. Approx Theory, 104(2000), 287-301.

[14] Xu, Y., Summability of Fourier orthogonal series for Jacobi weight on a ball in R^d, Trans. Amer. Math. Soc.

[15] Xu, Y.,Representation of reproducing Kernels and the Lebesque constants on the Ball, J. Approx. Theory,112(2001), 295-310.

V. Goldis¸ 51 A, 510018, Alba-Iulia, Romania E-mail address: emil [email protected]