# Weighted Approximation for Jackson-Matsuoka Polynomials on the Sphere

Authors: Guo Feng
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider the best approximation by Jackson-Matsuoka polynomials in the weightedLp space on the unit sphere of ℝd. Using the relation between K-functionals and modulus of smoothness on the sphere, we obtain the direct and inverse estimate of approximation by these polynomials for the h-spherical harmonics.


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## Body

## 1. Introduction and Notations

Let𝕊:=𝕊d-1={x:∥x∥=1} denote the unit sphere in ℝd(d≥3),d∈ℕ, where ∥x∥ denotes the usual Euclidean norm, ℝ the set of real numbers. For a nonzero vector v∈ℝd, let σv denote the reflection with respect to the hyperplane perpendicular to v,xσv:=x-2(〈x,v〉/∥v∥2)v,x∈ℝd, where 〈x,v〉 denote the usual Euclidean inner product. Let G be a finite reflection group on ℝd with a fixed positive root system ℝ+, normalized so that 〈v,v〉=2 for all v∈ℝ+. Then G is a subgroup of the orthogonal group generated by the reflections {σv:v∈ℝ+}. Let κ be a nonnegative multiplicity function v↦κv defined on ℝ+ with the property that κu=κv whenever σu is conjugate to σv in G, then v↦κv is a G-invariant function. We consider the weighted best Lp approximation with respect to the measure hκ2dω on 𝕊, where hκ2 is defined by(1.1)hκ=∏v∈R+|〈x,v〉|κv,x∈Rd,dω is the surface (Lebesgue) measure on 𝕊. The function hκ is a positive homogeneous function of degree γκ:=∑v∈R+κv, and it is invariant under the reflection group. We denote by aκ the normalization constant of hκ,aκ-1=∫𝕊hκ2(y)dω and denote by Lp(hκ2), 1≤p≤∞, the space of functions defined on 𝕊 with the finite norm(1.2)‖f‖κ,p:=(aκ∫S|f(y)|phκ2(y)dω(y))1/p,1≤p<∞,
and for p=∞ we assume that L∞ is replaced by C(𝕊) the space of continuous functions on 𝕊 with the usual uniform norm ∥f∥∞.Δh denote the h-Laplacian. Δh,0 is the Laplace-Beltrami operator on the sphere. 𝒫nd denote the subspace of homogeneous polynomials of degree n in d variables. The h-harmonics are defined as the homogeneous polynomials satisfying the equation ΔhP=0,P∈𝒫nd. Furthermore, let ℋnd(hκ2) denote the space of h-spherical harmonics of degree n in d variables. The spherical h-harmonics are the restriction of h-harmonics on the unit sphere. It is well known that spherical h-harmonics are eigenfunctions of Δh,0; that is,(1.3)Δh,0Y(x)=-n(n+2λ)Y(x),x∈S,Y∈Hnd(hκ2).The standard Hilbert space theory shows thatL2(hκ2)=∑n=0∞⊕ℋnd(hκ2). That is, with each f∈L2(hκ2) we can associate its h-harmonic expansion(1.4)f(x)=∑n=0∞Yn(hκ2;f,x),x∈S,
in L2(hκ2) norm. For the surface measure (κ=0), such a series is called the Laplace series (see [1]). The orthogonal projection Yn(hκ2):L2(hκ2)→ℋnd(hκ2) takes the form(1.5)Yn(hκ2;f,x):=∫Sf(y)Pn(hκ2;x,y)hκ2(y)dω(y),
where Pn(hκ2;x,y) is the reproducing kernel of the space of h-harmonics ℋnd(hκ2), which is given by (see [2])(1.6)Pn(hκ2;x,y)=n+λλVκ[Cnλ(〈⋅,y〉)](x).Cnλ is the ultraspherical polynomial of degree n,λ:=γκ+(d-2)/2, γκ=∑v∈ℝ+κv, and the intertwining operator Vκ is a linear operator uniquely determined by(1.7)VκPn⊂Pn,Vκ1=1,DiVκ=Vκ∂i,1≤i≤d.The spherical means are denoted by(1.8)Tθ(f)=1|Sd-2|(sin⁡θ)d-2∫〈x,y〉=cos⁡θf(y)dω(y),
where |𝕊d-2|=∫𝕊d-2dω=2π(d-1)/2/Γ((d-1)/2).The spherical means associated withhκ2dω, in which Tθκ(f) is defined by(1.9)cλ∫0πTθκ(f,x)g(cos⁡θ)(sin⁡θ)2λdθ=aκ∫Sf(y)Vκg(〈x,y〉)hκ2(y)dω(y),
where g is any function [-1,1]↦ℝ such that the integral in the right-hand side is finite, cλ-1=∫-11(1-t2)λ-1/2dt=Γ(λ+1/2)π/Γ(λ+1). Tθκ(f) is a proper extension of Tθ(f), since Tθ(f) satisfies Tθκ(f) when κ=0 and Vκ=id, and the properties of Tθκ are well known (see [2]). In particular, the function Tθκf(x) has the expansion
(1.10)Tθκ(f)∼∑n=0∞Cnλ(cos⁡θ)Cnλ(1)Yn(hκ2;f):=∑n=0∞Qnλ(cos⁡θ)Yn(hκ2;f).
Simultaneously, they lead to the following definition of an analog of the modulus of smoothness.Definition 1.1 (see [2]).
Forf∈Lp(hκ2), 1≤p<∞, or f∈C(𝕊), the modulus of smoothness on the sphere is given by
(1.11)ω(f;t)κ,p:=sup⁡0<θ≤t⁡‖f-Tθκ(f)‖κ,p.
The K-functional of the sphere is given by
(1.12)K(f;t2)κ,p=inf⁡g∈Wp(hκ2)⁡{‖f-g‖κ,p+t2‖Δh,0g‖κ,p},
where Wp(hκ2):={f:f∈Lp(hκ2),-k(k+2λ)Pk(hκ2;f)=Pk(hκ2;g)forsomeg∈Lp(hκ2)}, 0<t<t0, t0 is a positive constant.In [2], Xu proved the weak equivalence relation(1.13)C-1ω(f;t)κ,p≤K(f;t2)κ,p≤Cω(f;t)κ,p.
Throughout this paper, C denotes a positive constant independent on n and f and C(a) denotes a positive constant dependent on a, which may be different according to the circumstances.Based on the classical Jackson-Matsuoka kernel (see [3]), we define a new kernel(1.14)Mn;j,i,s(θ):=1Ωn;j,i,s(sin⁡2jnθ/2sin⁡2iθ/2)2s,n=1,2,…,θ∈R,
where j,i,s∈ℕ,Ωn;j,i,s is a constant chosen such that cλ∫0πMn;j,i,s(θ)sin2λθdθ=1. It is known that Mn;j,i,s(θ) is an even nonnegative operator. In particular, it is an even nonnegative trigonometric polynomial of degree at most 2s(nj+2j-2i) for j>i and the Jackson polynomial for j=i. Using Mn;j,i,s(θ) we consider the spherical convolution(1.15)Jn;j,i,s(f;x):=(f*Mn;j,i,s)(x):=cλ∫0πTθκ(f;x)Mn;j,i,s(θ)(θ)sin⁡2λθdθ.
It is called the Jackson-Matsuoka polynomials on the sphere based on the Jackson-Matsuoka kernel. In particular, (f0*Mn;j,i,s)(x)=1 for f0(x)=1. The classical Jackson-Matsuoka polynomials in the classical Lp space have been studied by many authors (see [3, 4]).The purpose of this paper is to consider approximation byh-harmonic polynomials, which in the Lp metric can be viewed as weighted approximation, in which the measure dω on the sphere is replaced by hκ2dω. It is well known that the situation can be quite different from that of ordinary harmonics; the weighted approximation is not a simple extension. Since the orthogonal group acts transitively on the sphere 𝕊, much of the results for the ordinary harmonics can be proved by considering just one point; the reflection groups do not act transitively on the sphere.In this paper, we consider weighted approximation of the Jackson-Matsuoka polynomials on the sphere. With the help of the relation betweenK-functionals and modulus of smoothness of sphere and the properties of the spherical means, we obtain the direct and inverse estimate for the best approximation by Jackson-Matsuoka polynomials in the weighted Lp space on the unit sphere of ℝd. We only consider best weighted approximation by Jackson-Matsuoka polynomials, and for the other polynomials on the unit sphere of ℝd, the methods and the results are similar.


## 2. Auxiliary Lemmas

We need the following lemmas.Lemma 2.1.
LetΩn;,j,i,s=∫0π((sin2jnθ/2)/(sin2iθ/2))2ssin2λθdθ. Then, the weak equivalence
(2.1)Ωn;j,i,s≍n4is-2λ-1
holds true for 4si>2λ+1, j≥i, where the weak equivalence relation A(n)≍B(n) means that A(n)≪B(n) and B(n)≪A(n), and relation An≪Bn means that there is a positive constant C independent on n such that A(n)≤CB(n) holds.The proof is similar to that of Lemma2.2 and we omit it.Lemma 2.2.
For4is>r+2λ+1,  j≥i,  r∈ℝ, there is a constant  C(λ,j,i,s) such that
(2.2)∫0πθrMn;j,i,s(θ)sin⁡2λθdθ≤C(λ,j,i,s)n-r.Proof.
Sinceθ/π≤sin(θ/2)≤θ/2 and sinθ≤θ hold for 0≤θ≤π, by Ωn;j,i,s≍n4is-2λ-1, we have
(2.3)∫0πθrMn;j,i,s(θ)sin⁡2λθdθ≤C(λ,j,i,s)n-4is+2λ+1∫0πθr(sin⁡2jnθ/2sin⁡2iθ/2)2ssin⁡2λθdθ≤C(λ,j,i,s)n-4is+2λ+1n4is-r-2λ-1∫0nπ/2tr+2λ(sin⁡2jtt2i)2sdt≤C(λ,j,i,s)n-r(∫0π/2tr+2λ(sin⁡2jtt2i)2sdt+∫π/2∞tr+2λ(sin⁡2jtt2i)2sdt)≤C(λ,j,i,s)C2nλ≤C(λ,j,i,s)nλ,
where
(2.4)C2=∫0π/2tλ(sin⁡2jtt2i)2sdt+∫π/2∞tλ(sin⁡2jtt2i)2sdt,4is>r+2λ+1,j≥i.
Lemma 2.2 has been proved.Lemma 2.3 (see [2]).
For0≤θ≤π, one has
(2.5)Tθκ(g;x)-g(x)=∫0θsin⁡-2λtdt∫0tTuκ(Δh,0g)sin⁡2λudu=∫0θsin⁡-2λtΦ(t)Bt(Δh,0g,x)dt,
where
(2.6)Bt(Δh,0g,x)=1Φ(t)∫0tTuκ(Δh,0g)sin⁡2λudu,
and Φ(t)=cλ-1∫0tsin2λudu.Lemma 2.4.
Letg,Δh,0g,Δh,02g∈Lp(hκ2),1≤p≤∞,Jn;j,i,s(f;x) be the Jackson-Matsuoka polynomials on the sphere based on the Jackson-Matsuoka kernel,  4is>2λ+5,j≥i. Then, there is a constant  C(λ,j,i,s) such that
(2.7)‖Jn;j,i,sg-g-α(n)Δh,0g‖κ,p≤C(λ,j,i,s)n-4‖Δh,02g‖κ,p,
where α(n)≍n-2.Proof.
By Lemma2.3, we have
(2.8)Jn;j,i,s(g;x)-g(x)=cλ∫0πMn;j,i,s(θ)(Tθκ(g;x)-g(x))sin⁡2λθdθ=cλ∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θΦ(t)sin⁡2λtBt(Δh,0g,x)dt=cλΔh,0g(x)∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θΦ(t)sin⁡2λtdt+cλ∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θΦ(t)sin⁡2λt(Bt(Δh,0g,x)-Δh,0g(x))dt=Δh,0g(x)∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θdtsin⁡2λt∫0tsin⁡2λudu+∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θdtsin⁡2λt∫0tsin⁡2λu(Bt(Δh,0g,x)-Δh,0g(x))du:=α(n)Δh,0g(x)+∫0πMn;j,i,s(θ)sin⁡2λθΨθ(g,x)dθ,
where
(2.9)α(n):=∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θdtsin⁡2λt∫0tsin⁡2λudu,Ψθ(g,x):=∫0θdtsin⁡2λt∫0tsin⁡2λu(Bt(Δh,0g,x)-Δh,0g(x))du.
By Lemma 2.1, we have
(2.10)α(n)=∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θdtsin⁡2λt∫0tsin⁡2λudu≍∫0πMn;j,i,s(θ)sin⁡2λθdθ∫0θtsin⁡2λξsin⁡2λtdt≍∫0πθ2Mn;j,i,s(θ)sin⁡2λθdθ≍n-2,(0<ξ<t).
We now estimate, using Lemma 2.3 again, the expression Bt(Δh,0g,x)-Δh,0g(x), and obtain
(2.11)‖Ψθ(g)‖κ,p≤C(λ,j,i,s)θ4‖Δh,02g‖κ,p.
By Lemma 2.2 and Hölder-Minkowski inequality shows that
(2.12)‖∫0πMn;j,i,s(θ)sin⁡2λθΨθ(g,x)dθ‖κ,p≤C(λ,j,i,s)‖Δh,02g‖κ,p∫0πθ4Mn;j,i,s(θ)sin⁡2λθdθ≤C(λ,j,i,s)n-4‖Δh,02g‖κ,p.
Consequently, by (2.8), (2.10), and (2.12) we complete the proof of this lemma.Lemma 2.5.
Fort≥0, there is a constant  C such that
(2.13)ω(f;tδ)κ,p≤Cmax⁡{1,t2}ω(f;δ)κ,p.Proof.
By the equivalence relation between the modulus of smoothness andK-functional, and the definition of K(f;t2)κ,p, we have
(2.14)ω(f;tδ)κ,p≤CK(f;(tδ)2)κ,p≤C(‖f-g‖κ,p+t2δ2‖Δh,0g‖κ,p)≤Cmax⁡{1,t2}(‖f-g‖κ,p+δ2‖Δh,0g‖κ,p)≤Cmax⁡{1,t2}K(f;δ2)κ,p≤Cmax⁡{1,t2}ω(f;δ)κ,p.
Lemma 2.5 has been proved.


## 3. Main Results

Our main results are the following.Theorem 3.1.
Suppose thatf∈Lp(hκ2),1≤p≤∞,Jn;j,i,s(f;x) is the Jackson-Matsuoka polynomials on the sphere based on the Jackson-Matsuoka kernel,  4is>2λ+5,  j≥i. Then
(3.1)‖Jn;j,i,s(f)-f‖κ,p≍ω(f;n-1)κ,p.Proof.
First we prove∥Jn;j,i,s(f)-f∥κ,p≪ω(f;n-1)κ,p. Since (f0*Mn;j,i,s)(x)=1 for f0(x)=1, therefore, we have that
(3.2)‖Jn;j,i,s(f)-f‖κ,p=‖∫0πMn;j,i,s(θ)(f(x)-Tθκ(f;x))sin⁡2λθdθ‖κ,p≤∫0π‖f-Tθκ(f)‖κ,pMn;j,i,s(θ)sin⁡2λθdθ.
Splitting the integral over [0,π] into two integrals over [0,1/n] and [1/n,π], respectively, and using the definition of ω(f;t)κ,p, we conclude that
(3.3)‖f-Tθκ(f)‖κ,p≤ω(f;n-1)κ,p+∫1/nπω(f;θ)κ,pMn;j,i,s(θ)sin⁡2λθdθ.
From Lemma 2.5 it follows that, for θ≥n-1,
(3.4)ω(f;θ)κ,p=ω(f;nθn)κ,p≤Cmax⁡{1,n2θ2}ω(f;θ)κ,p≤Cn2θ2ω(f;θ)κ,p.
Therefore, it follows that
(3.5)‖Jn;j,i,s(f)-f‖κ,p≤ω(f;θ)κ,p(1+Cn2∫1/nπθ2Mn;j,i,s(θ)sin⁡2λθdθ).
From Lemma 2.2, we get
(3.6)‖Jn;j,i,s(f)-f‖κ,p≤C(λ,j,i,s)ω(f;n-1)κ,p.
Next we prove ω(f;n-1)κ,p≪∥Jn;j,i,s(f)-f∥κ,p. Let m be a fixed positive integer Denote by
(3.7)Jn;j,i,sm(f):=∑k=0m(∫0πMn;j,i,s(θ)Qkλ(cos⁡θ)sin⁡2λθdθ)mYk(hκ2;f).
By orthogonality of the orthogonal projector Yk, we have that
(3.8)Jm+l(f)=∑k=0m(∫0πMn;j,i,s(θ)Qkλ(cos⁡θ)sin⁡2λθdθ)m×Yk(hκ2;∑v=0m(∫0πMn;j,i,s(θ)Qvλ(cos⁡θ)sin⁡2λθdθ)lYv(hκ2;f))=Jn;j,i,sm(Jn;j,i,sl(f)).
Leting g=Jn;j,i,sm(f), by (3.8) we get
(3.9)‖f-g‖κ,p=‖f-Jn;j,i,sm(f)‖κ,p≤∑k=1m‖Jn;j,i,sk-1(f)-Jn;j,i,sk(f)‖κ,p≤C(λ,j,i,s)∑k=1m‖Jn;j,i,sk-1((f)-Jn;j,i,s(f))‖κ,p≤C(λ,j,i,s)m‖f-Jn;j,i,s(f)‖κ,p
where Jn;j,i,s0(f)=f.
On the other hand,(3.10)‖Δh,0Jn;j,i,sm(f)‖κ,p≤∑k=0mk(k+2λ)(∫0πMn;j,i,s(θ)|Qkλ(cos⁡θ)|sin⁡2λθdθ)mYk(hκ2;f).
Note that [5]
(3.11)|Qkλ(cos⁡θ)|≡|Ckλ(cos⁡θ)Ckλ(1)|≤Cmin⁡{(kθ)-1,1}.
For kθ≥1, from (2.2) it follows that
(3.12)‖Δh,0Jn;j,i,sm(f)‖κ,p≤C(λ,j,i,s)‖∑k=0mk(k+2λ)k-mλ(∫0πMn;j,i,s(θ)θ-λsin⁡2λθdθ)mYk(hκ2;f)‖κ,p≤C(λ,j,i,s)nmλ‖f‖κ,p∑k=0∞k2-mλ≤C(λ,j,i,s)nmλ‖f‖κ,p.
holds for m>3/λ. For kθ<1, by (2.2), we get
(3.13)‖Δh,0Jn;j,i,sm(f)‖κ,p≤‖∑k=0m(∫0πMn;j,i,s(θ)θ-2/m(θ2k(k+2λ))1/m|Qkλ(cos⁡θ)|sin⁡2λθdθ)mYk(hκ2;f)‖κ,p≤C(λ,j,i,s)‖∑k=0m(∫0πMn;j,i,s(θ)θ-2/m((kθ)2)2/msin⁡2λθdθ)mYk(hκ2;f)‖κ,p≤C(λ,j,i,s)‖∑k=0m(∫0πMn;j,i,s(θ)θ-2/msin⁡2λθdθ)mYk(hκ2;f)‖κ,p≤C(λ,j,i,s)n2‖∑k=0∞Yk(hκ2;f)‖≤Cn2κ,p‖f‖κ,p.
Consequently, the inequality
(3.14)‖Δh,0Jn;j,i,sm(f)‖κ,p≤C(λ,j,i,s)n2‖f‖κ,p
holds uniformly for m>3/λ. Without loss of generality, we may assume m1>3/λ, m>m1+3/λ. Using Lemma 2.4 and (3.8), we have
(3.15)α(n)‖Δh,0Jn;j,i,sm(f)‖κ,p=‖α(n)Δh,0Jn;j,i,sm(f)‖κ,p≤‖Jn;j,i,sm(f)-f‖κ,p+C(λ,j,i,s)n-4‖Δh,02Jn;j,i,sm(f)‖κ,p≤m‖Jn;j,i,s(f)-f‖κ,p+C(λ,j,i,s)n-2‖Δh,02Jn;j,i,sm-m1(f)‖κ,p≤m‖Jn;j,i,s(f)-f‖κ,p+C(λ,j,i,s)(n-2‖Δh,0Jn;j,i,sm(f)‖κ,p+n-2‖Jn;j,i,sm(f)-Jn;j,i,sm-m1(f)‖κ,p)≤m‖Jn;j,i,s(f)-f‖κ,p+C(λ,j,i,s)(n-2‖Δh,0Jn;j,i,sm(f)‖κ,p+‖Jn;j,i,sm1(f)-f‖κ,p)≤C(λ,j,i,s)(‖Jn;j,i,s(f)-f‖κ,p+n-2‖Δh,0Jn;j,i,sm(f)‖κ,p)≤C(λ,j,i,s)(‖Jn;j,i,s(f)-f∥κ,p+∥(f)‖κ,p).
Consequently, n-2∥Δh,0Jn;j,i,sm(f)∥κ,p≤C(λ,j,i,s)∥f-Jn;j,i,s(f)∥κ,p, by the definition of K(f;t2)κ,p and (1.13) shows that
(3.16)ω(f;n-1)κ,p≤CK(f;n-2)κ,p≤C(‖f-Jn;j,i,sm(f)‖κ,p+n-2‖Δh,0Jn;j,i,sm(f)‖κ,p)≤C(λ,j,i,s)‖f-Jn;j,i,s(f)‖κ,p,
that is, ω(f;n-1)κ,p≪∥f-Jn;j,i,s(f)∥κ,p.
The proof is completed.



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