Weighted Approximation for Jackson-Matsuoka Polynomials on the Sphere

Abstract

We consider the best approximation by Jackson-Matsuoka polynomials in the weighted 𝐿_𝑝 space on the unit sphere of ℝ^𝑑. 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.

1. Introduction and Notations

Let 𝕊∶=𝕊^𝑑−1={𝑥∶‖𝑥‖=1} denote the unit sphere in ℝ^𝑑(𝑑≥3),𝑑∈ℕ, where ‖𝑥‖ denotes the usual Euclidean norm, ℝ the set of real numbers. For a nonzero vector 𝑣∈ℝ^𝑑, let 𝜎_𝑣 denote the reflection with respect to the hyperplane perpendicular to 𝑣,𝑥𝜎_𝑣∶=𝑥−2(⟨𝑥,𝑣⟩/‖𝑣‖²)𝑣,𝑥∈ℝ^𝑑, where ⟨𝑥,𝑣⟩ denote the usual Euclidean inner product. Let 𝐺 be a finite reflection group on ℝ^𝑑 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 {𝜎_𝑣∶𝑣∈ℝ₊}. Let 𝜅 be a nonnegative multiplicity function 𝑣↦𝜅_𝑣 defined on ℝ₊ with the property that 𝜅_𝑢=𝜅_𝑣 whenever 𝜎_𝑢 is conjugate to 𝜎_𝑣 in 𝐺, then 𝑣↦𝜅_𝑣 is a 𝐺-invariant function. We consider the weighted best 𝐿_𝑝 approximation with respect to the measure ℎ²_𝜅𝑑𝜔 on 𝕊, where ℎ²_𝜅 is defined byℎ_𝜅=_𝑣∈ℝ+|⟨𝑥,𝑣⟩|^𝜅𝑣,𝑥∈ℝ^𝑑, (1.1)𝑑𝜔 is the surface (Lebesgue) measure on 𝕊. The function ℎ_𝜅 is a positive homogeneous function of degree 𝛾_𝜅∶=∑_𝑣∈𝑅+𝜅_𝑣, and it is invariant under the reflection group. We denote by 𝑎_𝜅 the normalization constant of ℎ_𝜅,𝑎_𝜅⁻¹=∫_𝕊ℎ²_𝜅(𝑦)𝑑𝜔 and denote by 𝐿_𝑝(ℎ²_𝜅), 1≤𝑝≤∞, the space of functions defined on 𝕊 with the finite norm𝑓_𝜅,𝑝∶=𝑎_𝜅𝕊|𝑓(𝑦)|^𝑝ℎ²_𝜅(𝑦)𝑑𝜔(𝑦)^1/𝑝,1≤𝑝<∞, (1.2) 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‖_∞.

Δ_ℎ denote the ℎ-Laplacian. Δ_ℎ,0 is the Laplace-Beltrami operator on the sphere. 𝒫^𝑑_𝑛 denote the subspace of homogeneous polynomials of degree 𝑛 in 𝑑 variables. The ℎ-harmonics are defined as the homogeneous polynomials satisfying the equation Δ_ℎ𝑃=0,𝑃∈𝒫^𝑑_𝑛. Furthermore, let ℋ^𝑑_𝑛(ℎ²_𝜅) 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 Δ_ℎ,0; that is,Δ_ℎ,0𝑌(𝑥)=−𝑛(𝑛+2𝜆)𝑌(𝑥),𝑥∈𝕊,𝑌∈ℋ^𝑑_𝑛ℎ²_𝜅. (1.3)

The standard Hilbert space theory shows that 𝐿₂(ℎ²_𝜅)=∑^∞_𝑛=0⊕ℋ^𝑑_𝑛(ℎ²_𝜅). That is, with each 𝑓∈𝐿₂(ℎ²_𝜅) we can associate its ℎ-harmonic expansion𝑓(𝑥)=^∞_𝑛=0𝑌_𝑛ℎ²_𝜅;𝑓,𝑥,𝑥∈𝕊, (1.4) in 𝐿₂(ℎ²_𝜅) norm. For the surface measure (𝜅=0), such a series is called the Laplace series (see [1]). The orthogonal projection 𝑌_𝑛(ℎ²_𝜅)∶𝐿₂(ℎ²_𝜅)→ℋ^𝑑_𝑛(ℎ²_𝜅) takes the form𝑌_𝑛ℎ²_𝜅;𝑓,𝑥∶=_𝕊𝑓𝑦𝑃_𝑛ℎ²_𝜅;𝑥,𝑦ℎ²_𝜅𝑦𝑑𝜔𝑦, (1.5) where 𝑃_𝑛(ℎ²_𝜅;𝑥,𝑦) is the reproducing kernel of the space of ℎ-harmonics ℋ^𝑑_𝑛(ℎ²_𝜅), which is given by (see [2])𝑃_𝑛ℎ²_𝜅;𝑥,𝑦=𝑛+𝜆𝜆𝑉_𝜅𝐶^𝜆_𝑛⋅,𝑦(𝑥). (1.6)𝐶^𝜆_𝑛 is the ultraspherical polynomial of degree 𝑛,𝜆∶=𝛾_𝜅+(𝑑−2)/2, 𝛾_𝜅=∑_𝑣∈ℝ+𝜅_𝑣, and the intertwining operator 𝑉_𝜅 is a linear operator uniquely determined by𝑉_𝜅𝒫_𝑛⊂𝒫_𝑛,𝑉_𝜅1=1,𝒟_𝑖𝑉_𝜅=𝑉_𝜅𝜕_𝑖,1≤𝑖≤𝑑. (1.7)

The spherical means are denoted by𝑇_𝜃𝑓=1𝕊^𝑑−2(sin𝜃)^𝑑−2_{⟨𝑥,𝑦⟩=cos𝜃}𝑓𝑦𝑑𝜔𝑦, (1.8) where |𝕊^𝑑−2|=∫_𝕊𝑑−2𝑑𝜔=2𝜋^(𝑑−1)/2/Γ((𝑑−1)/2).

The spherical means associated with ℎ²_𝜅𝑑𝜔, in which 𝑇^𝜅_𝜃(𝑓) is defined by𝑐_𝜆^𝜋₀𝑇^𝜅_𝜃𝑓,𝑥𝑔(cos𝜃)(sin𝜃)^2𝜆𝑑𝜃=𝑎_𝜅𝕊𝑓𝑦𝑉_𝜅𝑔𝑥,𝑦ℎ²_𝜅𝑦𝑑𝜔𝑦, (1.9) where 𝑔 is any function [−1,1]↦ℝ such that the integral in the right-hand side is finite, 𝑐_𝜆⁻¹=∫¹₋₁(1−𝑡²)^𝜆−1/2𝑑𝑡=Γ(𝜆+1/2)√𝜋/Γ(𝜆+1). 𝑇^𝜅_𝜃(𝑓) is a proper extension of 𝑇_𝜃(𝑓), since 𝑇_𝜃(𝑓) satisfies 𝑇^𝜅_𝜃(𝑓) when 𝜅=0 and 𝑉_𝜅=𝑖𝑑, and the properties of 𝑇^𝜅_𝜃 are well known (see [2]). In particular, the function 𝑇^𝜅_𝜃𝑓(𝑥) has the expansion 𝑇^𝜅_𝜃𝑓∼^∞_𝑛=0𝐶^𝜆_𝑛(cos𝜃)𝐶^𝜆_𝑛(1)𝑌_𝑛ℎ²_𝜅;𝑓∶=^∞_𝑛=0𝑄^𝜆_𝑛(cos𝜃)𝑌_𝑛ℎ²_𝜅;𝑓. (1.10) Simultaneously, they lead to the following definition of an analog of the modulus of smoothness.

Definition 1.1 (see [2]). For 𝑓∈𝐿_𝑝(ℎ²_𝜅), 1≤𝑝<∞, or 𝑓∈𝐶(𝕊), the modulus of smoothness on the sphere is given by 𝜔𝑓;𝑡_𝜅,𝑝∶=sup_{0<𝜃≤𝑡}𝑓−𝑇^𝜅_𝜃𝑓_𝜅,𝑝. (1.11) The 𝐾-functional of the sphere is given by 𝐾𝑓;𝑡²_𝜅,𝑝=inf_{𝑔∈𝑊𝑝ℎ2𝜅}𝑓−𝑔_𝜅,𝑝+𝑡²Δ_ℎ,0𝑔_𝜅,𝑝, (1.12) where 𝑊_𝑝(ℎ²_𝜅)∶={𝑓∶𝑓∈𝐿_𝑝(ℎ²_𝜅),−𝑘(𝑘+2𝜆)𝑃_𝑘(ℎ²_𝜅;𝑓)=𝑃_𝑘(ℎ²_𝜅;𝑔)forsome𝑔∈𝐿_𝑝(ℎ²_𝜅)}, 0<𝑡<𝑡₀, 𝑡₀ is a positive constant.

In [2], Xu proved the weak equivalence relation𝐶⁻¹𝜔𝑓;𝑡_𝜅,𝑝≤𝐾𝑓;𝑡²_𝜅,𝑝≤𝐶𝜔𝑓;𝑡_𝜅,𝑝. (1.13) 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 kernel𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)1∶=Ω_{𝑛;𝑗,𝑖,𝑠}sin^2𝑗𝑛𝜃/2sin^2𝑖𝜃/2^2𝑠,𝑛=1,2,…,𝜃∈ℝ, (1.14) where 𝑗,𝑖,𝑠∈ℕ,Ω_{𝑛;𝑗,𝑖,𝑠} is a constant chosen such that 𝑐_𝜆∫^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃=1. It is known that 𝑀_{𝑛;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 𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃) we consider the spherical convolution𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓;𝑥∶=𝑓∗𝑀_{𝑛;𝑗,𝑖,𝑠}(𝑥)∶=𝑐_𝜆^𝜋₀𝑇^𝜅_𝜃𝑓;𝑥𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)(𝜃)sin^2𝜆𝜃𝑑𝜃. (1.15) It is called the Jackson-Matsuoka polynomials on the sphere based on the Jackson-Matsuoka kernel. In particular, (𝑓₀∗𝑀_{𝑛;𝑗,𝑖,𝑠})(𝑥)=1 for 𝑓₀(𝑥)=1. The classical Jackson-Matsuoka polynomials in the classical 𝐿_𝑝 space have been studied by many authors (see [3, 4]).

The purpose of this paper is to consider approximation by ℎ-harmonic polynomials, which in the 𝐿_𝑝 metric can be viewed as weighted approximation, in which the measure 𝑑𝜔 on the sphere is replaced by ℎ²_𝜅𝑑𝜔. 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 between 𝐾-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 𝐿_𝑝 space on the unit sphere of ℝ^𝑑. We only consider best weighted approximation by Jackson-Matsuoka polynomials, and for the other polynomials on the unit sphere of ℝ^𝑑, the methods and the results are similar.

2. Auxiliary Lemmas

We need the following lemmas.

Lemma 2.1. Let Ω_{𝑛;,𝑗,𝑖,𝑠}=∫^𝜋₀((sin^2𝑗𝑛𝜃/2)/(sin^2𝑖𝜃/2))^2𝑠sin^2𝜆𝜃𝑑𝜃. Then, the weak equivalence Ω_{𝑛;𝑗,𝑖,𝑠}≍𝑛^{4𝑖𝑠−2𝜆−1} (2.1) 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 𝐴_𝑛≪𝐵_𝑛 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 Lemma 2.2 and we omit it.

Lemma 2.2. For 4𝑖𝑠>𝑟+2𝜆+1, 𝑗≥𝑖, 𝑟∈ℝ, there is a constant 𝐶(𝜆,𝑗,𝑖,𝑠) such that ^𝜋₀𝜃^𝑟𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃≤𝐶𝜆,𝑗,𝑖,𝑠𝑛^−𝑟. (2.2)

Proof. Since 𝜃/𝜋≤sin(𝜃/2)≤𝜃/2 and sin𝜃≤𝜃 hold for 0≤𝜃≤𝜋, by Ω_{𝑛;𝑗,𝑖,𝑠}≍𝑛^{4𝑖𝑠−2𝜆−1}, we have ^𝜋₀𝜃^𝑟𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃≤𝐶𝜆,𝑗,𝑖,𝑠𝑛^{−4𝑖𝑠+2𝜆+1𝜋}₀𝜃^𝑟sin^2𝑗𝑛𝜃/2sin^2𝑖𝜃/2^2𝑠sin^2𝜆𝑛𝜃𝑑𝜃≤𝐶𝜆,𝑗,𝑖,𝑠^{−4𝑖𝑠+2𝜆+1}𝑛^{4𝑖𝑠−𝑟−2𝜆−1}₀^𝑛𝜋/2𝑡^𝑟+2𝜆sin^2𝑗𝑡𝑡^2𝑖2𝑠𝑛𝑑𝑡≤𝐶𝜆,𝑗,𝑖,𝑠^−𝑟₀^𝜋/2𝑡^𝑟+2𝜆sin^2𝑗𝑡𝑡^2𝑖2𝑠𝑑𝑡+^∞_𝜋/2𝑡^𝑟+2𝜆sin^2𝑗𝑡𝑡^2𝑖2𝑠𝐶𝑑𝑡≤𝐶𝜆,𝑗,𝑖,𝑠₂𝑛^𝜆≤𝐶𝜆,𝑗,𝑖,𝑠𝑛^𝜆, (2.3) where 𝐶₂=₀^𝜋/2𝑡^𝜆sin^2𝑗𝑡𝑡^2𝑖2𝑠𝑑𝑡+^∞_𝜋/2𝑡^𝜆sin^2𝑗𝑡𝑡^2𝑖2𝑠𝑑𝑡,4𝑖𝑠>𝑟+2𝜆+1,𝑗≥𝑖. (2.4) Lemma 2.2 has been proved.

Lemma 2.3 (see [2]). For 0≤𝜃≤𝜋, one has 𝑇^𝜅_𝜃𝑔;𝑥−𝑔(𝑥)=^𝜃₀sin^−2𝜆𝑡𝑑𝑡^𝑡₀𝑇^𝜅_𝑢Δ_ℎ,0𝑔sin^2𝜆=𝑢𝑑𝑢_𝜃0sin^−2𝜆𝑡Φ(𝑡)𝐵_𝑡Δ_ℎ,0𝑔,𝑥𝑑𝑡, (2.5) where 𝐵_𝑡Δ_ℎ,0=1𝑔,𝑥Φ(𝑡)^𝑡₀𝑇^𝜅_𝑢Δ_ℎ,0𝑔sin^2𝜆𝑢𝑑𝑢, (2.6) and Φ(𝑡)=𝑐_𝜆⁻¹∫^𝑡₀sin^2𝜆𝑢𝑑𝑢.

Lemma 2.4. Let 𝑔,Δ_ℎ,0𝑔,Δ²_ℎ,0𝑔∈𝐿_𝑝(ℎ²_𝜅),1≤𝑝≤∞,𝐽_{𝑛;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 𝐽_{𝑛;𝑗,𝑖,𝑠}𝑔−𝑔−𝛼(𝑛)Δ_ℎ,0𝑔_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝑛⁻⁴Δ²_ℎ,0𝑔_𝜅,𝑝, (2.7) where 𝛼(𝑛)≍𝑛⁻².

Proof. By Lemma 2.3, we have 𝐽_{𝑛;𝑗,𝑖,𝑠}𝑔;𝑥−𝑔(𝑥)=𝑐_𝜆^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝑇^𝜅_𝜃𝑔;𝑥−𝑔(𝑥)sin^2𝜆𝜃𝑑𝜃=𝑐_𝜆^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀Φ(𝑡)sin^2𝜆𝑡𝐵_𝑡Δ_ℎ,0𝑔,𝑥𝑑𝑡=𝑐_𝜆Δ_ℎ,0𝑔(𝑥)^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀Φ(𝑡)sin^2𝜆𝑡𝑑𝑡+𝑐_𝜆^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀Φ(𝑡)sin^2𝜆𝑡𝐵_𝑡Δ_ℎ,0𝑔,𝑥−Δ_ℎ,0𝑔(𝑥)𝑑𝑡=Δ_ℎ,0𝑔(𝑥)^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀𝑑𝑡sin^2𝜆𝑡^𝑡₀sin^2𝜆+𝑢𝑑𝑢_𝜋0𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀𝑑𝑡sin^2𝜆𝑡^𝑡₀sin^2𝜆𝑢𝐵_𝑡Δ_ℎ,0𝑔,𝑥−Δ_ℎ,0𝑔(𝑥)(𝑛)Δ𝑑𝑢∶=𝛼_ℎ,0𝑔(𝑥)+^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃Ψ_𝜃𝑔,𝑥𝑑𝜃, (2.8) where 𝛼(𝑛)∶=^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀𝑑𝑡sin^2𝜆𝑡^𝑡₀sin^2𝜆Ψ𝑢𝑑𝑢,_𝜃𝑔,𝑥∶=^𝜃₀𝑑𝑡sin^2𝜆𝑡^𝑡₀sin^2𝜆𝑢𝐵_𝑡Δ_ℎ,0𝑔,𝑥−Δ_ℎ,0𝑔(𝑥)𝑑𝑢. (2.9) By Lemma 2.1, we have 𝛼(𝑛)=^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀𝑑𝑡sin^2𝜆𝑡^𝑡₀sin^2𝜆𝑢𝑑𝑢≍^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃^𝜃₀𝑡sin^2𝜆𝜉sin^2𝜆𝑡≍𝑑𝑡_𝜋0𝜃²𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃≍𝑛⁻²,(0<𝜉<𝑡). (2.10) We now estimate, using Lemma 2.3 again, the expression 𝐵_𝑡(Δ_ℎ,0𝑔,𝑥)−Δ_ℎ,0𝑔(𝑥), and obtain Ψ_𝜃𝑔_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝜃⁴Δ²_ℎ,0𝑔_𝜅,𝑝. (2.11) By Lemma 2.2 and Hölder-Minkowski inequality shows that ^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃Ψ_𝜃𝑔,𝑥𝑑𝜃_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠Δ²_ℎ,0𝑔_𝜅,𝑝^𝜋₀𝜃⁴𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝑛𝜃𝑑𝜃≤𝐶𝜆,𝑗,𝑖,𝑠⁻⁴Δ²_ℎ,0𝑔_𝜅,𝑝. (2.12) Consequently, by (2.8), (2.10), and (2.12) we complete the proof of this lemma.

Lemma 2.5. For 𝑡≥0, there is a constant 𝐶 such that 𝜔𝑓;𝑡𝛿_𝜅,𝑝≤𝐶max1,𝑡²𝜔𝑓;𝛿_𝜅,𝑝. (2.13)

Proof. By the equivalence relation between the modulus of smoothness and 𝐾-functional, and the definition of 𝐾(𝑓;𝑡²)_𝜅,𝑝, we have 𝜔𝑓;𝑡𝛿_𝜅,𝑝≤𝐶𝐾𝑓;(𝑡𝛿)²_𝜅,𝑝≤𝐶𝑓−𝑔_𝜅,𝑝+𝑡²𝛿²Δ_ℎ,0𝑔_𝜅,𝑝≤𝐶max1,𝑡²𝑓−𝑔_𝜅,𝑝+𝛿²Δ_ℎ,0𝑔_𝜅,𝑝≤𝐶max1,𝑡²𝐾𝑓;𝛿²_𝜅,𝑝≤𝐶max1,𝑡²𝜔𝑓;𝛿_𝜅,𝑝. (2.14) Lemma 2.5 has been proved.

3. Main Results

Our main results are the following.

Theorem 3.1. Suppose that 𝑓∈𝐿_𝑝(ℎ²_𝜅),1≤𝑝≤∞,𝐽_{𝑛;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 𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝≍𝜔𝑓;𝑛⁻¹_𝜅,𝑝. (3.1)

Proof. First we prove ‖𝐽_{𝑛;𝑗,𝑖,𝑠}(𝑓)−𝑓‖_𝜅,𝑝≪𝜔(𝑓;𝑛⁻¹)_𝜅,𝑝. Since (𝑓₀∗𝑀_{𝑛;𝑗,𝑖,𝑠})(𝑥)=1 for 𝑓₀(𝑥)=1, therefore, we have that 𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝=^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝑓(𝑥)−𝑇^𝜅_𝜃(𝑓;𝑥)sin^2𝜆𝜃𝑑𝜃_𝜅,𝑝≤^𝜋₀𝑓−𝑇^𝜅_𝜃𝑓_𝜅,𝑝𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃. (3.2) 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 𝑓−𝑇^𝜅_𝜃𝑓_𝜅,𝑝≤𝜔𝑓;𝑛⁻¹_𝜅,𝑝+^𝜋_1/𝑛𝜔𝑓;𝜃_𝜅,𝑝𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃. (3.3) From Lemma 2.5 it follows that, for 𝜃≥𝑛⁻¹, 𝜔𝑓;𝜃_𝜅,𝑝𝜃=𝜔𝑓;𝑛𝑛_𝜅,𝑝≤𝐶max1,𝑛²𝜃²𝜔𝑓;𝜃_𝜅,𝑝≤𝐶𝑛²𝜃²𝜔𝑓;𝜃_𝜅,𝑝. (3.4) Therefore, it follows that 𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝≤𝜔𝑓;𝜃_𝜅,𝑝1+𝐶𝑛^2𝜋_1/𝑛𝜃²𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)sin^2𝜆𝜃𝑑𝜃. (3.5) From Lemma 2.2, we get 𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝜔𝑓;𝑛⁻¹_𝜅,𝑝. (3.6) Next we prove 𝜔(𝑓;𝑛⁻¹)_𝜅,𝑝≪‖𝐽_{𝑛;𝑗,𝑖,𝑠}(𝑓)−𝑓‖_𝜅,𝑝. Let 𝑚 be a fixed positive integer Denote by 𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓∶=^𝑚_𝑘=0^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝑄^𝜆_𝑘(cos𝜃)sin^2𝜆𝜃𝑑𝜃^𝑚𝑌_𝑘ℎ²_𝜅;𝑓. (3.7) By orthogonality of the orthogonal projector 𝑌_𝑘, we have that 𝐽^𝑚+𝑙𝑓=^𝑚_𝑘=0^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝑄^𝜆_𝑘(cos𝜃)sin^2𝜆𝜃𝑑𝜃^𝑚×𝑌_𝑘ℎ²_𝜅;^𝑚_𝑣=0^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝑄^𝜆_𝑣(cos𝜃)sin^2𝜆𝜃𝑑𝜃^𝑙𝑌_𝑣ℎ²_𝜅;𝑓=𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝐽^𝑙_{𝑛;𝑗,𝑖,𝑠}𝑓. (3.8) Leting 𝑔=𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}(𝑓), by (3.8) we get 𝑓−𝑔_𝜅,𝑝=𝑓−𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤^𝑚_𝑘=1𝐽^𝑘−1_{𝑛;𝑗,𝑖,𝑠}𝑓−𝐽^𝑘_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠^𝑚_𝑘=1𝐽^𝑘−1_{𝑛;𝑗,𝑖,𝑠}𝑓−𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝑚𝑓−𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝 (3.9) where 𝐽⁰_{𝑛;𝑗,𝑖,𝑠}(𝑓)=𝑓.
On the other hand, Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤^𝑚_𝑘=0𝑘(𝑘+2𝜆)^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝑄^𝜆_𝑘(cos𝜃)sin^2𝜆𝜃𝑑𝜃^𝑚𝑌_𝑘ℎ²_𝜅;𝑓. (3.10) Note that [5] 𝑄^𝜆_𝑘(≡𝐶cos𝜃)^𝜆𝑘(cos𝜃)𝐶^𝜆_𝑘(1)≤𝐶min(𝑘𝜃)⁻¹,1. (3.11) For 𝑘𝜃≥1, from (2.2) it follows that Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠^𝑚_𝑘=0𝑘(𝑘+2𝜆)𝑘^{−𝑚𝜆𝜋}₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝜃^−𝜆sin^2𝜆𝜃𝑑𝜃^𝑚𝑌_𝑘ℎ²_𝜅;𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝑛^𝑚𝜆𝑓_𝜅,𝑝^∞_𝑘=0𝑘^2−𝑚𝜆≤𝐶𝜆,𝑗,𝑖,𝑠𝑛^𝑚𝜆𝑓_𝜅,𝑝. (3.12) holds for 𝑚>3/𝜆. For 𝑘𝜃<1, by (2.2), we get Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤^𝑚_𝑘=0^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝜃^−2/𝑚(𝜃²𝑘(𝑘+2𝜆))^1/𝑚|𝑄^𝜆_𝑘(cos𝜃)|sin^2𝜆𝜃𝑑𝜃^𝑚𝑌_𝑘ℎ²_𝜅;𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠^𝑚_𝑘=0^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝜃^−2/𝑚((𝑘𝜃)²)^2/𝑚sin^2𝜆𝜃𝑑𝜃^𝑚𝑌_𝑘ℎ²_𝜅;𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠^𝑚_𝑘=0^𝜋₀𝑀_{𝑛;𝑗,𝑖,𝑠}(𝜃)𝜃^−2/𝑚sin^2𝜆𝜃𝑑𝜃^𝑚𝑌_𝑘ℎ²_𝜅;𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝑛^2∞_𝑘=0𝑌_𝑘ℎ²_𝜅;𝑓_𝜅,𝑝≤𝐶𝑛²𝑓_𝜅,𝑝. (3.13) Consequently, the inequality Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝑛²𝑓_𝜅,𝑝 (3.14) 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 𝛼(𝑛)Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝=𝛼(𝑛)Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝+𝐶𝜆,𝑗,𝑖,𝑠𝑛⁻⁴Δ²_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝑚𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝+𝐶𝜆,𝑗,𝑖,𝑠𝑛⁻²Δ²_ℎ,0𝐽^𝑚−𝑚1_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝑚𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝+𝐶𝜆,𝑗,𝑖,𝑠𝑛⁻²Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝+𝑛⁻²𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓−𝐽^𝑚−𝑚1_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝑚𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝+𝐶𝜆,𝑗,𝑖,𝑠𝑛⁻²Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝+𝐽^𝑚1_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓_𝜅,𝑝+𝑛⁻²Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓−𝑓‖_𝜅,𝑝+‖𝑓_𝜅,𝑝. (3.15) Consequently, 𝑛⁻²‖Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}(𝑓)‖_𝜅,𝑝≤𝐶(𝜆,𝑗,𝑖,𝑠)‖𝑓−𝐽_{𝑛;𝑗,𝑖,𝑠}(𝑓)‖_𝜅,𝑝, by the definition of 𝐾(𝑓;𝑡²)_𝜅,𝑝 and (1.13) shows that 𝜔𝑓;𝑛⁻¹_𝜅,𝑝≤𝐶𝐾𝑓;𝑛⁻²_𝜅,𝑝≤𝐶𝑓−𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝+𝑛⁻²Δ_ℎ,0𝐽^𝑚_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝≤𝐶𝜆,𝑗,𝑖,𝑠𝑓−𝐽_{𝑛;𝑗,𝑖,𝑠}𝑓_𝜅,𝑝, (3.16) that is, 𝜔(𝑓;𝑛⁻¹)_𝜅,𝑝≪‖𝑓−𝐽_{𝑛;𝑗,𝑖,𝑠}(𝑓)‖_𝜅,𝑝.
The proof is completed.

Acknowledgments

The author would like to thank Professor Norimichi Hirano and the anonymous referees for their valuable comments, remarks, and suggestions which greatly help us to improve the presentation of this paper and make it more readable. Project is supported by the Natural Science Foundation of China (Grant no. 10671019), Zhejiang Provincial Natural Science Foundation (Grant no. Y12A010022), and Cultivation Fund of Taizhou University.