The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap

Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is, so called Jackson-type operator $\{J_{k,s}^m\}_{k=1}^{\infty}$ is constructed to approximate the function defined on the spherical cap $D(x_0,\gamma)$. We thus establish the direct and inverse inequalities and obtain saturation theorems for $\{J_{k,s}^m\}_{k=1}^{\infty}$ on the cap $D(x_0,\gamma)$. Using methods of $K$-functional and multiplier, we obtain the inequality \begin{eqnarray*} C_1\:\| J_{k,s}^m(f)-f\|_{D,p}\leq \omega^2\left(f,\:k^{-1}\right)_{D,p} \leq C_2 \max_{v\geq k}\| J_{v,s}^m(f) - f\|_{D,p} \end{eqnarray*} and that the saturation order of these operators is $O(k^{-2})$, where $\omega^2\left(f,\:t\right)_{D,p}$ is the modulus of smoothness of degree 2, the constants $C_1$ and $C_2$ are independent of $k$ and $f$.


Introduction
In the past decades, many mathematicians dedicated to establish the Jackson and Bernstein-type theorems on the sphere. Early works, such as Butzer [3], Nikol'skiǐ [14] and [15], Lizorkin [6] had successfully established the direct and inverse theorems on the sphere. In early 1990s, Li and Yang [5] constructed Jackson operators on the sphere and obtained the Jackson and Bernstein-type theorems for the Jackson operators.
Jackson operator on the sphere is defined by (see [6], [5]) is the classical Jackson kernel, f is measurable function of degree p on the sphere S n−1 in R n , dω(y) is the elementary surface piece, |S n−1 | is the measurement of S n−1 . For f ∈ L p (S n−1 ), (1 ≤ p ≤ ∞) (L ∞ (S n−1 ) is the collection of continuous functions on S n−1 ), Li and Yang [5] proved that and the saturation order for J k,s is k −2 , where C 1 and C 2 are independent of positive integer k and f , and ω 2 (f, t) is the modulus of smoothness of degree 2 on the unit sphere S n−1 .
On the spherical caps, we desire to obtain the same results. The main difficulty we face is to establish the Bernstein-type inequality (for polynomials) on the cap. Recently, Belinsky, Dai and Ditzian [1] constructed m-th translation operator S m θ when discussing the averages of functions on the sphere. This inspires us and allows us to construct the m-th Jackson-type operator J m k, s on the spherical cap. Fortunately, the Bernsteintype inequality holds for J m k, s , which helps us get the direct and inverse inequalities of approximation on the spherical cap. Finally, we obtain that the saturation order for the constructed Jackson-type operator is k −2 , the same to that of the Jackson operator on the sphere.

Definitions and Auxiliary Notations
Throughout out this paper, we denote by the letters C and C i (i is either positive integers or variables on which C depends only) positive constants depending only on the dimension n. Their value may be different at different occurrences, even within the same formula. We shall denote the points in S n−1 by x, x 0 , x 1 , . . . , y, y 0 , y 1 , . . . , and z, z 0 , z 1 , . . . , and the elementary surface piece on S n−1 by dω. If it is necessary, we shall write dω(x) referring to the variable of the integration. The notation a ≈ b means that there exists a positive constant C such that C −1 b ≤ a ≤ Cb where C is independent of some variable n on which a and b both depend.
Next, we introduce some concepts and properties of sphere as well as caps (see [8], [12]). The volume of S n−1 is Corresponding to dω, the inner product on S n−1 is defined by We denote by D(x 0 , γ) the spherical cap with center x 0 and angle 0 < γ ≤ π 2 , i.e., |f (x)| p dω(x) For any f ∈ L p (D(x 0 , γ)), we note and clearly, f * ∈ L p (S n−1 ) and f * p = f D,p . This allows us to introduce some operators on spherical cap using existing operators on the sphere.
is an operator on S n−1 , then is called the operator on D(x 0 , γ) introduced by T. We may use the notation T instead of T x 0 ,γ for convenience if without mixing up.
We now make a brief introduction of projection operators Y j (·) by ultraspherical for discussion of saturation property of Jackson operators.
Besides, for any j = 0, 1, 2, . . ., and |t| ≤ 1, |P n j (t)| ≤ 1 (see [8]). The projection operators is defined by In the same way, we define the inner product on D(x 0 , γ) as follows, We denote by ∆ the Laplace-Beltrami operator by which we define a K-function on D(x 0 , γ) as For f ∈ L 1 (D(x 0 , γ)), the translation operator is defined by where dω ′ (y) denotes the the elementary surface piece on the sphere {y ∈ D(x 0 , γ) : x · y = cos θ}. Then we have The modulus of smoothness of f is defined by Using the method of [3], we have We introduce m-th translation operator in terms of multipliers (see [7], [9], [12]) It has been proved that (see [12]) With the help of S m θ , we can construct Jackson-type operators on D(x 0 , γ). . This difference will help us to prove Bernstein inequality for J m k,s . For sake of ensuring that Bernstein inequality for J m k,s holds, γ has to be no more than π 2 . Particularly, for m = 1, we have f (y) D k,s (arctan(x · y))dω(y).
Finally, we introduce the definition of saturation for operators (see [2]).

Definition 2.3
Let ϕ(ρ) be a positive function with respect to ρ, 0 < ρ < ∞, tending monotonely to zero as ρ → ∞. For ρ > 0, I ρ is a sequence of operators. If there exists K L p (D(x 0 , γ)) such that: if and only if f ∈ K; then I ρ is said to be saturated on L p (D(x 0 , γ)) with order O(ϕ(ρ)) and K is called its saturation class.

Some Lemmas
In this section, we show some lemmas on both S m θ and J m k,s as the preparation for the main results. For S m θ , we have For m ≥ 1, and f which satisfies ∆f ∈ L p (D(x 0 , γ)), ∆S m θ (f ) D,p ≤ ∆f D,p .
Proof. (i), (ii) and (iii) are clear. Using Remark 3.5 of [1], we can obtain (iv). For We need the following lemma.
Proof. A simple calculation gives, for β ≥ −2, For Jackson-type operators, we have We just have to add the proof of (iii). In fact, using Minkowski inequality, (iv) of Lemma 3.1 and Lemma 3.2, we have where the constant in the approximation is independent of m and k.
The following lemma is useful in the proof of Bernstein-type inequality for Jacksontype operators.

Lemma 3.4 ([13]) Suppose that for nonnegative sequences {σ
is satisfied for any positive integer n. Then one has The following lemma gives the multiplier representation of J m k,s (f ), which follows from Definition 2.2 and (2.7).
The following lemma is useful for determining the saturation order. It can be deduced by the methods of [2] and [4].

Main Results and Their Proofs
In this section, we shall discuss the main results, that is, the lower and upper bounds as well as the saturation order for Jackson-type operators on L p (D(x 0 , γ)).
The following theorem gives the Jackson-type inequality for J m k,s .
Theorem 4.1 For any integer m ≥ 1 and 0 < γ ≤ π 2 , {J m k,s } ∞ k=1 is the series of Jackson-type operators on L p (D(x 0 , γ)) defined above, and g ∈ L Therefore, for f ∈ L p (D(x 0 , γ)), where C is independent of k and f .
Next, we prove the Bernstein-type inequality for J m k,s (f )(x) for f ∈ L p (D(x 0 , γ)).
are m-th Jackson-type operators on D(x 0 , γ). For f ∈ L p (D(x 0 , γ)), 0 < γ ≤ π 2 , then there exits a constant C independent of k and f such that holds for every f ∈ L p (D(x 0 , γ)) and every integer k.