Approximation by boolean sums of Jackson operators on the sphere

Approximation by Boolean Sums of Jackson Operators on the Sphere ∗ Feilong Cao†

arXiv:1409.3923v1 [math.CA] 13 Sep 2014

Yuguang Wang

Institute of Metrology and Computational Science, China Jiliang University, Hangzhou 310018, Zhejiang Province, P. R. China.

Abstract This paper concerns the approximation by the Boolean sums of Jackson operators ⊕r Jk,s (f ) on the unit sphere Sn−1 of Rn . We prove the following the direct and inverse theorem for ⊕r Jk,s (f ): there are constants C1 and C2 such that C1 k ⊕r Jk,s f − f kp ≤ ω 2r (f, k −1 )p ≤ C2 max k ⊕r Jk,s f − f kp v≥k

for any positive integer k and any pth Lebesgue integrable functions f defined on Sn−1 , where ω 2r (f, t)p is the modulus of smoothness of degree 2r of f . We also prove that the saturation order for ⊕r Jk,s is k −2r . MSC(2000): 41A17 Key words: approximation; Jackson operator; Boolean sums; saturation; sphere

1

Introduction

In past decades, many mathematicians have dedicated to establish the Jackson and Bernstein-type theorems on the sphere. Lizorkin and Nikol’skiˇı [7] constructed Boolean sums of Jackson operators ⊕r Jk,s (which will be defined in the next section) for proving the direct and inverse theorems on a special Banach space Hpr (Sn−1 ). Later, by using a modulus of smoothness as metric, Lizorkin and Nikol’skiˇı [8] obtained the direct estimate for Jackson operators (i.e., the 1-th Boolean sums of Jackson operators) approximating continuous function defined on the unit sphere Sn−1 . In 1991, Li and Yang [6] used the equivalent relation between the K-functional and the smoothness (see [2]) and established an inverse inequality of weak type, which was also called Steckin-Marchaud type inequality. Recently, Ditzian [5] proved an equivalence relation between K-functional and modulus of smoothness of high order, which will provide a tool and allow us to make a proof for direct and inverse theorems for the Boolean sums of Jackson operators. ∗

The research was supported by the National Natural Science Foundation of China (No. 60873206), the Natural Foundation of Zhejiang Province of China (No. Y7080235) and the Innovation Foundation of Post-Graduates of Zhejiang Province of China (No. YK2008066) † Corresponding author. E-mail: [email protected]

1

Actually, after improving a Steckin-Marchaud type inequality, we will prove the direct and inverse theorem of approximation for arbitrary r-th Boolean sums of Jackson operators ⊕r Jk,s approximating p-th Lebesgue integrable function on the sphere. Particularly, a converse inequality of strong type (see [4]) for ⊕r Jk,s will be established. Moreover, we will use the method of multipliers and obtain the saturation order of ⊕r Jk,s .

2

Definitions and Auxiliary Notations

Let Sn−1 be the unit sphere in Euclidean space Rn . We denote by the letters C and Ci positive constants, where i is either positive integers or variables on which C depends only. Their values may be different at different occurrences, even within the same formula. We shall denote by x and y the points of Sn−1 . The notation a ≈ b means that there exists a positive constant C such that C −1 b ≤ a ≤ Cb. We now introduce some concepts and properties of sphere (see also [9], [?]). The volume of Sn−1 is n Z n−1 2π 2 := S . dω = Γ( n2 ) Sn−1 Denote by Lp (Sn−1 ) the space of p-th integrable functions on Sn−1 endowed with the norms kf k∞ := kf kL∞ (Sn−1 ) := ess sup |f (x)| x∈Sn−1

and kf kp := kf kLp (Sn−1 ) :=

Z

1/p < ∞, |f (x)| dω(x) p

Sn−1

1 ≤ p < ∞.

For f ∈ L1 (Sn−1 ), the translation operator is defined by (see for instance [?]) Z 1 Sθ (f )(x) := n−2 f (y)dω ′ (y), 0 < θ < π |S | sinn−2 θ x·y=cos θ where dω ′ (y) denotes the elementary surface piece on Sn−2 . We set (j) (j) (0) Sθ (f ) := f, Sθ (f ) := Sθ Sθ (f ), j = 1, 2, 3, . . . and introduce the spherical differences (see [7]) ∆1θ (f ) := ∆θ (f ) := Sθ (f ) − f and ∆rθ (f )

:=

∆θ ∆θr−1 (f )

=

r X

r+j

(−1)

j=0

  r (j) S (f ) = (Sθ − I)r f, j θ

r = 2, 3, . . .

where I is the identity operator on Lp (Sn−1 ). Then the modulus of smoothness of degree 2r of f ∈ Lp (Sn−1 ) is defined by (see [11]) ω 2r (f, t) := sup k∆rθ f kp ,

0 < t < π, r = 1, 2, . . . .

0 0, we have, using (2.1),

e (λt)2r )p ≤ C1 max{1, λ2r }K2r (f, ∆, e t2r )p ω 2r (f, λt)p ≤ C1 K2r (f, ∆, ≤ C2 max{1, λ2r }ω 2r (f, t)p

(2.2)

where C1 and C2 are independent of t and λ. Spherical polynomials of order k on Sn−1 is defined by (see [7]) Pk (x) :=

k X

Hj (x)

j=0

where Hj (x) is the spherical harmonic of order j, the trace of some homogeneous polynomials on Rn X xi11 · · · xinn Qj (x) := i1 +i2 +···+in =m

of order j, where xi (i = 1, 2, . . . , n) is the i-th coordinate of x ∈ Rn . We denote by Πnk the collection of all spherical polynomials on Sn−1 of degree no more than k. The known Jackson operators are defined by (see [7]) Z 1 f (y)Dk,s (arccos x · y)dω(y), f ∈ Lp (Sn−1 ) Jk,s (f )(x) := n−2 |S | Sn−1 where k and s are positive integers, dω(y) is the elementary surface piece of Sn−1 , Dk,s (θ) :=

A−1 k,s

sin kθ 2 sin θ2

!2s

is the classical Jackson kernel where A−1 k,s is a constant connected with k and s such that Z π n−2 . Dk,s (θ) sin2λ θ dθ = 1, λ = 2 0

3

We observe that Jk,s (f )(x) = =

1

0

=

f (y)Dk,s (arccos x · y)dω(y)  Z 1 ′ f (y)dω (y) Dk,s (θ) sin2λ θ dθ |Sn−2 | sin2λ θ x·y=cos θ

|Sn−2 | Z π

Z

π

Z

Sn−1

Sθ (f )(x)Dk,s (θ) sin2λ θ dθ.

(2.3)

0

We introduce the definition of r-th Boolean sums of Jackson operator as follows (see [7]). Definition 2.1 The r-th (r ≥ 1) Boolean sum of Jackson operator of degree k (k ≥ 1) on Sn−1 is defined by ⊕r Jk,s (f ) := (−(I − Jk,s )r + I) (f ),

f ∈ Lp (Sn−1 ),

(2.4)

where s is a positive integer. It is clear that   r X i r (−1) (Jk,s )i (f ). ⊕ Jk,s (f ) = − i r

(2.5)

i=1

We now make a brief introduction of projection operators Yj (·) by ultraspherical n−2 (Gegenbauer) polynomials {Gλj }∞ j=1 (λ = 2 ) for discussion of saturation property of ⊕r Jk,s . Ultraspherical polynomials {Gλj }∞ j=1 are defined in terms of the generating function (see [12]): ∞ X 1 Gλj (t)r j = (1 − 2tr + r 2 )λ j=0

where |r| < 1, |t| ≤ 1. For any λ > 0, we have (see [12])

and

When λ =

n−2 2

Gλ1 (t) = 2λt

(2.6)

d λ G (t) = 2λGλ+1 j−1 (t). dt j

(2.7)

(see [?]), Gλj (t) =

Γ(j + 2λ) P n (t), Γ(j + 1)Γ(2λ) j

j = 0, 1, 2, . . .

where Pjn (t) is the Legendre polynomial of degree j (see [9]). Particularly, Gλj (1) =

Γ(j + 2λ) Γ(j + 2λ) Pjn (1) = , Γ(j + 1)Γ(2λ) Γ(j + 1)Γ(2λ) 4

j = 0, 1, 2, . . . ,

therefore, Pjn (t) =

Gλj (t) Gλj (1)

.

(2.8)

Besides, for any j = 0, 1, 2, . . ., and |t| ≤ 1, |Pjn (t)| ≤ 1 (see [9]). The projection operators are defined by Z Γ(λ)(n + λ) Yj (f )(x) := Gλj (x · y)f (y) dω(y). n 2π 2 Sn−1 It follows from (2.6), (2.7) and (2.8) that 1 − Pjn (t) j(j + 2λ) = , n t→1− 1 − P1 (t) 2λ + 1 lim

j = 0, 1, 2, . . . .

(2.9)

Finally, we introduce the definition of saturation for operators (see [1]). Definition 2.2 Let ϕ(ρ) be a positive function with respect to ρ, 0 < ρ < ∞, tending monotonely to zero as ρ → ∞. For a sequence of operators {Iρ }ρ>0 if there exists K j Lp (Sn−1 ) such that (i) If kIρ (f ) − f kp = o(ϕ(ρ)), then Iρ (f ) = f ; (ii) kIρ (f ) − f kp = O(ϕ(ρ)) if and only if f ∈ K;

then Iρ is said to be saturated on Lp (Sn−1 ) with order O(ϕ(ρ)) and K is called its saturation class.

3

Some Lemmas

In this section, we show some lemmas as the preparation for the proof of the main results. Lemma 3.1 For any f ∈ Lp (Sn−1 ) and any positive integers k, r, u, s, we have, (i) ⊕u Jk,s (f ) is a spherical polynomial of degree no more than s(k − 1), which e r (⊕u Jk,s (f )) ∈ Lp (Sn−1 ); implies ∆ (ii)

(iii)

(iv)

k ⊕u Jk,s (f )kp ≤ 2u kf kp ; e r (⊕u Jk,s (f ))kp ≤ Ck 2r kf kp ; k∆

e r g ∈ Lp (Sn−1 ), then k∆ e r (⊕u Jk,s (g))kp ≤ 2u k∆ e r gkp . If ∆

Proof. (i) Since Dk,s (θ) is an even trigonometric polynomial of degree no more than k(s − 1), then Jk,s (f ) is a spherical polynomial of degree no more than s(k − 1). Thus we can prove by induction that ⊕u Jk,s (f ) is a spherical polynomial of degree no more than s(k − 1). (ii) Using the contraction of translation operator that (see for instance [?]) kSθ (f )kp ≤ kf kp ,

0 1. Then there exists positive a ≤ k < mN +1 . Take sv ≥ 0 such that km−v−1 < sv ≤ km−v ,

Proof. First we take m > 1, such that ln integer N , such that mN v = 0, 1, . . . , N as well as

(km−v−1 < j ≤ km−v ).

τs v ≤ τj

Set sN +1

ln m a = 1, bm = < 1. Clearly bm → 1, as m → ∞. Then ln m  as p 0 σs 0 + τ s 0 σk ≤ k     N X asv+1 p v+1 p −p (a sv ) σsv − σsv+1 + τs0 ≤ k sv ≤ mp ≤ mp

v=0 N  X

v=0 N +1 X

m −p(v+1) τsv+1 + τs0 a

m

v=0

−





pv

τs v

N +1 X

≤ Cm k−bm p  = Cm k−bm p



ln m a ln m

X

v=0 km−v−1 0 is a sequence of operators on Lp (Sn−1 ), and there exists series {λρ (j)}∞ j=1 with respect to ρ, such that Iρ (f )(x) =

∞ X j=0

7

λρ (j)Yj (f )(x)

for every f ∈ Lp (Sn−1 ). If for any j = 0, 1, 2, . . . , there exists ϕ(ρ) → 0+ (ρ → ρ0 ) such that 1 − λρ (j) = τj 6= 0, lim ρ→ρ0 ϕ(ρ) then {Iρ }ρ>0 is saturated on Lp (Sn−1 ) with the order O(ϕ(ρ)) and the collection of all constants is the invariant class for {Iρ }ρ>0 on Lp (Sn−1 ). 

4

Main Results and Their Proof

In this section, we shall state and prove the main results, that is, the lower and upper bounds as well as the saturation order for Boolean sums of Jackson operators on Lp (Sn−1 ). Theorem 4.1 Let 2s ≥ n, and let {⊕r Jk,s }∞ k=1 be the sequence of Boolean sums of Jackson operators defined above. Then for any positive integers k and r as well as e r g ∈ Lp (Sn−1 ), we have sufficiently smoothing g ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞ such that ∆ e r gkp , k ⊕r Jk,s (g) − gkp ≤ C1 k−2r k∆

(4.13)

k ⊕r Jk,s (f ) − f kp ≤ C2 ω 2r (f, k−1 )p ,

(4.14)

therefore, for any f ∈ Lp (Sn−1 ), we have

where C1 and C2 are constants independent of f and k. Proof. By Definition 2.1, we have ⊕r Jk,s (g)(x) − g(x) = −(I − Jk,s )r (g)(x). Now we prove (4.13) by induction. For r = 1, Sθ (g)(x) − g(x) =

Z

θ

−2λ

sin

0

τ

Z

τ 0

e sin2λ uSu (∆g)(x)dudτ

(see [10]) implies (explained below)

Z π

2λ

kJk,s (g) − gkp = Dk,s (θ) (Sθ (g)(·) − g(·)) sin θdθ

Z

Z

0

Z

p



e sin2λ u Su (∆g) sin−2λ τ Dk,s (θ) sin2λ θ ≤

dudτ dθ p 0 0 0   Z π  Z τ Z θ 2 2λ −2 −2λ −2λ e p ≤ sup θ θ Dk,s (θ) sin θdθ k∆gk sin udu dτ sin τ π

θ>0 −2

≤ Ck

0

θ

τ

0

0

e p, k∆gk

(4.15)

where the Minkowski inequality is used in the first inequality, the second one by (3.10) and the third one is deduced from Lemma 3.2.

8

Assume that for any fixed positive integer u, e u gkp . k ⊕u Jk,s (g) − gkp ≤ Ck−2u k∆

Then

e u Jk,s (g) − g)kp k ⊕u+1 Jk,s (g) − gkp = k(Jk,s − I)(⊕u Jk,s (g) − g)kp ≤ Ck −2 k∆(⊕ e − ∆gk e p ≤ Ck −2u−2 k∆ e u+1 gkp , = Ck −2 k ⊕u Jk,s (∆g)

where the first inequality is by (4.15), the second one by (3.11), the last by induction assumption. Therefore, (4.13) holds. Using (2.1) and noticing that ⊕u Jk,s is a linear operator, we obtain (4.14). This completes the proof of the theorem.  Next, we establish an inverse inequality of strong type for ⊕r Jk,s on Lp (Sn−1 ). Theorem 4.2 For positive r ≥ 1 and f ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞, there exists a constant C independent of f and k such that ω 2r (f, k−1 )p ≤ C max k ⊕r Jv,s (f ) − f kp .

(4.16)

v≥k

Proof. We first establish a Steckin-Marchaud type inequality, that is, for f ∈ Lp (Sn−1 ), k X v 2bm r−1 k ⊕r Jv,s (f ) − f kp , ω 2r (f, k−1 )p ≤ Cm k−2bm r v=1

where 0 < bm < 1, bm → 1, as m → ∞. Set e r (⊕r Jv,s (f )) kp , τv = k ⊕r Jv,s (f ) − f kp , σv = v −2r k∆

v ≥ 1.

Using Lemma 3.1, we have

e r (⊕r Jk,s (⊕r Jv,s (f ))) kp + k−2r k∆ e r (⊕r Jk,s (⊕r Jv,s (f ) − f )) kp σk ≤ k−2r k∆   v 2r  e r (⊕r Jv,s (f )) kp + C k ⊕r Jv,s (f ) − f kp v −2r k∆ ≤ 2r k √ !2r 2v = σv + Cτv . k By Lemma 3.3, we have σk ≤ Cm k

−2bm r

k X

v 2bm r−1 τv

v=1

for some large enough m. That is,

e r (⊕u Jk,s (f )) kp ≤ Cm k−2bm r k−2r k∆ 9

k X v=1

v 2bm r−1 k ⊕r Jv,s (f ) − f kp .

For k ≥ 1, there exists a positive integer k0 ,

k ≤ k0 ≤ k, such that 2

k ⊕r Jk0 ,s (f ) − f kp ≤ k ⊕r Jv,s (f ) − f kp ,

k ≤ v ≤ k. 2

Thus e k−2r )p ≤ k ⊕r Jk ,s (f ) − f kp + k−2r k∆ e r (⊕r Jk ,s (f )) kp K2r (f, ∆, 0 0 X ≤ 22r k−2r v 2r−1 k ⊕r Jv,s (f ) − f kp k ≤v≤k 2

+Cm k−2bm r

k X v=1

k X

≤ Cm k−2bm r

v=1

v 2bm r−1 k ⊕r Jv,s (f ) − f kp

v 2bm r−1 k ⊕r Jv,s (f ) − f kp .

From (2.1) it follows that 2r

ω (f, k

−1

)p ≤ Cm k

−2bm r

k X v=1

v 2bm r−1 k ⊕u Jv,s (f ) − f kp .

To finish our proof, we need the following inequalities. ω 2r (f, k−1 )p ≈ ≈

1 max v 2r k ⊕r Jv,s (f ) − f kp k2r 1≤v≤k 1 1 max v 2r+ 4 k ⊕r Jv,s (f ) − f kp . 2r+ 41 1≤v≤k k

(4.17)

In the first place, we prove the former inequality of (4.17) (explained below). 2r

ω (f, k

−1

)p ≤ C1 k ≤ C1 ≤ ≤

−2bm r

k X v=1

k

−2bm r

v 2bm r−1 k ⊕r Jv,s (f ) − f kp

k X

v

−2(1−bm )r−1

!

v=1 C2 k max v 2r k ⊕r Jv,s (f ) − 1≤v≤k C3 k−2r max v 2r ω 2r (f, v −1 )p 1≤v≤k

≤ C4

−2r

k

−2r

max v

1≤v≤k

≤ C4 ω 2r (f, k−1 )p ,

2r

max v 2r k ⊕r Jv,s (f ) − f kp

1≤v≤k

f kp

 2r ! k ω 2r (f, k−1 )p v

where the fourth inequality is deduced by Theorem 4.1 and the fifth is by (2.2). Thus 1 ω 2r (f, k−1 )p ≈ 2r max v 2r k ⊕r Jv,s (f ) − f kp . k 1≤v≤k 10

In the same way, we have 2r

ω (f, k

−1

)p ≤ C1 k ≤ C1

−2bm r

k X v=1

k

−2bm r

v 2bm r−1 k ⊕r Jv,s (f ) − f kp

k X

v

−2(1−bm )r− 41 −1

v=1

1

!

1

max v 2r+ 4 k ⊕r Jv,s (f ) − f kp

1≤v≤k

1

≤ C5 k−2r− 4 max v 2r+ 4 k ⊕r Jv,s (f ) − f kp 1≤v≤k

1

1

≤ C6 k−2r− 4 max v 2r+ 4 ω 2r (f, v −1 )p 1≤v≤k  2r ! k 2r+ 41 −2r− 14 max v ≤ C6 k ω 2r (f, k−1 )p 1≤v≤k v ≤ C7 ω 2r (f, k−1 )p , that is, ω 2r (f, k−1 )p ≈

1 k

1

2r+ 14

max v 2r+ 4 k ⊕r Jv,s (f ) − f kp . v≥k

Therefore ω 2r (f, k−1 )p ≈ ≈

1 max v 2r k ⊕r Jv,s (f ) − f kp k2r v≥k 1 2r+ 14 k ⊕r Jv,s (f ) − f kp . 1 max v 2r+ 4 v≥k k

(4.18)

Now we can complete the proof of (4.16). Clearly, there exists 1 ≤ k1 ≤ k such that 2r+ 14

k1

1

k ⊕r Jk1 ,s (f ) − f kp = max v 2r+ 4 k ⊕r Jv,s (f ) − f kp . 1≤v≤k

Then it is deduced from (4.18) that 1 max v 2r k ⊕r Jv,s (f ) − f kp k2r 1≤v≤k 1 1 max v 2r+ 4 k ⊕r Jv,s (f ) − f kp ≤ C8 2r+ 41 1≤v≤k k

k−2r k12r k ⊕r Jk1 ,s (f ) − f kp ≤

1

2r+ 41

= C8 k−2r− 4 k1

k ⊕r Jk1 ,s (f ) − f kp .

This implies k1 ≈ k. Applying (4.18) again implies ω 2r (f, k−1 )p ≤ C5 = C5

1 k

2r+ 41

1 2r+ 41

1

max v 2r+ 4 k ⊕r Jv,s (f ) − f kp

1≤v≤k

2r+ 14

(k1

k ⊕r Jk1 ,s (f ) − f kp )

k ≤ C5 max k ⊕r Jv,s (f ) − f kp . k1 ≤v≤k

11

Noticing that k1 ≈ k, we may rewrite the above inequality as ω 2r (f, k−1 )p ≤ C max k ⊕r Jv,s (f ) − f kp . v≥k

This completes the proof of Theorem 4.2.



p n−1 ) with order k −2r and the collecTheorem 4.3 {⊕r Jk,s }∞ k=1 are saturated on L (S tion of constants is their invariant class.

Proof. We first prove for j = 0, 1, 2, . . . , 1 − 1 ξk (j) k→∞ 1 − 1 ξk (1) lim

=

j(j + 2λ) . 2λ + 1

(4.19)

In fact, for any 0 < δ < π, it follows from (3.12) that Z

π

Dk,s (θ) sin

δ

2λ

 3 θ Dk,s (θ) sin2λ θ dθ θ dθ ≤ δ δ Z π ≤ δ−3 θ 3 Dk,s (θ) sin2λ θ dθ ≤ Cδ,s k−3 . Z

π

0

For v = 1, 2, . . . , we have, using (3.12) again,   Z π Z π Gλ1 (cos θ) θ 1 2λ Dk,s (θ) 1 − 1 − ξk (1) = Dk,s (θ) sin2 sin2λ θ dθ sin θ dθ ≈ λ 2 G1 (1) 0 0 ≈ k−2 .

(4.20)

We deduce from (2.9) that for any ǫ > 0, there exists δ > 0, for 0 < θ < δ, such that
j(j + 2λ) n n 1 − Pj (cos θ) − (1 − P1 (cos θ)) ≤ ǫ (1 − P1n (cos θ)) . 2λ + 1

Then it follows that

1 − 1 ξk (j) − j(j + 2λ) 1 − 1 ξk (1) 2λ + 1 Z π
= Dk,s (θ) 1 − Pjn (cos θ) sin2λ θ dθ 0 Z π j(j + 2λ) 2λ n sin θ dθ Dk,s (θ) (1 − P1 (cos θ)) − 2λ + 1 Z π0  

j(j + 2λ) = Dk,s (θ) 1 − Pjn (cos θ) − 1 − P1n (cos θ) sin2λ θ dθ 2λ + 1 0   Z π Z δ j(j + 2λ) dθ Dk,s (θ) sin2λ θ 1 + Dk,s (θ) ǫ (1 − P1n (cos θ)) sin2λ θ dθ + 2 ≤ 2λ + 1 δ 0 ≤ Cǫ k −2 + Cδ,s k−3 .

12

So, (4.19) holds. By Lemma 3.4, ⊕r Jk,s (f ) − f = and for j = 1, 2, . . . , Z r 1 − ξk,s(j) =

0

γ

Dk,s (θ) 1 −

∞ X j=0

(1 −r ξk,s (j)) Yj (f )




Pjn (cos θ)

Combining with (4.19) and (4.20), we have, 1 − r ξk,s(j) = lim k→∞ 1 − r ξk,s (1)



2λ

sin

θ dθ

j(j + 2λ) 2λ

r

r


r = 1 − 1 ξk,s(j) .

6= 0

and 1 − r ξk,s (1) ≈ k−2r . Using of Lemma 3.5, we finish the proof of Theorem 4.3.  We obtain the following corollary from Theorem 4.1, Theorem 4.2 and Theorem 4.3. Corollary 4.1 For positive integers r and s, 2s ≥ n, 0 < α ≤ 2r, f ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞, and the sequence of Boolean sums of Jackson operators {⊕r Jk,s }∞ k=1 given by (2.4), the following statements are equivalent. (i) k ⊕r Jk,s (f ) − f kp = O(k−α ) (k → ∞); (ii) ω 2r (f, δ)p = O(δ α ) (δ → 0). 

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[8] S. M. Nikol’skiˇı, I. P. Lizorkin, Approximation by spherical functions, Proc. Steklov Inst. Math., 173 (1987), 195-203. [9] C. M¨ uller, Spherical harmonics, Lecture Notes in Mathematics, 17, Springer, Berlin, 1966. ¨ [10] S. Pawelke, Uber die approximationsordnung bei kugelfunktionen und algebraischen polynomen, Tohoku Math. J., 24 (3) (1972), 473-486. [11] K. V. Rustamov, On equivalence of different moduli of smoothness on the sphere, Proc. Stekelov Inst. Math., 204 (3) (1994), 235-260. [12] E. M. Stein, G. Weiss, An introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, New Jersey, 1971. [13] K. Wang, L. Li, Harmonic analysis and approximation on the unit sphere, Science Press, Beijing, 2006. [14] E. van Wickeren, Steckin-Mauchaud type inequalities in connection with Bernstein polynomials, Constr. Approx., 2 (1986), 331-337.

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