direct and converse results in the ba space for jackson matsuoka polynomials on the unit sphere

R E S E A R C H Open AccessDirect and converse results in the Ba space for Jackson-Matsuoka polynomials on the unit sphere Guo Feng1*and Yuan Feng2 * Correspondence: gfeng@tzc.edu.cn 1 D

R E S E A R C H Open Access

Direct and converse results in the Ba space

for Jackson-Matsuoka polynomials on the

unit sphere

Guo Feng1*and Yuan Feng2

* Correspondence:

gfeng@tzc.edu.cn

1 Department of Mathematics,

Taizhou University, Taizhou,

Zhejiang 317000, China

Full list of author information is

available at the end of the article

Abstract

In this paper, we introduce K-functional and modulus of smoothness of the unit sphere in the Ba space, establish their relations and obtain the direct and converse theorem of approximation in the Ba space for Jackson-Matsuoka polynomials on the

unit sphere ofRd

MSC: 41A25; 41A35; 41A63

Keywords: Jackson-Matsuoka polynomials; Ba space; modulus of smoothness;

K-functional; spherical means

1 Introduction

LetS := Sd–={x : x = } denote the unit sphere in R d (d ≥ ), d ∈ N, where x denotes

the usual Euclidean norm,Z+ the set of nonnegative integers, andN the set of positive

integers We denote by L p := L p(S),  ≤ p ≤ ∞, the space of functions defined on S with

the finite norm

f  p:=

⎧

⎨

⎩

(

S|f ( )| p d)p, ≤ p < ∞,

where  ∈ S, and d is the measure element on S, and |S d–| =Sd=π d

(d)is the surface area ofS

The conception of Ba space was first put forward by Ding and Luo (see []) in their

discussion of the prior estimate of Laplace operator in some classical domains and in their study of the embedding theorem of Orlicz-Sobolev spaces, higher dimensional singular

integrals, and harmonic function etc.

Definition . (see []) Let B = {B, B, , B m, } be a sequence of linear normed

function spaces, a = {a, a, , a m, } be a sequence of nonnegative numbers For f ∈

∞

m=B m, we form the power series of

I (f , α) :=

∞



m=

a m α m f  m

If I(f , α) has a non-zero radius of convergence, we say f ∈ Ba.

© 2014 Feng and Feng; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction

in any medium, provided the original work is properly cited.

The norm in Ba is defined by

f  Ba:= inf

α>





α : I(f , α)≤ 

As proved in [], Ba is a Banach space if B m is a Banach space Evidently, if B m = L m, then

Ba space is an Orlicz space If B m = L p , a = {, , , , }, then a Ba space is a classical

Lebesgue space

Hereafter the space of spherical harmonics of degree k is denoted by H d

k The Laplace-Beltrami operator on the unit sphere is denoted by

Df ( ) := f 

| |

∈S

which has eigenvalue λ k := –k(k + d – ) corresponding to the eigenspace H d

k with k∈

Z+, namely,H d

k={ ∈ C(S) : D = –k(k + d – )} For the properties of the space of

spherical harmonics and the Laplace-Beltrami operators, see [–] The standard Hilbert

space theory shows that L(S) = ∞k=H d

k The orthogonal projection Y k : L(S) →H d

k

takes the form

Y k (f ;  ) :=  (λ)(k + λ)

π λ+



where λ = d – , P λ

k denotes hyperspherical polynomials of degree k which satisfies ( –

r cos θ + r)–r= ∞k=r k P λ

k (cos θ ),  ≤ θ ≤ π.

The spherical means are denoted by

T θ (f ) := T θ (f ;  ) := 

|Sd–|(sin θ) d–



f (ϑ) dϑ,

where|Sd–| is the surface area of Sd–,

The properties of the spherical means are well known (see [, ])

Based on the classical Jackson-Matsuoka kernel (see []) we define a new kernel

M n ;j,i,s (θ ) := 

n ;j,i,s

sinj nθ/

sini θ/

s

, n = , , , θ∈ R,

where j, i, s ∈ N, n ;j,i,sis chosen such thatπ

 M n ;j,i,s (θ ) sin λ θ dθ=  It is well known that

M n ;j,i,s (θ ) is an even nonnegative operator In particular, it is an even and nonnegative

trigonometric polynomial of degree at most s(nj + j – i) for j ≥ i and the Jackson

poly-nomial for j = i Using M n ;j,i,s (θ ) we consider spherical convolution:

J n ;j,i,s (f ;  ) := (f ∗ M n ;j,i,s )( ) :=

 π



T θ (f ;  )M n ;j,i,s (θ )(θ ) sin λ θ dθ (.)

It is called the Matsuoka polynomial on the unit sphere based on the

Jackson-Matsuoka kernel In particular, (f∗ M n ;j,i,s )( ) =  for f( ) =  The classical

Jackson-Matsuoka polynomial in classical L pspace has been studied by many authors (see [, ])

In this paper, we consider the approximation of the Jackson-Matsuoka polynomial on

the unit sphere in the Ba space Firstly, we introduce K -functionals, modulus of

smooth-ness on the unit sphere in the Ba space, establish their relations Then with the help of

the relation between K -functionals and modulus of smoothness on the sphere in the Ba

space and the properties of the spherical means, we obtain the direct and converse best

approximation in the Ba space by Jackson-Matsuoka polynomial on the unit sphere ofRd

2 K-Functionals and modulus of smoothness

Definition . For f ∈ Ba, the modulus of smoothness on the unit sphere is given by

ω (f ; t) Ba:= sup

<θ≤t

f – T θ (f )

The K -functional of the unit sphere is given by

K

f ; t

Ba:= inf

g ∈W Ba(S)



f – g Ba + tDg Ba



where W Ba(S) := {f : f ∈ Ba, Df ∈ Ba},  < t < t, tis a positive constant, Df denotes the

Laplace-Beltrami operator on the unit sphere

To prove the weak equivalence between the K -functional and the modulus of

smooth-ness on the unit sphere, we need the following lemma

Lemma . Let B={L p, L p, , L p m, } be a sequence of Lebesgue spaces, p m ≥ , m =

, , , a = {a, a, , a m, } be a sequence of nonnegative numbers, {a m

m } ∈ l∞,{a–m

m } ∈

l∞ If f ∈ Ba :=∞m=L p m , then

f  p m≤ 

where μ= infm≥{a m

m}

Proof Since{a m

m } ∈ l∞, we may let  < q = sup m≥{a m

m } ∈ L∞ From{a–m

m } ∈ l∞, we may

let μ = inf m≥{a m

m } Then  < μ < ∞.

In view of the ∞m=a m α m f  m

p m≤ , the supm≥f  p mexists Let

u= sup

m≥



f  p m



By the definition of supremum, for any δ > , there exists K ≥ , such that f  p K > u – δ.

By the definition off  Ba= infα>{

α : I(f , α) ≤ }, for any ε > , there exists 

α, such that

∞

m=a m α m

f  m

p m ≤  holds Therefore f  Ba= infα>{

α : I(f , α)} > 

α– ε Namely

≥

∞



m=

a m α mf  m

p m ≥ a K α Kf  K

p K >

a



K

K (u – δ)K≥sα(u – δ)K

By the arbitrariness of δ,



α ≥ μ · u = μ · sup

m≥



f  p m

 ,

f  p m> 

α – ε ≥ μ · sup

m≥



f  p m



– ε,

and also ε is arbitrary, therefore

sup

m≥



f  p m



≤ 

μ f  Ba,

which implies that for any p m, we have

f  p m≤ 

μ f  Ba

We will establish the weak equivalence between the K -functional and the modulus of smoothness on the unit sphere in the Ba space.

Theorem . Let B={L p, L p, , L p m, } be a sequence of Lebesgue spaces, p m ≥ , m =

, , , a = {a, a, , a m, } be a sequence of nonnegative numbers If {a m

m } ∈ l∞,{a–m

m } ∈

l∞ Then for f ∈ Ba,  < t < π

, the weak equivalence

ω (f ; t) Ba Kf ; t

holds , where the weakly equivalent relation A(n)

on n such that A (n) ≤ CB(n) holds.

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

cir-cumstances

Proof For m = , , , g ∈ W Ba(S), note that []

T θ g – gp m ≤ CθDg p m,

T θ fp m ≤ f  p m

By the definition of the Ba-norm · Baand (.), we have

T θ g – g Ba= inf

α>



α:

∞



m=

a m

α m T θ g – g m

p m≤ 



≤ inf

α>



α:

∞



m=

a m

α m C m θ m Dg m

p m≤ 



≤ inf

α>



α:

∞



m=

C m q m

α m θ m Dg m

p m≤ 



≤ inf

α>



α:

∞

α m

C · q · θ

m

≤ 



Let α =  C ·q·θ μ Dg Ba, then ∞m=αm(C ·q·θ μ Dg Ba)m=  Consequently ∞m=a m

α m T θ g–

gm

p m≤  Therefore, we have

The proof is similar to that of (.), we get

T θ (f – g)

The triangle inequality gives

T θ f – fBa ≤ f – g Ba + C(q, μ)θDg Ba,

which shows that ω(f ; t) Ba ≤ C(q, μ)K(f ; t)Ba On the other hand, we define

g (x) = v

 θ



(sin u) –λ du

 u



T t f (x)(sin t) λ dt

with v–

θ =θ

(sin u) –λ duu

(sin t) λ dt Then Dg = v θ (T θ f – f ), this also gives

Since for ≤ θ ≤ π

, the inequality πθ ≤ sin θ ≤ θ shows that v–

θ θ Moreover,

f – g = v–θ

 θ



(sin u) –λ du

 u



(T t – f )(sin t) λ dt Consequently, we get

By (.) and (.), similar to the proof of (.), we obtain

and

Combining (.), (.), and the definition of K -functional, we have

K

f ; θ

Ba ≤ f – g Ba + θDg Ba

≤ CT θ f – fBa + Cθ–θT θ f – fBa

Thus

K

f ; t

Corollary . For t ≥ , there is a constant C such that

ω (f ; tδ) Ba ≤ C max, t

Proof By the weakly equivalent relation between the modulus of smoothness and K

-func-tional, and the definition of K (f ; t)Ba, we have

ω (f ; tδ) Ba ≤ CKf ; (tδ)

Ba ≤ Cf – g Ba + tδDg Ba



≤ C max, t

f – g Ba + δDg Ba



≤ C max, t

K

f ; δ

Ba ≤ C max, t

ω (f ; δ) Ba

3 Some lemmas

Lemma . Let n ;j,i,s=π

(sinj nθ

sini θ



)ssinλ θ dθ Then the weak equivalence

n ;j,i,s n is–λ– (.)

holds for si > λ + , j ≥ i.

π ≤ sinθ

≤θ

, and sin θ ≤ θ for  ≤ θ ≤ π, we have

n ;j,i,s =

 π



sinj nθ sini θ

s

sinλ θ dθ

n is–λ–

 nπ/



t λ sin

j t

t i

s

dt

n is–λ– π/



t λ sin

j t

t i

s

dt+

 ∞

π/

t λ sin

j t

t i

s

dt

n is–λ–, (.)

Lemma . For is > r + λ + , j ≥ i, r ∈ R, there is a constant C(λ, j, i, s) such that

 π



θ r M n ;j,i,s (θ ) sin λ θ dθ ≤ C(λ, j, i, s)n –r (.)

Proof Since θ

π ≤ sinθ

≤θ

, and sin θ ≤ θ for  ≤ θ ≤ π, by n ;j,i,s n is–λ–, we have

 π



θ r M n ;j,i,s (θ ) sin λ θ dθ

≤ C(λ, i, j, s)n –is+λ+

 π



θ r sin

j nθ



sini θ

s

sinλ θ dθ

≤ C(λ, i, j, s)n –is+λ+ n is–r–λ–

 nπ/

t r +λ sin

j t

t i

s

dt

≤ C(λ, i, j, s)n –r

π/



t r +λ sin

j t

t i

s

dt+

 ∞

π/

t r +λ sin

j t

t i

s

dt

≤ C(λ, j, i, s)Cn λ ≤ C(λ, j, i, s)n λ

, where

C=

 π/



t λ sinj t

t i

s

dt+

 ∞

π/

t λ sinj t

t i

s

dt, is > r + λ + , j ≥ i. 

Lemma .(see []) Suppose that g ∈ C(S) Then, for  ∈ (S) and  < t <π

, we have

B t (g,  ) – g( ) = 

 (t)

 t



sind–θ dθ

 θ



 sind–u  (u)B u (Dg,  ) du, (.)

T θ (g;  ) – g( ) = (

d–

 )

π d–

 θ



 (t)

where

B t (f ,  ) = 

 (t)



cost

f (ϑ) dϑ, t > ,  , ϑ∈ Sd–,

 (t) = π

d–



(d– )

t

sind–u du

Lemma . Let g , Dg, Dg ∈ Ba, Ba :=∞m=L p m(S), m = , , ,  ≤ p m ≤ ∞, J n ;j,i,s (f ;  )

be the Jackson-Matsuoka polynomial on the unit sphere based on the Jackson-Matsuoka

kernel , is > d +  Then there is a constant C(d, j, i, s) such that

J n ;j,i,s g – g – α(n)Dg

Ba ≤ C(d, j, i, s)n–Dg

where α (n) n–

Proof For m∈ N, by (.), we have

J n ;j,i,s (g;  ) – g( )

=

 π



M n ;j,i,s (θ )

T θ (g;  ) – g( )

sind–θ dθ

=

 π



M n ;j,i,s (θ ) sin d–θ dθ (

d–

 )

π d–

 θ



 (t)

sind–t B t (Dg,  ) dt

= Dg( )

 π



M n ;j,i,s (θ ) sin d–θ dθ (

d–

 )

π d–

 θ



 (t)

sind–t dt

+

 π



M n ;j,i,s (θ ) sin d–θ dθ (

d–

 )

π d–

 θ



 (t)

sind–t



B t (Dg,  ) – Dg( )

dt

×

 t



sind–u

B t (Dg,  ) – Dg( )

du

= Dg( )

 π

M n ;j,i,s (θ ) sin d–θ dθ

 θ

dt

sind–t

 t sind–u du

 π



M n ;j,i,s (θ ) sin d–θ dθ

 θ



dt

sind–t

 t



sind–u

B t (Dg,  ) – Dg( )

du

:= α(n)Dg( ) +

 π



M n ;j,i,s (θ ) sin d–θ  θ (g,  ) dθ , (.) where

α (n) :=

 π



M n ;j,i,s (θ ) sin d–θ dθ

 θ



dt

sind–t

 t



sind–u du

and

 θ (g,  ) :=

 θ



dt

sind–t

 t



sind–u

B t (Dg,  ) – Dg( )

du,

α (n) =

 π



M n ;j,i,s (θ ) sin d–θ dθ

 θ



dt

sind–t

 t



sind–u du

 π



M n ;j,i,s (θ ) sin d–θ dθ

 θ



t sin d–ξ

sind–t dt

 π



θM n ;j,i,s (θ ) sin d–θ dθ n– ( < ξ < t). (.)

Using Lemma ., and the expression of B t (Dg,  ) – Dg, we obtain

 θ (g)

p m ≤ C(d, j, i, s)θDg

p m

By Lemma ., and the Hölder-Minkowski inequality we get



π M n ;j,i,s (θ ) sin d–θ  θ (g,  ) dθ



p m

≤ C(d, j, i, s)Dg

p m

 π



θM n ;j,i,s (θ ) sin d–θ dθ ≤ C(d, j, i, s)n–Dg

p m (.) Consequently, by (.), (.), and (.), we get

J n ;j,i,s g – g – α(n)Dg

p m ≤ C(d, j, i, s)n–Dg

By Lemma ., we have

J n ;j,i,s g – g – α(n)Dg

Ba

= inf

α>



α:

∞



m=

a m

α mJ n ;j,i,s g – g – α(n)Dgm

p m≤ 



≤ inf

α>



α:

∞



m=

a m

α m C (d, j, i, s)n–Dgm

p m≤ 



≤ inf

α>



α:

∞



m=

q m · C m

α m C (d, j, i, s)n–Dgm

Ba≤ 



≤ C(d, j, i, s, q, μ)n–Dg

Ba

4 Main results

Theorem . Suppose that f ∈ Ba :=∞m=L p m(S), m = , , ,  ≤ p m ≤ ∞, J n ;j,i,s (f ;  )

be the Jackson-Matsuoka polynomial on the unit sphere based on the Jackson-Matsuoka

kernel , is > d + , λ = d – , j ≥ i Then

J n ;j,i,s (f ) – f

Ba ≤ C(d, j, i, s)ωf ; n–

Proof Since (f∗ M n ;j,i,s )( ) =  for f( ) = , Therefore, we have

J n ;j,i,s (f ) – f

Ba

= inf

α>



α:

∞



m=

a m

α mJ n ;j,i,s (f ) – fm

p m≤ 



≤ inf

α>



α:

∞



m=

a m

α m



π M n ;j,i,s (θ )

f (x) – T θ (f ; x)

sinλ θ dθ

m

p m

≤ 



≤ inf

α>



α:

∞



m=

a m

α m π



f – T θ (f )

p m M n ;j,i,s (θ ) sin λ θ dθ

m

≤ 



≤ inf

α>



α:

∞



m=

q m · C m

α m

π



f – T θ (f )

Ba M n ;j,i,s (θ ) sin λ θ dθ

m

≤ 



Splitting the integral on [, π ] into two integrals on [, /n] and [/n, π ], respectively, and

using the definition of ω(f ; t) Ba, we conclude that

f – T θ (f )

Ba ≤ ωf ; n–

Ba+

 π

/n

ω (f ; θ ) Ba M n ;j,i,s (θ ) sin λ θ dθ (.)

From Corollary . we have, for θ ≥ n–,

ω (f ; θ ) Ba = ω

f ; n–

Ba ≤ C max, nθ

ω

f ; n–

Ba ≤ Cnθω

f ; n–

By (.), (.), and Lemma ., we get

f – T θ (f )

Therefore, by (.), (.), we have

J n ;j,i,s (f ) – f

Ba

≤ inf

α>



α:

∞



m=

q m · C m

α m

π



ω

f ; n–

Ba M n ;j,i,s (θ ) sin λ θ dθ

m

≤ 



= inf

α>



α:

∞



m=

q m · C m

α m



ω

f ; n–

Ba

m≤ 



≤ C(d, j, i, s, q, μ)ωf ; n–



Theorem . Suppose that f ∈ Ba :=∞m=L p m(S),  ≤ pm ≤ ∞, J n ;j,i,s (f ; x) is the

Jackson-Matsuoka polynomial on the unit sphere based on the Jackson-Jackson-Matsuoka kernel , is > d + ,

λ = d – , j ≥ i,  < α <  Then the following statements are equivalent:

() J n ;j,i,s (f ) – f

Ba = O

n –α

() ω

f ; n–

Ba = O

t α

Proof By Theorem ., we have ()⇒ () Now, we prove () ⇒ () Let r be a fixed positive

integer, defined by

J n r ;j,i,s (f ;  ) :=

r



k=

π



M n ;j,i,s (θ )Q λ

k (cos θ ) sin λ θ dθ

r

Y k (f ;  ).

By orthogonality of the orthogonal projector Y k, we have

J r +l (f ) =

r



k=

π



M n ;j,i,s (θ )Q λ k (cos θ ) sin λ θ dθ

r

× Y k

 r



v=

π



M n ;j,i,s (θ )Q λ v (cos θ ) sin λ θ dθ

l

Y v (f )



= J n r ;j,i,s

J n l ;j,i,s (f )

Let g = J r

n ;j,i,s (f ), by (.) we get

f – g Ba= inf

α>



α:

∞



m=

a m

α m f – g m

p m≤ 



= inf

α>



α:

∞



m=

a m

α mf – J r

n ;j,i,s (f )m

p m≤ 



≤ inf

α>



α:

∞



m=

a m

α m

 r



k=

J k–

n ;j,i,s (f ) – J n k ;j,i,s (f )

p m

m

≤ 



≤ inf

α>



α:

∞



m=

a m

α m



C (d, j, i, s)

r



k=

J k–

n ;j,i,s



f – J n ;j,i,s (f )

p m

m

≤ 



≤ inf

α>



α:

∞



m=

a m

α m C (d, j, i, s)rf – J n ;j,i,s (f )m

p m≤ 



≤ inf

α>



α:

∞



m=

q m · C m

(d, j, i, s, r)

α m f – J n ;j,i,s (f )m

Ba≤ 



≤ C(d, j, i, s, r, q, μ)f – J n ;j,i,s (f )

where J n;j,i,s (f ) = f

On the other hand,

DJ r

n ;j,i,s (f )

p m≤

r



k (k + d – )

π



M n ;j,i,s (θ ) Q λ

k (cos θ ) sinλ θ dθ

r

Y k (f ).

space and the properties of the spherical means, we obtain the direct and converse best

approximation in the Ba space by Jackson- Matsuoka. .. class="page_container" data-page="3">

smooth-ness on the unit sphere in the Ba space, establish their relations Then with the help of

the relation between K -functionals and modulus... x) is the

Jackson- Matsuoka polynomial on the unit sphere based on the Jackson- Jackson -Matsuoka kernel , is > d + ,

λ = d – , j ≥ i,  < α <  Then the following