THÔNG TIN TÀI LIỆU
Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 RESEARCH 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 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 of Rd MSC: 41A25; 41A35; 41A63 Keywords: Jackson-Matsuoka polynomials; Ba space; modulus of smoothness; K-functional; spherical means Introduction Let S := Sd– = {x : x = } denote the unit sphere in Rd (d ≥ ), d ∈ N, where x denotes the usual Euclidean norm, Z+ the set of nonnegative integers, and N the set of positive integers We denote by Lp := Lp (S), ≤ p ≤ ∞, the space of functions defined on S with the finite norm ⎧ ⎨( |f ( )|p d ) p , ≤ p < ∞, S f p := (.) ⎩ess sup |f ( )|, p = ∞, ∈S where ∈ S, and d is the measure element on S, and |Sd– | = d Sd = π ( d ) is the surface area of S 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 , , Bm , } be a sequence of linear normed function spaces, a = {a , a , , am , } be a sequence of nonnegative numbers For f ∈ ∞ m= Bm , we form the power series of ∞ am α m f I(f , α) := m Bm (.) 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 Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page of 13 The norm in Ba is defined by f Ba : I(f , α) ≤ α := inf α> (.) As proved in [], Ba is a Banach space if Bm is a Banach space Evidently, if Bm = Lm , then Ba space is an Orlicz space If Bm = Lp , a = {, , , , }, then a Ba space is a classical Lebesgue space Hereafter the space of spherical harmonics of degree k is denoted by Hkd The LaplaceBeltrami operator on the unit sphere is denoted by Df ( ) := f , | | (.) ∈S which has eigenvalue λk := –k(k + d – ) corresponding to the eigenspace Hkd with k ∈ Z+ , namely, Hkd = { ∈ 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) = ∞ Hkd The orthogonal projection Yk : L (S) → Hkd k= takes the form Yk (f ; ) := (λ)(k + λ) π λ+ S Pkλ ( , ϑ)f (ϑ) dϑ, (.) where λ = d – , Pkλ denotes hyperspherical polynomials of degree k which satisfies ( – k λ r cos θ + r )–r = ∞ k= r Pk (cos θ ), ≤ θ ≤ π The spherical means are denoted by Tθ (f ) := Tθ (f ; ) := |Sd– |(sin θ )d– f (ϑ) dϑ, ,ϑ =cos θ where |Sd– | is the surface area of Sd– , x, y denotes the usual Euclidean inner product The properties of the spherical means are well known (see [, ]) Based on the classical Jackson-Matsuoka kernel (see []) we define a new kernel Mn;j,i,s (θ ) := n;j,i,s sinj nθ / sini θ / s n = , , , θ ∈ R, , π where j, i, s ∈ N, n;j,i,s is chosen such that Mn;j,i,s (θ ) sinλ θ dθ = It is well known that Mn;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 polynomial for j = i Using Mn;j,i,s (θ ) we consider spherical convolution: Jn;j,i,s (f ; π ) := (f ∗ Mn;j,i,s )( ) := Tθ (f ; )Mn;j,i,s (θ )(θ ) sinλ θ dθ (.) It is called the Jackson-Matsuoka polynomial on the unit sphere based on the JacksonMatsuoka kernel In particular, (f ∗ Mn;j,i,s )( ) = for f ( ) = The classical JacksonMatsuoka polynomial in classical Lp space 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- Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page of 13 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 of Rd 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 f – Tθ (f ) , there exists K ≥ , such that f pK > u – δ By the definition of f Ba = infα> { α : I(f , α) ≤ }, for any ε > , there exists α , such that ∞ m m m= am α f pm ≤ holds Therefore f Ba = infα> { α : I(f , α)} > α – ε Namely ∞ ≥ am αm f m pm ≥ aK αK f K pK m= By the arbitrariness of δ, ≥ μ · u = μ · sup f α m≥ f pm > pm – ε ≥ μ · sup f α m≥ , pm > aKK (u – δ) – ε, K K ≥ sα (u – δ) Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page of 13 and also ε is arbitrary, therefore sup f m≥ ≤ pm f μ Ba , which implies that for any pm , we have f pm ≤ f μ Ba The proof is completed 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 = {Lp , Lp , , Lpm , } be a sequence of Lebesgue spaces, pm ≥ , m = – , , , a = {a , a , , am , } be a sequence of nonnegative numbers If {amm } ∈ l∞ , {amm } ∈ l∞ Then for f ∈ Ba, < t < π , the weak equivalence ω(f ; t)Ba K f ; t (.) Ba holds, where the weakly equivalent 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 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 Proof For m = , , , g ∈ WBa (S), note that [] Tθ g – g Tθ f pm pm ≤ Cθ Dg ≤ f pm , pm By the definition of the Ba-norm · ∞ Tθ g – g Ba = inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= Ba and (.), we have am Tθ g – g αm m pm ≤ am m m C θ Dg αm m pm ≤ C m qm m θ Dg αm m pm ≤ C · q · θ Dg αm μ m Ba ≤ (.) Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 ∞ C·q·θ m= α m ( μ Let α = C·q·θ Dg Ba , then μ g m ≤ Therefore, we have pm Tθ g – g Ba ≤ C(q, μ)θ Dg Page of 13 Dg Ba ) m = Consequently ∞ am m= α m Ba Tθ g – (.) The proof is similar to that of (.), we get Tθ (f – g) Ba ≤ C(q, μ) f – g Ba (.) The triangle inequality gives Tθ f – f ≤ f –g Ba 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 u θ (sin u)–λ du g(x) = v Tt f (x)(sin t)λ dt θ u –λ du (sin t)λ dt (sin u) with v– θ = Dg pm ≤ Cθ – Tθ f – f Then Dg = vθ (Tθ f – f ), this also gives pm (.) Since for ≤ θ ≤ π , the inequality π θ ≤ sin θ ≤ θ shows that v– θ u θ f – g = v– θ θ Moreover, (sin u)–λ du (Tt – f )(sin t)λ dt Consequently, we get f –g pm ≤ C Tθ f – f pm (.) By (.) and (.), similar to the proof of (.), we obtain Dg Ba ≤ Cθ – Tθ f – f (.) Ba and f –g Ba ≤ C Tθ f – f Ba (.) Combining (.), (.), and the definition of K -functional, we have K f ; θ Ba ≤ f –g Ba + θ Dg ≤ C Tθ f – f Ba ≤ C Tθ f – f Ba Thus K f ; t Ba ≤ Cω(f ; t)Ba Ba – + Cθ θ Tθ f – f Ba (.) Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page of 13 Corollary . For t ≥ , there is a constant C such that ω(f ; tδ)Ba ≤ C max , t ω(f ; δ)Ba (.) Proof By the weakly equivalent relation between the modulus of smoothness and K -functional, and the definition of K(f ; t )Ba , we have ω(f ; tδ)Ba ≤ CK f ; (tδ) ≤ C max , t Ba ≤C f –g f –g ≤ C max , t K f ; δ Ba + t δ Dg Ba Ba + δ Dg Ba ≤ C max , t ω(f ; δ)Ba Ba Corollary . has been proved Some lemmas Lemma . Let n;j,i,s = π sinj nθ λ s ( sini θ ) sin θ dθ Then the weak equivalence nis–λ– n;j,i,s (.) holds for si > λ + , j ≥ i Proof As ≤ sin θ ≤ θ , and sin θ ≤ θ for ≤ θ ≤ π , we have θ π π n;j,i,s = sinj nθ sini θ s sinλ θ dθ sinj t t i s t λ sinj t t i s t λ nπ / nis–λ– π / nis–λ– dt ∞ dt + t λ π / sinj t t i s dt nis–λ– , (.) since si > λ + , j ≥ i Lemma . has been proved Lemma . For is > r + λ + , j ≥ i, r ∈ R, there is a constant C(λ, j, i, s) such that π θ r Mn;j,i,s (θ ) sinλ θ dθ ≤ C(λ, j, i, s)n–r (.) Proof Since θ π ≤ sin θ ≤ θ , and sin θ ≤ θ for ≤ θ ≤ π , by n;j,i,s π θ r Mn;j,i,s (θ ) sinλ θ dθ π ≤ C(λ, i, j, s)n–is+λ+ θr sinj nθ sini θ s sinλ θ dθ nπ / ≤ C(λ, i, j, s)n–is+λ+ nis–r–λ– t r+λ sinj t t i s dt nis–λ– , we have Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 t r+λ ∞ s sinj t t i π / ≤ C(λ, i, j, s)n–r Page of 13 dt + t r+λ π / sinj t t i s dt ≤ C(λ, j, i, s)C nλ ≤ C(λ, j, i, s)nλ , where tλ ∞ s sinj t t i π / C = dt + s sinj t t i tλ π / is > r + λ + , j ≥ i dt, ∈ (S) and < t < π , we have Lemma . (see []) Suppose that g ∈ C (S) Then, for Bt (g, ) – g( ) = Tθ (g; t (t) θ sind– θ dθ ) ( d– ) – g( ) = π θ d– (t) sind– t (u)Bu (Dg, sind– u Bt (Dg, ) du, ) dt, (.) (.) where Bt (f , d– (t) = (t) )= π ( d– ) t ,ϑ ≤ cos t≤ f (ϑ) dϑ, , ϑ ∈ Sd– , t > , sind– u du ) Lemma . Let g, Dg, D g ∈ Ba, Ba := ∞ m= Lpm (S), m = , , , ≤ pm ≤ ∞, Jn;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 Jn;j,i,s g – g – α(n)Dg Ba ≤ C(d, j, i, s)n– D g Ba , (.) n– where α(n) Proof For m ∈ N, by (.), we have Jn;j,i,s (g; ) – g( ) π Mn;j,i,s (θ ) Tθ (g; = ) – g( ) sind– θ dθ π Mn;j,i,s (θ ) sind– θ dθ = ) ( d– π d– θ ( d– ) π Mn;j,i,s (θ ) sind– θ dθ = Dg( ) π Mn;j,i,s (θ ) sind– θ dθ + (t) sind– t π π θ d– ( d– ) (t) d– sin θ d– Bt (Dg, (t) d– sin t ) dt dt Bt (Dg, t ) – Dg( ) dt t × sind– u Bt (Dg, ) – Dg( ) du π θ Mn;j,i,s (θ ) sind– θ dθ = Dg( ) t dt d– sin t sind– u du Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 π θ t dt Mn;j,i,s (θ ) sind– θ dθ + Page of 13 sind– u Bt (Dg, sind– t ) – Dg( ) du π Mn;j,i,s (θ ) sind– θ := α(n)Dg( ) + θ (g, ) dθ , (.) where π θ t dt Mn;j,i,s (θ ) sind– θ dθ α(n) := d– sin t sind– u du and θ θ (g, ) := t dt d– sin sind– u Bt (Dg, t π θ d– sin π θ Mn;j,i,s (θ ) sind– θ dθ t dt Mn;j,i,s (θ ) sind– θ dθ α(n) = ) – Dg( ) du, sind– u du t t sind– ξ sind– t dt π θ Mn;j,i,s (θ ) sind– θ dθ n– ( < ξ < t) (.) Using Lemma ., and the expression of Bt (Dg, θ (g) pm ≤ C(d, j, i, s)θ D g pm ) – Dg, we obtain By Lemma ., and the Hölder-Minkowski inequality we get π Mn;j,i,s (θ ) sind– θ θ (g, ) dθ pm ≤ C(d, j, i, s) D g π pm θ Mn;j,i,s (θ ) sind– θ dθ ≤ C(d, j, i, s)n– D g pm (.) Consequently, by (.), (.), and (.), we get Jn;j,i,s g – g – α(n)Dg pm ≤ C(d, j, i, s)n– D g pm By Lemma ., we have Jn;j,i,s g – g – α(n)Dg ∞ = inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= Ba am Jn;j,i,s g – g – α(n)Dg αm am C(d, j, i, s)n– D g αm m pm m pm ≤ qm · C m C(d, j, i, s)n– D g αm ≤ C(d, j, i, s, q, μ)n– D g The proof is completed Ba ≤ m Ba ≤ (.) Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page of 13 Main results Theorem . Suppose that f ∈ Ba := ∞ ) m= Lpm (S), m = , , , ≤ pm ≤ ∞, Jn;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 Jn;j,i,s (f ) – f Ba ≤ C(d, j, i, s)ω f ; n– Ba (.) Proof Since (f ∗ Mn;j,i,s )( ) = for f ( ) = , Therefore, we have Jn;j,i,s (f ) – f Ba ∞ = inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= am Jn;j,i,s (f ) – f αm m pm ≤ m π am αm ≤ Mn;j,i,s (θ ) f (x) – Tθ (f ; x) sinλ θ dθ pm m π am αm f – Tθ (f ) qm · C m αm M (θ ) sinλ θ dθ pm n;j,i,s ≤ m π f – Tθ (f ) M (θ ) sinλ θ dθ Ba n;j,i,s ≤ (.) 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 – Tθ (f ) ≤ ω f ; n– Ba π ω(f ; θ )Ba Mn;j,i,s (θ ) sinλ θ dθ + Ba (.) /n 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– Ba (.) By (.), (.), and Lemma ., we get f – Tθ (f ) Ba ≤ Cω(f ; θ )Ba (.) Therefore, by (.), (.), we have Jn;j,i,s (f ) – f Ba ∞ ≤ inf α : α> m= ∞ = inf α : α> m= qm · C m αm m π ω f ; n– qm · C m ω f ; n– αm ≤ C(d, j, i, s, q, μ)ω f ; n– Ba m Ba M (θ ) sinλ θ dθ Ba n;j,i,s ≤ ≤ (.) Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page 10 of 13 Theorem . Suppose that f ∈ Ba := ∞ m= Lpm (S), ≤ pm ≤ ∞, Jn;j,i,s (f ; x) is the JacksonMatsuoka polynomial on the unit sphere based on the Jackson-Matsuoka kernel, is > d + , λ = d – , j ≥ i, < α < Then the following statements are equivalent: () Jn;j,i,s (f ) – f () ω f ; n– = O n–α , Ba = O tα , Ba n ≥ ; (.) < t < (.) Proof By Theorem ., we have () ⇒ () Now, we prove () ⇒ () Let r be a fixed positive integer, defined by r r (f ; Jn;j,i,s r π Mn;j,i,s (θ )Qλk (cos θ ) sinλ θ dθ ) := k= Yk (f ; ) By orthogonality of the orthogonal projector Yk , we have r r π Mn;j,i,s (θ )Qλk (cos θ ) sinλ θ dθ J r+l (f ) = k= r l π × Yk v= Mn;j,i,s (θ )Qλv (cos θ ) sinλ θ dθ Yv (f ) r l Jn;j,i,s (f ) = Jn;j,i,s (.) r (f ), by (.) we get Let g = Jn;j,i,s ∞ f –g Ba = inf α : α> m= ∞ = inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> m= am f –g αm m pm ≤ am r (f ) f – Jn;j,i,s αm am αm m pm ≤ m r k– k Jn;j,i,s (f ) – Jn;j,i,s (f ) ≤ pm k= am C(d, j, i, s) αm m r k– Jn;j,i,s f – Jn;j,i,s (f ) pm ≤ k= am C(d, j, i, s)r f – Jn;j,i,s (f ) αm m pm ≤ qm · Cm (d, j, i, s, r) f – Jn;j,i,s (f ) αm m Ba ≤ C(d, j, i, s, r, q, μ) f – Jn;j,i,s (f ) Ba ≤ , (.) (f ) = f where Jn;j,i,s On the other hand, r r (f ) DJn;j,i,s pm ≤ k= r π k(k + d – ) Mn;j,i,s (θ ) Qλk (cos θ ) sinλ θ dθ Yk (f ) Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page 11 of 13 Note that [] Ckλ (cos θ ) ≤ C (kθ )– , Ckλ () Qλk (cos θ ) ≡ For kθ ≥ , from (.) we have r DJn;j,i,s (f ) pm r r π ≤ C(d, j, i, s) k(k + d – )k –rλ Mn;j,i,s (θ )θ –λ sinλ θ dθ Yk (f ) k= pm ∞ ≤ C(d, j, i, s)nrλ f k –rλ ≤ C(d, j, i, s)nrλ f pm (.) pm k= holds for r > d– r DJn;j,i,s (f ) For kθ < , by Lemma ., we get pm r π ≤ Mn;j,i,s (θ )θ – r θ k(k + d – ) r r Qλk (cos θ ) sinλ θ dθ k= Yk (f ) pm r π ≤ C(d, j, i, s) Mn;j,i,s (θ )θ – r (kθ ) r r sinλ θ dθ Yk (f ) k= r pm π ≤ C(d, j, i, s) r Mn;j,i,s (θ )θ – r sinλ θ dθ Yk (f ) k= pm ∞ ≤ C(d, j, i, s)n ≤ C(d, j, i, s)n f Yk (f ) k= pm (.) pm Consequently, the inequality r (f ) DJn;j,i,s pm ≤ C(d, j, i, s)n f holds uniformly for r > r (f ) DJn;j,i,s Thereby Ba ∞ = inf α : α> m= ∞ ≤ inf α : α> m= ∞ ≤ inf α : α> d– (.) pm m= am r DJn;j,i,s (f ) αm pm am C(d, j, i, s)n f αm ≤ pm ≤ qm · C m C(d, j, i, s)n f αm ≤ C(d, j, i, s, q, μ)n f Ba Ba ≤ (.) Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Page 12 of 13 Without loss of generality, we may assume r > (.), we have r (f ) α(n) DJn;j,i,s Ba , d– r > r + d– Using Lemma . and r = α(n)DJn;j,i,s (f ) Ba ≤ r + C(d, j, i, s)n– D Jn;j,i,s (f ) r (f ) – f Jn;j,i,s Ba ≤ r Jn;j,i,s (f ) – f Ba ≤ r Jn;j,i,s (f ) – f Ba – + C(d, j, i, s)n r + C(d, j, i, s) n– DJn;j,i,s (f ) ≤ r Jn;j,i,s (f ) – f Ba r–r D Jn;j,i,s (f ) Ba Ba r–r r + n– Jn;j,i,s (f ) – Jn;j,i,s (f ) Ba r + Jn;j,i,s (f ) – f Ba – + C(d, j, i, s) n r DJn;j,i,s (f ) ≤ C(d, j, i, s, r) Jn;j,i,s (f ) – f – Ba +n ≤ C(d, j, i, s, r, q, μ) Jn;j,i,s (f ) – f Ba Ba r DJn;j,i,s (f ) Ba + f Ba r Consequently, considering n– DJn;j,i,s (f ) Ba ≤ C(d, j, i, s, r, q, μ) f – Jn;j,i,s (f ) inition of K(f ; t )Ba , and Theorem ., we have ω f ; n– Ba ≤ CK f ; n– Ba (.) Ba , by the def- Ba r ≤ C f – Jn;j,i,s (f ) Ba r + n– DJn;j,i,s (f ) ≤ C(d, j, i, s, r, q, μ) f – Jn;j,i,s (f ) Ba Ba (.) In view of (.), we get ω f ; n– Ba ≤ C(d, j, i, s, r, q, μ)n–α (.) Let (n + )– < t ≤ n– , we have ω(f ; t)Ba ≤ ω f ; n– ≤ C(d, j, i, s, r, q, μ) Ba n n+ –α (n + )–α ≤ C(d, j, i, s)(n + )–α ≤ C(d, j, i, s, r, q, μ)t α (.) The proof is completed Competing interests The authors declare that they have no competing interests Authors’ contributions All authors contributed equally to the writing of this paper All authors read and approved the final manuscript Author details Department of Mathematics, Taizhou University, Taizhou, Zhejiang 317000, China School of Science, China University of Mining and Technology, Beijing, 100083, China Feng and Feng Journal of Inequalities and Applications 2014, 2014:419 http://www.journalofinequalitiesandapplications.com/content/2014/1/419 Acknowledgements The authors would like to thank the anonymous referees for their valuable comments, remarks, and suggestions, which greatly helped us to improve the presentation of this paper and made it more readable Project was supported by the Natural Science Foundation of China (Grant No 10671019), the Zhejiang Provincial Natural Science Foundation (Grant No LY12A01008), and the Cultivation Fund of Taizhou University Received: 26 April 2014 Accepted: 10 October 2014 Published: 21 October 2014 References Ding, XX, Luo, PZ: Ba spaces and some estimates of Laplace operator J Syst Sci Math Sci 1, 9-33 (1981) Freedden, W, Gervens, T, Schreiner, M: Constructive Approximation on the Sphere Oxford University Press, New York (1998) Wang, KY, Li, LQ: Harmonic Analysis and Approximation on the Unit Sphere Science Press, Beijing (2000) Müller, C: Spherical Harmonics Lecture Notes in Mathematics, vol 17 Springer, Berlin (1971) Berens, H, Butzer, PL, Pawelke, S: Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten Publ Res Inst Math Sci., Ser A 4, 201-268 (1968) Pawelke, S: Über die Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen Tohoku Math J 24, 473-486 (1972) Matsuoka, Y: On the approximation of functions by some singular integrals Tohoku Math J 18, 13-43 (1966) Chen, WZ: Approximation Theory of Operators Xiamen University Publishing House, Xiamen (1989) (in Chinese) Ditzian, Z, Runovskii, K: Averages on caps of Sd–1 J Math Anal Appl 248, 260-274 (2000) 10 Belinsky, E, Dai, F, Ditzian, Z: Multivariate approximating averages J Approx Theory 125, 85-105 (2003) doi:10.1186/1029-242X-2014-419 Cite this article as: Feng and Feng: Direct and converse results in the Ba space for Jackson-Matsuoka polynomials on the unit sphere Journal of Inequalities and Applications 2014 2014:419 Page 13 of 13 ... 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. .. Cite this article as: Feng and Feng: Direct and converse results in the Ba space for Jackson- Matsuoka polynomials on the unit sphere Journal of Inequalities and Applications 2014 2014:419 Page 13... approximation of the Jackson- Matsuoka polynomial on the unit sphere in the Ba space Firstly, we introduce K -functionals, modulus of smooth- Feng and Feng Journal of Inequalities and Applications 2014,
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