Pointwise weighted approximation of functions with endpoint singularities by combinations of Bernstein operators
aa r X i v : . [ m a t h . F A ] A ug POINTWISE WEIGHTED APPROXIMATION OF FUNCTIONS WITHENDPOINT SINGULARITIES BY COMBINATIONS OF BERNSTEINOPERATORS
WEN-MING LU AND LIN ZHANG
Abstract.
We give direct and inverse theorems for the weighted approximation of functionswith endpoint singularities by combinations of Bernstein operators. Introduction
The set of all continuous functions, defined on the interval I , is denoted by C ( I ). For any f ∈ C ([0 , Bernstein operators are defined as follows: B n ( f, x ) := n X k =0 f ( kn ) p n,k ( x ) , where p n,k ( x ) := (cid:18) nk (cid:19) x k (1 − x ) n − k , k = 0 , , , . . . , n, x ∈ [0 , . Approximation properties of Bernstein operators have been studied very well (see [2], [3],[5]-[8], [12]-[14], for example). In order to approximate the functions with singularities, DellaVecchia et al. [3] and Yu-Zhao [12] introduced some kinds of modified Bernstein operators .Throughout the paper, C denotes a positive constant independent of n and x , which may bedifferent in different cases.Let w ( x ) = x α (1 − x ) β , α, β > , α + β > , x . and C w := { f ∈ C ((0 , x −→ ( wf )( x ) = lim x −→ ( wf )( x ) = 0 } . The norm in C w is defined by k wf k C w := k wf k = sup x | ( wf )( x ) | . Define W rw,λ := { f ∈ C w : f ( r − ∈ A.C. ((0 , , k wϕ rλ f ( r ) k < ∞} . For f ∈ C w , define the weighted modulus of smoothness by ω rϕ λ ( f, t ) w := sup Primary 41A10, Secondary 41A17. Key words and phrases. Combinations of Bernstein polynomials; Functions with endpoint singularities;Direct and inverse results. where ∆ rhϕ f ( x ) = r X k =0 ( − k (cid:18) rk (cid:19) f ( x + ( r − k ) hϕ ( x )) , −→ ∆ rh f ( x ) = r X k =0 ( − k (cid:18) rk (cid:19) f ( x + ( r − k ) h ) , ←− ∆ rh f ( x ) = r X k =0 ( − k (cid:18) rk (cid:19) f ( x − kh ) , and ϕ ( x ) = p x (1 − x ). Della Vecchia et al. firstly introduced B ∗ n ( f, x ) and ¯ B n ( f, x ) in [3],where the properties of B ∗ n ( f, x ) and ¯ B n ( f, x ) are studied. Among others, they prove that k w ( f − B ∗ n ( f )) k Cω ϕ ( f, n − / ) , f ∈ C w , k ¯ w ( f − ¯ B n ( f )) k Cn / √ n ] X k =1 k ω ϕ ( f, k ) ∗ ¯ w , f ∈ C ¯ w , where w ( x ) = x α (1 − x ) β , α, β > , α + β > , x . In [11], for any α, β > , n > r + α + β , there hold k wB ∗ n,r ( f ) k C k wf k , f ∈ C w , k w ( B ∗ n,r ( f ) − f ) k (cid:26) Cn r ( k wf k + k wϕ r f (2 r ) k ) , f ∈ W rw ,C ( ω rϕ ( f, n − / ) w + n − r k wf k ) , f ∈ C w . k wϕ r B ∗ (2 r ) n,r ( f ) k (cid:26) Cn r k wf k , f ∈ C w ,C ( k wf k + k wϕ r f (2 r ) k ) , f ∈ W rw . and for 0 < γ < r, k w ( B ∗ n,r ( f ) − f ) k = O ( n − γ/ ) ⇐⇒ ω rϕ ( f, t ) w = O ( t r ) . Ditzian and Totik [5] extended this method of combinations and defined the following combi-nations of Bernstein operators: B n,r ( f, x ) := r − X i =0 C i ( n ) B n i ( f, x ) . with the conditions(a) n = n < n < · · · < n r − Cn, (b) P r − i =0 | C i ( n ) | C, (c) P r − i =0 C i ( n ) = 1 , (d) P r − i =0 C i ( n ) n − ki = 0, for k = 1 , . . . , r − The main results Now, we can define our new combinations of Bernstein operators as follows: B ∗ n,r ( f, x ) := B n,r ( F n , x ) = r − X i =0 C i ( n ) B n i ( F n , x ) , (2.1)where C i ( n ) satisfy the conditions (a)-(d). For the details, it can be referred to [11]. Ourmain results are the following: OINTWISE WEIGHTED APPROXIMATION OF FUNCTIONS WITH ENDPOINT SINGULARITIES 3 Theorem 2.1. If α, β > , for any f ∈ C w , we have k wB ∗ ( r ) n,r − ( f ) k Cn r k wf k . (2.2) Theorem 2.2. For any α, β > , λ , we have | w ( x ) ϕ rλ ( x ) B ∗ ( r ) n,r − ( f, x ) | (cid:26) Cn r/ { max { n r (1 − λ ) / , ϕ r ( λ − ( x ) }}k wf k , f ∈ C w ,C k wϕ rλ f ( r ) k , f ∈ W rw,λ . (2.3) Theorem 2.3. For f ∈ C w , α, β > , α ∈ (0 , r ) , λ , we have w ( x ) | f ( x ) − B ∗ n,r − ( f, x ) | = O (( n − ϕ − λ ( x ) δ n ( x )) α ) ⇐⇒ ω rϕ λ ( f, t ) w = O ( t α ) . (2.4) 3. Lemmas Lemma 3.1. ( [13] ) For any non-negative real u and v , we have n − X k =1 ( kn ) − u (1 − kn ) − v p n,k ( x ) Cx − u (1 − x ) − v . (3.1) Lemma 3.2. ( [3] ) If γ ∈ R, then n X k =0 | k − nx | γ p n,k ( x ) Cn γ ϕ γ ( x ) . (3.2) Lemma 3.3. For any f ∈ W rw,λ , λ and α, β > , we have k wϕ rλ F ( r ) n k C k wϕ rλ f ( r ) k . (3.3) Proof. By symmetry, we only prove the above result when x ∈ (0 , / x ∈ (0 , /n ] , by [5], we have | L ( r ) r ( f, x ) | C |−→ ∆ r r f (0) | Cn − r +1 Z rn u r | f ( r ) ( u ) | du Cn − r +1 k wϕ rλ f ( r ) k Z rn u r w − ( u ) ϕ − rλ ( u ) du. So | w ( x ) ϕ rλ ( x ) F ( r ) n ( x ) | C k wϕ rλ f ( r ) k . If x ∈ [ n , n ] , we have | w ( x ) ϕ rλ ( x ) F ( r ) n ( x ) | | w ( x ) ϕ rλ ( x ) f ( r ) ( x ) | + | w ( x ) ϕ rλ ( x )( f ( x ) − F n ( x )) ( r ) | := I + I . For I , we have f ( x ) − F n ( x ) = ( ψ ( nx − 1) + 1)( f ( x ) − L r ( f, x )) .w ( x ) ϕ rλ ( x ) | ( f ( x ) − F n ( x )) ( r ) | = w ( x ) ϕ rλ ( x ) r X i =0 n i | ( f ( x ) − L r ( f, x )) ( r − i ) | . By [5], then | ( f ( x ) − L r ( f, x )) ( r − i ) | [ n , n ] C ( n r − i k f − L r k [ n , n ] + n − i k f ( r ) k [ n , n ] ) , < j < r. Now, we estimate I := w ( x ) ϕ rλ ( x ) | f ( x ) − L r ( x ) | . (3.4) WEN-MING LU AND LIN ZHANG By Taylor expansion, we have f ( in ) = r − X u =0 ( in − x ) u u ! f ( u ) ( x ) + 1( r − Z in x ( in − s ) r − f ( r ) ( s ) ds, (3.5)It follows from (3.5) and the identity r X i =1 ( in ) v l i ( x ) = Cx v , v = 0 , , · · · , r. We have L r ( f, x ) = r X i =1 r − X u =0 ( in − x ) u u ! f ( u ) ( x ) l i ( x ) + 1( r − r X i =1 l i ( x ) Z in x ( in − s ) r − f ( r ) ( s ) ds = f ( x ) + C r − X u =1 f ( u ) ( x )( u X v =0 C vu ( − x ) u − v r X i =1 ( in ) v l i ( x ))+ 1( r − r X i =1 l i ( x ) Z in x ( in − s ) r − f ( r ) ( s ) ds, which implies that w ( x ) ϕ rλ ( x ) | f ( x ) − L r ( f, x ) | = 1 r ! w ( x ) ϕ rλ ( x ) r X i =1 l i ( x ) Z in x ( in − s ) r − f ( r ) ( s ) ds, since | l i ( x ) | C for x ∈ [0 , n ] , i = 1 , , · · · , r . It follows from | in − s | r − w ( s ) | in − x | r − w ( x ) , s between in and x , then w ( x ) ϕ rλ ( x ) | f ( x ) − L r ( f, x ) | Cw ( x ) ϕ rλ ( x ) r X i =1 Z in x ( in − s ) r − | f ( r ) ( s ) | ds Cϕ rλ ( x ) k wϕ rλ f ( r ) k r X i =1 Z in x ( in − s ) r − ϕ − rλ ( s ) ds Cn r k wϕ rλ f ( r ) k . Thus I C k wϕ rλ f ( r ) k . So, we get I C k wϕ rλ f ( r ) k . Above all, we have | w ( x ) ϕ rλ ( x ) F ( r ) n ( x ) | C k wϕ rλ f ( r ) k . (cid:3) Lemma 3.4. If f ∈ W rw,λ , λ and α, β > , then | w ( x )( f ( x ) − L r ( f, x )) | [0 , n ] C ( δ n ( x ) √ nϕ λ ( x ) ) r k wϕ rλ f ( r ) k . (3.6) | w ( x )( f ( x ) − R r ( f, x )) | [1 − n , C ( δ n ( x ) √ nϕ λ ( x ) ) r k wϕ rλ f ( r ) k . (3.7) OINTWISE WEIGHTED APPROXIMATION OF FUNCTIONS WITH ENDPOINT SINGULARITIES 5 Proof. By Taylor expansion, we have f ( in ) = r − X u =0 ( in − x ) u u ! f ( u ) ( x ) + 1 r ! Z in x ( in − s ) r − f ( r ) ( s ) ds, (3.8)It follows from (3.8) and the identity r − X i =1 ( in ) v l i ( x ) = Cx v , v = 0 , , . . . , r. We have L r ( f, x ) = r X i =1 r − X u =0 ( in − x ) u u ! f ( u ) ( x ) l i ( x ) + 1( r − r X i =1 l i ( x ) Z in x ( in − s ) r − f ( r ) ( s ) ds = f ( x ) + C r − X u =1 f ( u ) ( x )( u X v =0 C vu ( − x ) u − v r X i =1 ( in ) v l i ( x ))+ 1( r − r X i =1 l i ( x ) Z in x ( in − s ) r − f ( r ) ( s ) ds, which implies that w ( x ) | f ( x ) − L r ( f, x ) | = 1( r − w ( x ) r X i =1 l i ( x ) Z in x ( in − s ) r − f ( r ) ( s ) ds, since | l i ( x ) | C for x ∈ [0 , n ] , i = 1 , , · · · , r .It follows from | in − s | r − w ( s ) | in − x | r − w ( x ) , s between in and x , then w ( x ) | f ( x ) − L r ( f, x ) | Cw ( x ) r X i =1 Z in x ( in − s ) r − | f ( r ) ( s ) | ds C ϕ r ( x ) ϕ rλ ( x ) k wϕ rλ f ( r ) k r X i =1 Z in x ( in − s ) r − ϕ − r ( s ) ds C δ rn ( x ) ϕ rλ ( x ) k wϕ rλ f ( r ) k r X i =1 Z in x ( in − s ) r − ϕ − r ( s ) ds C ( δ n ( x ) √ nϕ λ ( x ) ) r k wϕ rλ f ( r ) k . The proof of (3.7) can be done similarly. (cid:3) Lemma 3.5. ( [11] ) For every α, β > , we have k wB ∗ n,r − ( f ) k C k wf k . (3.9) Lemma 3.6. ( [15] ) If ϕ ( x ) = p x (1 − x ) , λ , β , then Z hϕλ ( x )2 − hϕλ ( x )2 · · · Z hϕλ ( x )2 − hϕλ ( x )2 ϕ − rβ ( x + r X k =1 u k ) du · · · du r Ch r ϕ r ( λ − β ) ( x ) . (3.10) WEN-MING LU AND LIN ZHANG Proof of Theorems Proof of Theorem 2.1. By symmetry, in what follows we will always assume that x ∈ (0 , ] . It is sufficient to prove the validity for B n,r − ( F n , x ) instead of B ∗ n,r − ( f, x ) . When x ∈ (0 , n ) , we have | w ( x ) B ∗ ( r ) n,r − ( f, x ) | w ( x ) r − X i =0 n i !( n i − r )! n i − r X k =0 |−→ ∆ r ni F n ( kn i ) | p n i − r,k ( x ) Cw ( x ) r − X i =0 n ri n i − r X k =0 |−→ ∆ r ni F n ( kn i ) | p n i − r,k ( x ) Cw ( x ) r − X i =0 n ri n i − r X k =0 r X j =0 C jr | F n ( k + r − jn i ) | p n i − r,k ( x ) Cw ( x ) r − X i =0 n ri r X j =0 C jr | F n ( r − jn i ) | p n i − r, ( x )+ Cw ( x ) r − X i =0 n ri r X j =0 C jr | F n ( n i − jn i ) | p n i − r,n i − r ( x )+ Cw ( x ) r − X i =0 n ri n i − r − X k =1 r X j =0 C jr | F n ( k + r − jn i ) | p n i − r,k ( x ):= H + H + H . We have H Cw ( x ) k wf k r − X i =0 n ri w − ( 1 n i ) p n i − r, ( x ) C k wf k r − X i =0 n ri ( n i x ) α (1 − x ) n i − r Cn r k wf k . When 1 k n i − r − , we have 1 k + 2 r − j n i − , and thus w ( kn i − r ) w ( k + r − jn i ) = ( n i n i − r ) α + β ( kk + r − j ) α ( n i − r − kn i − r − k + j ) β C. Thereof, by (3.1), we have H Cw ( x ) k wF n k r − X i =0 n ri n i − r − X k =1 r X j =0 w ( k + r − jn i ) p n i − r,k ( x ) Cw ( x ) k wF n k r − X i =0 n ri n i − r − X k =1 w ( kn i − r ) p n i − r,k ( x ) Cn r k wF n k Cn r k wf k . Similarly, we can get H Cn r k wf k . So | w ( x ) B ∗ ( r ) n,r − ( f, x ) | Cn r k wf k , x ∈ (0 , n ) . (4.1) OINTWISE WEIGHTED APPROXIMATION OF FUNCTIONS WITH ENDPOINT SINGULARITIES 7 When x ∈ [ n , ] , according to [5], we have | w ( x ) B ∗ ( r ) n,r − ( f, x ) | = | w ( x ) B ( r ) n,r − ( F n , x ) | w ( x )( ϕ ( x )) − r r − X i =0 r X j =0 | Q j ( x, n i ) | n ji n X k =0 | ( x − kn i ) j F n ( kn i ) | p n i ,k ( x ) . Then Q j ( x, n i ) = ( n i x (1 − x )) [ r − j ] , and ( ϕ ( x )) − r Q j ( x, n i ) n ji C ( n i /ϕ ( x )) r + j , we have | w ( x ) B ∗ ( r ) n,r − ( f, x ) | Cw ( x ) r − X i =0 r X j =0 ( n i ϕ ( x ) ) r + j n i X k =0 | ( x − kn i ) j F n ( kn i ) | p n i ,k ( x ) C k wF n k w ( x ) r − X i =0 r X j =0 ( n i ϕ ( x ) ) r + j n i X k =0 | x − kn i | j w ( k ∗ n i ) p n i ,k ( x ) , (4.2)where k ∗ = 1 for k = 0 , k ∗ = n i − k = n i and k ∗ = k for 1 < k < n i . Note that w ( x ) p n i , ( x ) w ( n i ) C ( n i x ) α (1 − x ) n i C, and w ( x ) p n i ,n i ( x ) w (1 − n i ) Cn βi x n i C n βi n i C. By (3.1), we have n i X k =0 w ( k ∗ n i ) p n i ,k ( x ) Cw − ( x ) . (4.3)Now, applying Cauchy’s inequality, by (3.2) and (4.3), we have n i X k =0 | x − kn i | j w ( k ∗ n i ) p n i ,k ( x ) ( n i X k =0 | x − kn i | j p n i ,k ( x )) / ( n i X k =0 w ( k ∗ n i ) p n i ,k ( x )) / Cn − j/ i ϕ j ( x ) w − ( x ) . Substituting this to (4.2), we have | w ( x ) B ∗ ( r ) n,r − ( f, x ) | Cn r k wf k , x ∈ [ 1 n , 12 ] . (4.4)We get Theorem 2.1 by (4.1) and (4.4). (cid:3) Proof of Theorem 2.2. (1) When f ∈ C w , we proceed it as follows: Case 1. If 0 ϕ ( x ) √ n , by (2.2), we have | w ( x ) ϕ rλ ( x ) B ∗ ( r ) n,r − ( f, x ) | Cn − rλ/ | w ( x ) B ∗ ( r ) n,r − ( f, x ) | Cn r (1 − λ/ k wf k . (4.5) WEN-MING LU AND LIN ZHANG Case 2. If ϕ ( x ) > √ n , we have | B ∗ ( r ) n,r − ( f, x ) | = | B ( r ) n,r − ( F n , x ) | ( ϕ ( x )) − r r − X i =0 r X j =0 | Q j ( x, n i ) C i ( n ) | n ji n i X k =0 | ( x − kn i ) j F n ( kn i ) | p n i ,k ( x ) ,Q j ( x, n i ) = ( n i x (1 − x )) [ r − j ] , and ( ϕ ( x )) − r Q j ( x, n i ) n ji C ( n i /ϕ ( x )) r + j .So | w ( x ) ϕ rλ ( x ) B ∗ ( r ) n,r − ( f, x ) | Cw ( x ) ϕ rλ ( x ) r − X i =0 r X j =0 ( n i ϕ ( x ) ) r + j n i X k =0 | ( x − kn i ) j F n ( kn i ) | p n i ,k ( x ) Cn r ϕ r ( λ − ( x ) . (4.6)It follows from combining with (4.5) and (4.6) that the first inequality is proved.(2) When f ∈ W rw,λ , we have B ( r ) n,r − ( F n , x ) = r − X i =0 C i ( n ) n ri n i − r X k =0 −→ ∆ r ni F n ( kn i ) p n i − r,k ( x ) . (4.7)If 0 < k < n i − r, we have |−→ ∆ r ni F n ( kn i ) | Cn − r +1 i Z rni | F ( r ) n ( kn i + u ) | du, (4.8)If k = 0 , we have |−→ ∆ r ni F n (0) | C Z rni u r − | F ( r ) n ( u ) | du, (4.9)Similarly |−→ ∆ r ni F n ( n i − rn i ) | Cn − r +1 i Z − rni (1 − u ) r | F ( r ) n ( u ) | du. (4.10)By (4.7)-(4.10), we have | w ( x ) ϕ rλ ( x ) B ∗ ( r ) n,r − ( f, x ) | Cw ( x ) ϕ rλ ( x ) k wϕ rλ F ( r ) n k r − X i =0 n i − r X k =0 ( wϕ rλ ) − ( k ∗ n i ) p n i − r,k ( x ) , (4.11)where k ∗ = 1 for k = 0 , k ∗ = n i − r − k = n i − r and k ∗ = k for 1 < k < n i − r. By (3.1),we have n i − r X k =0 ( wϕ rλ ) − ( k ∗ n i ) p n i − r,k ( x ) C ( wϕ rλ ) − ( x ) . (4.12)which combining with (4.12) give | w ( x ) ϕ rλ ( x ) B ∗ ( r ) n,r − ( f, x ) | C k wϕ rλ f ( r ) k . (cid:3) So we get the second inequality of the Theorem 2.2.4.3. Proof of Theorem 2.3. OINTWISE WEIGHTED APPROXIMATION OF FUNCTIONS WITH ENDPOINT SINGULARITIES 9 The direct theorem. We know F n ( t ) = F n ( x ) + F ′ n ( t )( t − x ) + · · · + 1( r − Z tx ( t − u ) r − F ( r ) n ( u ) du, (4.13) B n,r − (( · − x ) k , x ) = 0 , k = 1 , , · · · , r − . (4.14)According to the definition of W rw,λ , for any g ∈ W rw,λ , we have B ∗ n,r − ( g, x ) = B n,r − ( G n ( g ) , x ) , and w ( x ) | G n ( x ) − B n,r − ( G n , x ) | = w ( x ) | B n,r − ( R r ( G n , t, x ) , x ) | , thereof R r ( G n , t, x ) = R tx ( t − u ) r − G ( r ) n ( u ) du. It follows from | t − u | r − w ( u ) | t − x | r − w ( x ) , u between t and x , we have w ( x ) | G n ( x ) − B n,r − ( G n , x ) | C k wϕ rλ G ( r ) n k w ( x ) B n,r − ( Z tx | t − u | r − w ( u ) ϕ rλ ( u ) du, x ) C k wϕ rλ G ( r ) n k w ( x )( B n,r − ( Z tx | t − u | r − ϕ rλ ( u ) du, x )) · ( B n,r − ( Z tx | t − u | r − w ( u ) du, x )) . (4.15)also Z tx | t − u | r − ϕ rλ ( u ) du C | t − x | r ϕ rλ ( x ) , Z tx | t − u | r − w ( u ) du | t − x | r w ( x ) . (4.16)By (3.2), (4.15) and (4.16), we have w ( x ) | G n ( x ) − B n,r − ( G n , x ) | C k wϕ rλ G ( r ) n k ϕ − rλ ( x ) B n,r − ( | t − x | r , x ) Cn − r ϕ r ( x ) ϕ rλ ( x ) k wϕ rλ G ( r ) n k Cn − r δ rn ( x ) ϕ rλ ( x ) k wϕ rλ G ( r ) n k = C ( δ n ( x ) √ nϕ λ ( x ) ) r k wϕ rλ G ( r ) n k . (4.17)By (3.3), (3.6), (3.7) and (4.17), when g ∈ W rw,λ , then w ( x ) | g ( x ) − B ∗ n,r − ( g, x ) | w ( x ) | g ( x ) − G n ( g, x ) | + w ( x ) | G n ( g, x ) − B ∗ n,r − ( g, x ) | | w ( x )( g ( x ) − L r ( g, x )) | [0 , n ] + | w ( x )( g ( x ) − R r ( g, x )) | [1 − n , + C ( δ n ( x ) √ nϕ λ ( x ) ) r k wϕ rλ G ( r ) n k C ( δ n ( x ) √ nϕ λ ( x ) ) r k wϕ rλ g ( r ) k . (4.18)For f ∈ C w , we choose proper g ∈ W rw,λ , by (3.9) and (4.18), then w ( x ) | f ( x ) − B ∗ n,r − ( f, x ) | w ( x ) | f ( x ) − g ( x ) | + w ( x ) | B ∗ n,r − ( f − g, x ) | + w ( x ) | g ( x ) − B ∗ n,r − ( g, x ) | C ( k w ( f − g ) k + ( δ n ( x ) √ nϕ λ ( x ) ) r k wϕ rλ g ( r ) k ) Cω rϕ λ ( f, δ n ( x ) √ nϕ λ ( x ) ) w . (cid:3) The inverse theorem. We define the weighted main-part modulus for D = R + byΩ rϕ λ ( C, f, t ) w = sup Let δ > , by (4.20), we choose proper g so that k w ( f − g ) k C Ω rϕ λ ( f, δ ) w , k wϕ rλ g ( r ) k Cδ − r Ω rϕ λ ( f, δ ) w . (4.21)then | w ( x )∆ rhϕ λ f ( x ) | | w ( x )∆ rhϕ λ ( f ( x ) − B ∗ n,r − ( f, x )) | + | w ( x )∆ rhϕ λ B ∗ n,r − ( f − g, x ) | + | w ( x )∆ rhϕ λ B ∗ n,r − ( g, x ) | r X j =0 C jr ( n − δ n ( x + ( r − j ) hϕ λ ( x )) ϕ λ ( x + ( r − j ) hϕ λ ( x )) ) α + Z hϕλ ( x )2 − hϕλ ( x )2 · · · Z hϕλ ( x )2 − hϕλ ( x )2 w ( x ) B ∗ ( r ) n,r − ( f − g, x + r X k =1 u k ) du · · · du r + Z hϕλ ( x )2 − hϕλ ( x )2 · · · Z hϕλ ( x )2 − hϕλ ( x )2 w ( x ) B ∗ ( r ) n,r − ( g, x + r X k =1 u k ) du · · · du r := J + J + J . (4.22)Obviously J C ( n − ϕ − λ ( x ) δ n ( x )) α . (4.23)By (2.2) and (4.21), we have J Cn r k w ( f − g ) k Z hϕλ ( x )2 − hϕλ ( x )2 · · · Z hϕλ ( x )2 − hϕλ ( x )2 du · · · du r Cn r h r ϕ rλ ( x ) k w ( f − g ) k Cn r h r ϕ rλ ( x )Ω rϕ λ ( f, δ ) w . (4.24) OINTWISE WEIGHTED APPROXIMATION OF FUNCTIONS WITH ENDPOINT SINGULARITIES 11 By the first inequality of (2.3), we let λ = 1 , and (3.10) as well as (4.21), we have J Cn r k w ( f − g ) k Z hϕλ ( x )2 − hϕλ ( x )2 · · · Z hϕλ ( x )2 − hϕλ ( x )2 ϕ − r ( x + r X k =1 u k ) du · · · du r Cn r h r ϕ r ( λ − ( x ) k w ( f − g ) k Cn r h r ϕ r ( λ − ( x )Ω rϕ λ ( f, δ ) w . (4.25)By the second inequality of (2.3) and (4.21), we have J C k wϕ rλ g ( r ) k w ( x ) Z hϕλ ( x )2 − hϕλ ( x )2 · · · Z hϕλ ( x )2 − hϕλ ( x )2 w − ( x + r X k =1 u k ) ϕ − rλ ( x + r X k =1 u k ) du · · · du r Ch r k wϕ rλ g ( r ) k Ch r δ − r Ω rϕ λ ( f, δ ) w . (4.26)Now, by (4.22)-(4.26), we get | w ( x )∆ rhϕ λ f ( x ) | C { ( n − δ n ( x )) α + h r ( n − δ n ( x )) − r Ω rϕ λ ( f, δ ) w + h r δ − r Ω rϕ λ ( f, δ ) w } . When n > , we have n − δ n ( x ) < ( n − − δ n − ( x ) √ n − δ n ( x ) , Choosing proper x, n ∈ N, so that n − δ n ( x ) δ < ( n − − δ n − ( x ) , Therefore | w ( x )∆ rhϕ λ f ( x ) | C { δ α + h r δ − r Ω rϕ λ ( f, δ ) w } . By Borens-Lorentz lemma in [5], we getΩ rϕ λ ( f, t ) w Ct α . (4.27)So, by (4.27), we get ω rϕ λ ( f, t ) w C Z t Ω rϕ λ ( f, τ ) w τ dτ = C Z t τ α − dτ = Ct α . (cid:3) References [1] P.L. Butzer, Linear combinations of Bernstein polynomials, Canad. J. Math. 5 (1953), pp. 559-567.[2] H. Berens and G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J. 21 (1972),pp. 693-708.[3] D. Della Vechhia, G. Mastroianni and J. Szabados, Weighted approximation of functions with endpointand inner singularities by Bernstein operators, Acta Math. Hungar. 103 (2004), pp. 19-41.[4] Z. Ditzian, A global inverse theorem for combinations of Bernstein polynomials, J. Approx. Theory 26(1979), pp. 277-292.[5] Z. Ditzian and V. 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Math. 30 (2006), pp. 1178-1189.[13] D.X. Zhou, Rate of convergence for Bernstein operators with Jacobi weights, Acta Math. Sinica 35 (1992),pp. 331-338.[14] D.X. Zhou, On smoothness characterized by Bernstein type operators, J. Approx. Theory 81 (1994), pp.303-315.[15] J.J. Zhang, Z.B. Xu, Direct and inverse approximation theorems with Jacobi weight for combinations andhiger derivatives of Baskakov operators(in Chinese), Journal of systems science and mathematical sciences.2008 28 (1), pp. 30-39. Wen-ming LuSchool of Science, Hangzhou Dianzi University, Hangzhou, People’s Republic of China E-mail address : lu [email protected] Lin ZhangDepartment of Mathematics, Zhejiang University, Hangzhou, People’s Republic of China E-mail address ::