On Approximation Methods by using Orthogonal ... - CiteSeerX

On Approximation Methods by using Orthogonal Polynomial Expansions Rupert Lasser, Detlef H. Mache and Josef Obermaier

1 Introduction and Basic Facts The following investigations start from a general point of view. Therefore let (Pn )n2N0 be an orthogonal polynomial sequence (OPS) on the real line with respect to a probability measure  with compact support S and card(S ) = 1. The polynomials Pn are assumed to be real valued with deg(Pn ) = n. Then the sequence (Pn )n2N0 satis es a three term recurrence relation of the following type

P1 (x)Pn (x) = an Pn+1 (x) + bnPn (x) + cn Pn?1 (x); n  1;

(1)

with P0 (x) = q0 and P1 (x) = q0 (x ? b0 )=a0 , where the coecients are real numbers with c1 q0 > 0, cn an?1 > 0; n > 1, and (cn an?1 )n2N , (bn )n2N are bounded sequences. On the contrary, if we de ne (Pn )n2N0 by (1) we get an OPS with the assumed properties, see [3]. With Z (2) h(k) = ( Pk2 (x) d(x))?1 = (< Pk ; Pk >)?1 S

the corresponding orthonormal polynomials are given by pn(x) = By Christoel-Darboux formula, see [3], we have n X k=0

ph(n)P (x). n

Pn (x)Pn+1 (y) ; n  1: (3) Pk (x)Pk (y)h(k) = aq 0 an h(n) Pn+1 (x)Pn (yx) ? ?y 0

For f 2 L1 (S; ) one may form orthogonal expansion with respect to the OPS by 1 X f  f^(k)Pk h(k); (4) k=0

1

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R. Lasser, D.H. Mache & J. Obermaier

where the Fourier coecients are de ned by Z f^(k) = f (x)Pk (x) d(x) =< f; Pk > : S

(5)

In this paper we will focus on weighted expansions n X k=0

an;k f^(k)Pk h(k); n ! 1;

(6)

and study convergence properties in various norms, e.g. for 1  p < 1 the Lp -norm when f 2 Lp (S; )  L1 (S; ) or for p = 1 the sup-norm when f 2 C (S ). For essential parts of our investigations we make the additional assumption that there exists a point x0 2 S such that jPn (x)j  Pn (x0 ) = 1 for all x 2 S; n 2 N0 : (7) Property (7) implies that the coecients in (1) ful ll q0 = 1, a0 + b0 = x0 and an + bn + cn = 1; n > 0. If the linearization coecients g in

Pi Pj =

i+j X

k=ji?j j

g(i; j; k)Pk

(8)

are non-negative, then there exists a normalized version of (Pn )n2N0 with property (7). Those polynomials are associated with a so-called hypergroup structure on N0 and there exist a lot of examples which are well studied, see [6] and [1]. Further on we denote an OPS with property (7) by (Rn )n2N0 .

2 Approximate Identities

Denote by B one of the Banach spaces C(S) or Lp (S; ); 1  p < 1, with respect to the orthogonalization measure  and by k
kB the actual norm. Let (an;k )0n 0 with kAn f kB  C kf kB for all f 2 B and for all n 2 N0 . In [9] it is also shown that (An )n2N0 is an approximate identity with respect to L1 (S; ) if and only if it is an approximate identity with respect to C (S ). Moreover, if (An )n2N0 is an approximate identity with respect to C (S ) then also with respect to Lp (S; ); 1 < p < 1. Of course, the opposite direction is not true. For this reason one may focus on approximate identities with respect to C (S ). Many classical approximation processes concerning trigonometric polynomials are performed by a sequence of convolution operators, see [2] and [7]. In some special cases of algebraic polynomial systems, there also does exist a proper convolution structure on C (S ). De nition 2.3 If for all x; y 2 S there exists a complex Borel measure x;y with kx;y k  M; where M > 0 is independent of x and y, such that

Pn (x)Pn (y) =

Z

S

Pn (z ) dx;y (z ) for all n 2 N0 ;

(12)

then we say that for the OPS (Pn )n2N0 a product formula holds.

by

If a product formula holds, then we are able to de ne a convolution on C (S )

'  (y) =

Z Z S S

'(x) (z ) dx;y (z ) d(x) =

see [10], where '  2 C (S ) and

Z Z S S

'(z ) (x) dx;y (z ) d(x); (13)

k'  k1  M k'k1 k k1:

(14)

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R. Lasser, D.H. Mache & J. Obermaier

Then the operator An acts as a convolution operator, that is

An f =

n X k=0

an;k f^(k)Pk h(k) = An ? f:

(15)

Now, by (14), the uniform boundedness of kAn k1 implies (ii) of Theorem 2.2. So one may derive great bene t from the existence of a convolution structure.

3 Positive Approximate Identities The situation becomes more handsome, if we assume the operators An to be positive. For the remainder of this section we suppose (Rn )n2N0 to be an OPS with property (7). De nition 3.1 An operator G from C (S ) into C (S ) is called positive, if f  0 implies Gf  0. In case of positive operators there is a simpli cation of Theorem 2.2. Theorem 3.2 Let (Rn )n2N0 be an OPS with property (7) and (An)n2N0 be a sequence of positive operators. Then (An )n2N0 is an approximate identity with respect to B if and only if the following two conditions hold. (i) limn!1 an;0 = limn!1 an;1 = 1. (ii) There exists a constant C > 0 with kAn f kB  C kf kB for all f 2 B and for all n 2 N0 . Proof. De ne by Dk (x) = Pki=0 Ri (x)h(i) the so-called Dirichlet kernel. By Christoel-Darboux formula (3) and (7) we derive (1 ? R1 (x))Dk (x) = ak h(k)(Rk (x) ? Rk+1 (x)); k  1: Let x 2 S . By (7) again it holds jDk (x)j  Dk (x0 ) and therefore ?(1 ? R1 (x)) jDa kj(hx(0k))  Rk (x) ? Rk+1 (x)  (1 ? R1 (x)) jDa kj(hx(0k)) : k k In case n  m it is simple to deduce that An Rm = an;m Rm . Hence, if n > k, then the stated positivity of the operators implies ?(an;0 ? an;1 R1 (x)) jDa kj(hx(0k))  an;k Rk (x) ? an;k+1 Rk+1 (x) k  (an;0 ? an;1 R1 (x)) jDa kj(hx(0k)) : k

Approximation Methods by using Orthogonal Polynomial Expansions

5

With x = x0 we get

jan;k ? an;k+1 j  jDa kj(hx(0k)) (an;0 ? an;1 ) for all n > k  1: k

Now condition (i) yields limn!1 jan;k ? an;k+1 j = 0. By induction we get limn!1 an;k = 1 for all k 2 N0 and according to Theorem 2.2 the proof is complete. 2 In case of a product formula with corresponding positive measures we may achieve positive operators by a simple procedure.

Theorem 3.3 Let (Rn )n2N0 be an OPS with property (7) and suppose that a product formula holds, where x;y is a positive measure for all x; y 2 S . If the generating polynomials An are non-negative, i.e An (x)  0 for all x 2 S , then (An )n2N0 is a sequence of positive operators. Additionally, if limn!1 an;0 = limn!1 an;1 = 1; then (An )n2N0 is an approximate identity with respect to any B .

Proof. Since ARn(Rz)  0 on S and x;y  0, the positivity of the operators is shown by An f (y) = S S An (z )f (x) x;y (z ) (x). Moreover, we have kAn k1 = an;0 and therefore limn!1 an;0 = 1 implies the uniform boundedness of kAn k1. Now, by (14) and Theorem 3.2, (An )n2N0 is an approximate identity with respect to C (S ) and according to the remark after Theorem 2.2 with respect to any B . 2

4 Local Convergence Behaviour of An At rst we give a local error estimate for the approximation of functions f 2 B = Lp (S ) or C (S ), S = [0; 1], by An f de ned in (10). In the following we assume that An is a positive linear operator with an;0 = 1, i.e. that the operator An preserves constant functions (An R0 = 1).

Let us note that the estimates in the following theorem will show that the order of approximation depends on the rst weight-coecient an;1 . To do this, recall that the Lipschitz type maximal function of order  introduced by B. Lenze [11] is de ned as

f(x) = sup jf (jxx)??tfj(t)j ; t6=x;t2S Than we have

x 2 S;  2]0; 1] :

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R. Lasser, D.H. Mache & J. Obermaier

Theorem 4.1 There exists a constant C > 0 such that for each bounded function f 2 B and for all x 2 S



 2

_ ? an;1 ) j f (x) ? (An f ) (x) j  C f (x) maxfjg1(x)j; jg2 (x)jg(1

;

for  2]0; 1] by using the functions

g1(x) := a20 c1 ? (b0 ? x)(a0 b1 +(b0 ? x))

g2 (x) := (b0 ? x)(a0 b1 +2(b0 ? x)): (16)

Proof. For the sequence (Rn)n2N0 we have the recurrence relation (1) with property (7)

R1 (x)Rn (x) = an Rn+1 (x) + bn Rn (x) + cn Rn?1 (x); n  1; with R0 (x) = 1 and R1 (x) = (x ? b0 )=a0 . By using the eigenstructure of An , i.e. An Rk = an;k Rk and the known inequality

jf (t) ? f (x)j  jt ? xj f(x); x; t 2 S; for 0 <   1, one obtains with An ((t ? x)2 ; x) = (1 ? an;2 )g1 (x) + (1 ? an;1 )g2 (x)

and the Holder's inequality

j f (x) ? (An f ) (x) j  f (x) An (jt ? xj ; x)  f (x) An ((t ? x)2 ; x) 2



 2

_ ? an;1)  C f(x) maxfjg1 (x)j; jg2 (x)jg(1

;

which concludes the proof. 2 At this point we mention that the constant C denotes a positive constant which can be dierent at each occurrence. Now if f 2 B := C (S ), for every x 2 S; the k?th dierence
kh f (x) ?of
f with P k f the step h 2 R; h 6= 0 at the point x is given by
h (x) := m=0 (?1)m+k mk f (x + mh), provided that the arguments x + kh 2 S . For the sake of brevity one sets
h f (x) :=
1h f (x) = f (x + h) ? f (x). Now, if f : S ! R is a bounded real function and if  > 0, the k-th modulus of continuity !k (f ; ) of f is de ned by

!k (f ; ) :=

sup

jhj; x;x+h2S

j
kh f (x)j;

where for k = 1 we have the well-known modulus of continuity !(f ) := !1 (f ; ). Now one can formulate the following

Approximation Methods by using Orthogonal Polynomial Expansions

Theorem 4.2 (Local Direct Results) For f 2 B := C (S ); x 2 S , we have

p

j f (x) ? (An f ) (x) j  C!(f ; 1 ? an;1 ) and further in addition with the second modulus

?

(17)

p

j f (x) ? (An f ) (x) j  C !2 (f ; 1 ? an;1 )

p

7

p

(18)




+jb0 ? xj 1 ? an;1 !(f ; 1 ? an;1 ) ; where C is a positive constant.

Proof. Following the known arguments in [13] and [16] we have that the method An de ned as

(An f )(x) =

n X k=0

an;k f^(k)Rk (x)h(k) =

n X k=0

an;k Rk (x) k

k

satis es the inequality

j f (x) ? (An f ) (x) j  2 !(f ; An (jx ? tj; x)); which proves (17). The second inequality (18) follows by the estimates used in [13], i.e. for h 2 (0; 2] and x 2 S

j f (x) ? (An f ) (x) j 



p





Therefore, with h = 1 ? an;1



3 + h12 An ((t ? x)2 ; x) !2 (f ; h) + 2 jA (x ? t; x)j!(f ; h)

h n



3 + h12 maxfg1(x); g2 (x)g(1 ? an;1) !2 (f ; h) + 2 jb ? xj(1 ? a ) !(f ; h) :

h

n;1

0

p

j f (x) ? (An f ) (x) j  (3 + C )!2 (f ; 1 ? an;1 ) p p +2jb0 ? xj 1 ? an;1 !(f ; 1 ? an;1 ); which proves (18). 2 With the rst part of the above theorem we can prove the following equivalence result

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R. Lasser, D.H. Mache & J. Obermaier

Theorem 4.3 (Local Characterization Result) Let An , be a positive linear operator de ned as in (10). Under the assumption that 1 ? an;1 = O(n?2 ) we have for for f 2 B (S ) and 0 <  < 1 ?
j f (x) ? (An f )(x) j  C n1  (n 2 N) ()

!(f ; t)

=

O(t )

(t ! 0) :

Some approximation operators in a similar form like in (10) were investigated in [13] and [14].

5 Examples of Kernels Having in mind Theorem 3.3 let us now derive some important kernels, which are associated with a sequence of positive operators and investigate there convergence behaviour. We also lay stress on the relationship to the corresponding trigonometric kernels. The most outstanding examples of OPS are the Jacobi polynomials (Jn(; ) )n2N0 , ; > ?1, which are orthogonal with respect to (1 ? x) (1+ x) dx and normalized by Jn(; ) (1) = 1. They exactly t the conditions of Theorem 3.3, if   and either  +  0 or  ? 21 , see [4] and [6]. For Jacobi polynomials (Jn(; ) )n2N0 (; )  , a(n; ) = (+ +2)(n++ +1)(n++1) , we have a(0; ) = 2(+ +1) =  +? +2 +2 , b0 (+1)(2n++ +1)(2n++ ) (; ) (; ) (  + )(  + +2) n(n+  ? bn = 2(+1) 1 ? (2n++ )(2n++ +2) , cn = (+1)(2(n++ +2) + )(2n++ +1) und +1)?( +1)?(n++1)?(n++ +1) h(; )(n) = (2n+?(+ +1)?( + +2)?(n+1)?(n+ +1) , n  1. Another example are the generalized Chebyshev polynomials (Cn(; ) )n2N0 , ; > ?1, which are orthogonal with respect to (1 ? x2 ) jxj2 +1 dx and normalized by Cn(; ) (1) = 1. If (  and  + > ?1) or  ? 21 , then they are tting our conditions, too, see [5] and [6]. 5.1 Fejer kernel In the trigonometric case the Fejer kernel (Fn )n2N0 is de ned as (C,1)-series of the Dirichlet kernel (Dn )n2N0 by n n X X kj )eikt ; t 2 [0; 2[; (19) (1 ? nj+ Fn (t) = n +1 1 Dk (t) = 1 k=0 k=?n

Approximation Methods by using Orthogonal Polynomial Expansions

9

see [2, Sec. 1.2.2]. Moreover, in the even case n = 2p we have the representation D2 (t) (20) F2p (t) = D p(0) : p Following the even trigonometric case we de ne a general Fejer kernel (F2p )p2N0 for OPS (Rn )n2N0 by 2p D2 (x) X F2p (x) = p = '2p;k Rk (x)h(k); x 2 S; 2p;0 k=0

(21)

P

where Dp (x) = pk=0PRk (x)h(k), see [8] and [9]. The coecients 2p;k are uniquely de ned by Dp2 (x) = 2kp=0 2p;k Rk (x)h(k). More explicitly we get

2p;k =

p p X X

j =0 i=jk?j j

g(k; j; i)h(j ):

(22)

Hence, the Fejer weights are determined by

P P

p p  j =0 i=jk?j j g (k; j; i)h(j ) 2p;k Pp h(j ) '2p;k =  = : 2p;0 j =0 Obviously, our de nition yields F2p (x)  0 and kF2p k1 = '2p;0 = 1. In case p  1 we derive '2p;1 = 1 ? Papp h(hp()j ) : j =0

(23)

(24)

In [8] we have shown that limn!1 cn =an?1 = 1 implies limn!1 '2p;1 = 1. For Jacobi polynomial systems (Jn(; ))n2N0 we get p +  + + 2)?( + 1)?(p +  + 2) (+1; ) (x))2 (25) F2(p; ) (x) = ?(?(  + + 2)?(p + + 1)?( + 2)?(p + 1) (Jp and '2p;1 = 1 ? 2p ++ ++ 2+ 2 : (26)

Thus 1 ? '2p;1 = O(p?1 ): Especially for Chebyshev polynomials of the rst kind ( = = ?1=2) our Fejer kernel coincides with the trigonometric one and for Chebyshev polynomials of the second kind ( = = 1=2) the weights are given by

8 q(1+q)(6p2 +18p+13?2q?2q2 ) > < 1 ? (1+2q)(1+p)(2+p)(3+2p) '2p;k = > : 1 ? (1+q)(3p2 +6+9p?2q?q2 ) (1+p)(2+p)(3+2p)

if k = 2q; if k = 2q + 1:

(27)

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R. Lasser, D.H. Mache & J. Obermaier

5.2 Fejer-Korovkin kernel In the trigonometric case the Fejer-Korovkin kernel (FKn )n2N0 is de ned by





2 2 n + 2)t=2) FKn (t) = 2 sin (n=+(n2 + 2)) cos cos(( t ? cos(=(n + 2)) ; t 2 [0; 2[; (28) see [2, Sec. 1.6.1]. By substitution x = cos t we get sin2 (=(n + 2)) 1 + Tn+2 (x) (29) n+2 (x ? cos(=(n + 2)))2 ; where Tn(x) = cos(n arccos x) are the Chebyshev polynomials of the rst kind. Whereas in the even case n = 2p we achieve  2 Tp+1 (x) sin2 (=(2(p + 1))) (30) p+1 x ? cos(=(2(p + 1))) : Following the even case we de ne a general Fejer-Korovkin kernel (F2p )p2N0 for OPS (Rn )n2N0 by

FK2p (x) =  1

2p;0

R

p+1 (x) x ? zp+1

2 X 2p =

k=0

2p;k Rk (x)h(k); x 2 S;

(31)

where zp+1 is that zero of Rp+1 , which is as close  Rp+1to(x)x02as possible. P The coecients 2p;k are uniquely de ned by x?zp+1 = 2kp=0 2p;k Rk (x)h(k). It holds p p X X g(k; j; i)Ri (zp+1 ): 2p;k = (R (z )h(1p + 1)a a )2 Rj (zp+1 )h(j ) p+2 p+1 0 p+1 j =0 i=jk?j j (32) The Fejer-Korovkin weights are given by Pp R (z )h(j ) Pp g(k; j; i)R (z ) j p+1 i p+1  i=jk?j j 2p;k P 2p;k =  = j=0 : (33) p R2 (z )h(j ) 2p;0 j =0 j p+1 Obviously, our de nition yields FK2p(x)  0 and kFK2pk1 = 2p;0 = 1. In case p  1 we derive 2p;1 = R1 (zp+1 ): (34) Since it is well known that limp!1 zp = x0 we get limp!1 2p;1 = 1. For Jacobi polynomial systems (Jn(; ))n2N0 we also de ne kernel polynomials of odd degree by

0 (; +r) 12 J n (x) A; FKn(; )(x) = n(; )(x + 1)r @ b 2 c+1 (; +r) x ? zb n2 c+1

(35)

Approximation Methods by using Orthogonal Polynomial Expansions

11

(; +r) +r) where r = n mod 2, zb(; n c+1 is that zero of Jb n c+1 which is as close to 1 as 2 2 possible,

(; ) n

and

+r) (; +r) (; +r) n +r) 2   + + 2 1?r (Jb(; (b 2 c + 1)a(0; +r)a(b; n c+2 (zb n c+1 )h n c+1 ) 2 2 2 = 2( + 1) Pb n2 c h(; +r)(j )(J (; +r)(z(; +r) 2 )) n

b 2 c+1

j

j =0

) (; ) (; +r) +r) (n;; (zb n2 c+1 ) = 1 ? 2(+ ++1)2 (1 ? zb(; n c+1 ): 1 = J1 2

(36)

) ?2 This kernel is also known as general Jacobi kernel. It holds 1 ? (n;; 1 = O(n ), which for positive kernels is the best achievable rate of convergence, see [16]. Especially for Chebyshev polynomials of the rst kind ( = = ?1=2) our FejerKorovkin kernel coincides with the well-known kernel (29) and we have k+2 k cos(=(n + 2)) k (? 21 ;? 12 ) = n? (37) n;k n + 2 cos n + 2 + (n + 2) sin(=(n + 2)) sin n + 2 : For Chebyshev polynomials of the rst kind we may also de ne a nearby Fejer(? 21 ;? 12 ) Korovkin kernel (NFKn )n2N0 by

n X 1 1 (? 12 ;? 21 ) (? 21 ;? 21 ) FK NFKn(? 2 ;? 2 ) (x) = cos nk ( x ) = 1 + 2  Tk (x); (38) n n;k +2 1 1  (? 2 ;? 2 )

k=1

(? 21 ;? 12 ) = cos nk . +2 n;k

Of course, this nearby Fejer-Korovkin kernel is with n;k 1 1  = O(n?2 ). also a positive approximate identity and 1 ? n;(?12 ;? 2 ) = sin2 n+2 5.3 De la Vallee-Poussin kernel For trigonometric polynomials the de la Vallee-Poussin kernel (Vn )n2N0 is de ned by

Vn (t) = 1 +

n X

k=1

(n!)2 (n!)2 (cos t + 1)n ;  = n;k cos kt = (2 n;k n)! (n ? k)!(n + k)! ; (39)

t 2 [?; ), see [2, Sec. 2.5.2]. We are able to give the de nition of a general de la Vallee-Poussin kernel for OPS (Rn )n2N0 , see [18], where the assumptions on the

coecients in (1) are changed slightly. Let us x s = ? minx2S R1 (x) > 0. The de la Vallee-Poussin kernel (Vn )n2N0 and the de la Vallee-Poussin weights n;k are de ned by n

Vn (x) = (R1 (x) + s) = n;0

n X k=0

n;k Rk (x)h(k);

(40)

12

R. Lasser, D.H. Mache & J. Obermaier

where n;k are uniquely determined by (R1 (x) + s)n = Pn theR coecients ( x ) h ( k ). k=0 n;k k If we de ne the coecients i;j by

R1i (x) =

1 X

Xi

i;j Rj (x);

(41)

n n sn?i  n;k = n;k = Pin=0 ?ni
n?i i;k h(1k) : n;0 i=0 i s i;0

(42)

j =0

i;j Rj (x) =

j =0

P ?
then we get n;k = ni=0 ni sn?i i;k =h(k) and P ?


Obviously, our de nition yields Vn (x)  0, x 2 S and kVn k1 = n;0 = 1. Moreover, if in the non-symmetric case, i.e. 9n 2 N : bn 6= 0, holds limi!1 i+1;0 =i;0 = 1 and in the symmetric case, i.e. 8n 2 N : bn = 0, holds limi!1 2i+2;0 =2i;0 = 1, then limn!1 n;1 = 1: Especially for Jacobi polynomials we have

Vn (x) =

  + 1)?(n +  + + 2) 1 + x n n;k Rk (x)h(k) = ?( ?(n + + 1)?( + + 2) 2 k=0 n X

(43)

with de la Vallee Poussin weights !?(n +  + + 2) n;k = (n ? nk)!?( n + k +  + + 2) :

(44)

?1 Thus 1 ? n;1 = n+++ +2 +2 = O(n ).

References [1] BLOOM, W.R., HEYER, H., Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter, Berlin{New York, (1995). [2] BUTZER, P.L., NESSEL, R.J., Fourier analysis and approximation, Birkhauser, Basel{Stuttgart, (1971). [3] CHIHARA, T.S., An introduction to orthogonal polynomials, Gordon and Breach, New York, (1978).

Approximation Methods by using Orthogonal Polynomial Expansions

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[4] GASPER, G., Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math., Vol. 95, (1972), 261 { 280. [5] LAINE, T.P., The product formula and convolution structure for the generalized Chebyshev polynomials, SIAM J. Math. Anal., Vol. 11, (1980), 133 { 146. [6] LASSER, R., Orthogonal polynomials and hypergroups, Rend. Mat., Vol. 3, (1983), 185 { 209. [7] LASSER, R., Introduction to Fourier Series, Marcel Dekker, New York, (1996). [8] LASSER, R., OBERMAIER, J., On Fejer means with respect to orthogonal polynomials: A hypergroup theoretic approach, in: Progress of Approximation Theorie, Academic Press, Boston, (1991), 551 {565. [9] LASSER, R., OBERMAIER, J., On the convergence of weighted Fourier expansions, Acta. Sci. Math., Vol. 61, (1995), 345 { 355. [10] LASSER, R., OBERMAIER, J., Orthogonal Expansions for Lp and C-spaces, in: Special Functions, Proceedings of the International Workshop, World Scienti c Publishing, Singapore { London, (2000), 194 { 206. [11] LENZE, B., On Lipschitz-type maximal functions and their smoothness spaces, Proc. Netherl. Acad. Sci. A 91 (1988), 53 | 63 . [12] LUBINSKY, D.S. and MACHE, D.H. (C; 1) Means of Orthonormal Expansions for Exponential Weights, Journal of Approximation Theory, Vol. 103, No. 1, (2000), 151 { 182. [13] LUPAS, A. & MACHE, D.H.; The  { Transformation of Certain Positive Linear Operators, in: International Journal of Mathematics and Mathematical Sciences, Vol. 19, No. 4, (1996), 667 { 678. [14] MACHE, D.H.; Generalized Methods using Integral Transforms and Convolution Structures with Jacobi Orthogonal Polynomials, Journal Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 68 (2002), 625 - 640. [15] MACHE, D.H.; Summation of Durrmeyer Operators with Chebyshev Weights, in: Approximation and Optimization, Proceedings of the Int. Conf. on Approximation and Optimization - ICAOR, Cluj - Napoca (1996), Vol. I, 307 { 318. [16] MACHE, D.H.; Optimale Konvergenzordnung positiver Summationsverfahren der Durrmeyer Operatoren mit Jacobi - Gewichtungen; Habilitationsschrift, Dortmund (1997).

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R. Lasser, D.H. Mache & J. Obermaier

[17] MACHE, D.H. & ZHOU, D.X.; Characterization Theorems for the Approximation by a Family of Operators, in: Journal of Approximation Theory, Vol. 84, No. 2, (1996), 145 { 161. [18] OBERMAIER, J., The de la Vallee Poussin kernel for Orthogonal Polynomial Systems, Analaysis, Vol. 21, (2001), 277 { 288.

Rupert Lasser Josef Obermaier GSF-Forschungszentrum fur Umwelt und Gesundheit Institut f. Biomathematik und Biometrie Ingolstadter Landstrasse 1 D-85764 Neuherberg [email protected] [email protected]

Detlef H. Mache Universitat Dortmund Institut f. Angewandte Mathematik Vogelpothsweg 87 D-44221 Dortmund [email protected]