A Global Approximation Theorem for Meyer-König and Zeller operators

Math. Z. 160, 195 -206 (1978) Mathematische Zeitschrift

�9 by Springer-Verlag 1978

A Global Approximation Theorem for Meyer-Kiinig and Zeller Operators

Michael Becker and Rolf J. Nessel Lehrstuhl A ftir Mathematik, Technische Hochschule Aachen, Templergraben 55, D-5100 Aachen, Federal Republic of Germany

1. Introduction

Let the operators M, of Meyer-K6nig and Zeller [13] be given by

Mnf(x)=M,(f(t); x)'.=k~=o f mk,,(X), O<X<I,

M,f(1):=f(1), mk,,(x):=(n+kk)xk(1-x)"+l, (1.1)

with n~N, the set of natural numbers, and f e C[0, 1], the space of functions continuous on the interval [0, 1]. In fact, the operators (1.1) represent the slightly modified version introduced by Cheney and Sharma [6]. The approximation-theoretical behaviour of the operators (1.1) has been the subject of many investigations (cf. [6; 7; 9; 11-18]). Let us just mention that various results exist concerning convergence, variation-diminishing and shape-preserving properties, asymptotic formulae of Voronovskaja-type, direct approximation theorems, best asymptotic constants, (pointwise) saturation assertions, etc. Among other facts it was pointed out that the properties of the series (1.1) are quite analoguous to those of the Bernstein polynomials (in fact, the operators M, are also called Bernstein power series).

The purpose of this note is to prove an equivalence theorem concerning the global (i.e. on the whole interval [0, 1]) approximation by the operators M,, corresponding to the one obtained by Berens and Lorentz (cf. [4; 10]) for the Bernstein polynomials. To formulate the result, let (0 < h < 1/2)

A2f(x)'.=f(x+h)-2f(x)+f(x-h) (x~[h, 1 -hi), co2(f, 6):= sup sup IA~f(x)[,

O<h_-<3 x~[h,l--h]

Lip2 c~:--{fe C[0, 1]; co2(f, 6)= O(fi~); 6--,0+ }.

With ~o(x): = x ( 1 - x ) 2 one has

0025-5874/78/0160/0195/$ 02.40

196 M. Becket and R.J. Nessel

Theorem 1.1. For f 6 C [0, 1], ~ ~ (0, 2] the following statements are equivalent:

f ~ Lip2 c~, (1.2)

]M, f (x ) - f ( x ) l<A [cP(x)] ~/z (n_>3, x~[0, 1]), (1.3) - -

the constant A being independent of n and x.

For the proof of this equivalence theorem see Section 3. Let us observe that the proofs of the direct part, i.e. (1.2)~(1.3), as well as of the inverse part for

= 2, the saturation case, follow along arguments by now standard in approximation theory; they are given for the sake of completeness. In fact, the only point to mention in connection with the direct theorem is that its present form furnishes an appropriate candidate for the inverse theorem. Thus the proof of the inverse part for the nonoptimal cases, i.e. (1.3)~ (1.2) for 0 < c~ < 2, will be the main point of this note. Here we use the "intermediate space method" developed by Berens and Lorentz [4] in their proof of the inverse theorem for Bernstein polynomials. It should be noted that the "elementary method" seems to fail in the present situation (see Remark 3.4 for details).

2. S o m e Basic Properties

It is well-known that the operators M, preserve linear functions, i.e. (cf. [6])

M,(ti; x)=x i (i=0, 1). (2.1)

For the "second moment" (cf. [12; 17])

q)n(x):=M~((t-x)2; x)=Mn(t 2" x ) - x 2 = x ( 1 - x ) -- ~ mk'n-l(x) (2.2) ' k=O n + k + l

one has the following estimates (cf. [16; 173)

Lemma 2.1. For n>__2, x~[O, 1] there holds

( 2x ) ( 2 x ) x(1-x)2 (2.3) x(1-x)2<cp"(x)< 1 + n~-i-- 1 n + l - - '

the inequalities being proper for x s(O, 1). In particular, with cp(x): = x ( 1 - x ) 2

cp (x)/(n + 1) < q~, (x) < 2 (p (x)/(n + 1) (n > 3). (2.4)

Proof. For the sake of completeness, let us briefly indicate a proof (cf. [16]). For x=0 , 1 all expressions in (2.3) vanish. For xe(0, 1) one has by (2.1) for n > 2

A Global Approximation Theorem for Meyer-K6nig and Zeller Operators 197

(p,(x) 1 - x q~,(x) 1 x ( 1 - x ) n + l - x ( 1 - x ) n + l M " - l ( 1 - t ; x )

n + l k=O ( n + k - 1 ) ( n + k + l ) [ 3] - ( n ~ l ~ - - - 1 ) k=, n + k - 2 1 n + k + l "

N o w (2.3) is a consequence of (2.1) and

n - 1 3 3 - i - < 1 <1 ( k > 1).

n + 2 n + 2 = n + k + l

Thus the second m o m e n t q~,(x), which as usual determines the rate of convergence, does indeed behave like (p(x)/(n+l) (cf. (1.3)). This is also con- f irmed by the Voronovska ja - type relat ion (cf. [12; 17; 18])

q)(x) M, f ( x ) - f ( x ) - 2(n + 1)

But, since the function ~0 possesses a double zero at x = 1, the si tuation is con- siderably different to that known for e.g. the Bernstein polynomials or Szfisz- Mi rak jan opera tors (cf. R e m a r k 3.4).

L e m m a 2.2. For 0 < fl < 1, 0 < h < 1/8 one has

hi2 dsdt < Mph 2 (x~[h, 1 - h ] ) . (2.6) ~ qp(x+s+t)P = m a x {~o(x+h), ~0(x)} s --h/2

Proof. For (small values of) x ~ [h, 2 h I as well as for x E [2 h, 1 - 2 hl the p roo f follows as for (2.5) in [-3], where we were concerned with the endpoint weight x ( 1 - x ) . F o r x ~ [ 1 - 2 h , 1 - h i we have for fl+1/2

hi2 dsdt 1 hi2 dsdt

' -hi2 ~ , o ( x + s + t ) S = ( x - h ) s - _ 2 (1-x-s- t ) 2s

(x - h ) - ](1 - x + h)2- 2 s - 2(1 - x)2- 2P + (1 - x - h)2- 2s[

- - ( 2 - 2 f i ) 1 1 - 2 i l l

< MS [ ( l _x+h)2+2( l_x )2 -2S ( l_x+h)gp = ~o (x - h) s

+ ( 1 - x - h ) 2 - 2 s ( 1 - x + h ) 2~]

< M s h2/cp (x - h) s = M s h 2/max { cp (x __ h), ~0 (x)}S.

For fl = 1/2 there holds

1 h/2 dsdt < 1 hi2 dsdt [x h~ . . , (.~_ ,pal2, -!~2 1 - x - s - t = ( x - h ) 1/2 ~ 2 h - s - t

= (_2 log 2) h < (6 log 2) hZ/(p (x - h) a/2 ( x_h ) l / 2

198

Lemma 2.3. One has

M,,~(x)<2cp(x)

k=l q) ~ mk'"(x) < 12/cp(x)

Proof. For n > 2, k >___ 1 there holds

(n>2, xe[0 ,1] ) ,

(neN, xe(0, 1)).

M. Becker and R.J. Nessel

(2.7)

(2.8)

(2.11)

Let Hfll:= max If(x)[ for f ~ C[0, 1]. Then x~[O, 11

Theorem 2.4. For g, g" ~ C [0, 1] one has

H(M,g)"II <2 IIg"ll (n>2).

( k + l t k -2(n+k+ 1) f [ ~ ! +(n+k) f (77~)].

r ~ = (n+k)2(n-1) \ k -1 ]-- \ k -1 "

Hence by (2.1)

Y'~176 {n+k-3~ xk( l_x) ,+ , =2cp(x), M. (x) cp < 2 k~l= \ k - - 1 ]

which proves (2.7). For k > 1 one also has

qo ( k ) - 1 ( +,to k)=(n+k)3(n+l)(n+2)n-2(k+l) ( n + k + 3 ] (n+k+l)(n+k+2)(n+k+3)k k + l ]

( n + k + 3 ~ <12 \ k + l ]"

Therefore for x e (0, 1)

~P ~ mk'"(X)<x(1--X)2 k k = l k = l

so that (2.8) follows.

Next we note two representations of (M,f)" which follow by direct com- putations (cf. [6; 11])

~ ( k )rk.(x)mk.(x ) (xero, l]), (2.9) (M.f)"(x)= f ff-~ , , k=0

rk (n+l,x]] n + l rk'"(x):=-x ~ l~-~x J - k - l - x JJ x(l-x) z'

(M.f)"(x)=(1-x) -z ~ (4k,.f)mk,.(x) (xE[0, 1)), (2.10) k=0

[(n+k+2)f ( k + 2 Ak,.f:=(n+k + l) [ \n~VZ2 !


Furthermore, there holds the Jackson-type inequality

[M,g(x)-g(x)[<~~ I[g"ll (n>3). (2.12) - n + l

Proof. In view of (2.10) and

n 2 ( n ; k ) ( n + k - 1 ) n ( n - ~ + k ) < _ 2 ( n - 2 k + k ) , (2.13) (n+k)(n+k+2) - ( n + k + 2 ) ( n - 1 ) -

the estimate (2.11) follows from (cf. [9; 11])

n 2 IAk, ,g[<(n+k)(n+k+2) Jlg"l[. (2.14)

To prove (2.14) one uses the Taylor expansion

t s

g(t) - g(x) = (t - x) g'(x) + ~ ~ g"(u) duds x x

and has for xi: = (k + i)/(n + k + i), i = O, i, 2,

~2 i ~ ~ ds IAk , ,g l=(n+k+l) ( n + k + 2 ) S g ' (u)duds+(n+k) S S g'(u)du x 1 Jcl x o s

<-Ilg"ll (n + k + 1)[(n + k + 2)(x z - xa)N +(n + k)(xl -Xo)Z]/2 n 2

Hg"N. (n+k) (n+k+l )

Concerning (2.12), one deduces (ef. (2.1))

k/,ny+k) s mk,,,(X) IM, g(x) -g(x)[ = ~ g" (u) du d s k = 0 x x

< Jig" Jt M~ ( ( t - x)2; x)/2.

Hence (2.12) follows from Lemma 2.1.

Theorem 2.5. For f ~ C[0, 1] one has the Bernstein-type inequality

< 4 ( n + l ) I(M,f)"(x)[= qo(x) r l f l[ (n~N, xe(0, 1)). (2.15)

Proof. The representation (2.9) shows that we have to verify the estimate

B1 ~ . . . . . < 4 ( n + 1) : = k = 0 rk ' " t X ) r ink,, tX) = - - ~ . ( 2 . 1 6 )

TO this end, define for r=0 , 1, 2, ...

- ~ (k--(n+l)X~rmk,,(x). (2.17) S"'r(x):--k= o 1--x /

200 M. Becker and R.J. Nessel

These functions S,, r are a slight modification of those considered in [12] such that

r(n+ l)x d s.,r+ l(x)= -(-1_-~ s.,r_l(x)+ x ~ s.,.(x),

( n + l ) x S,,0(x)= 1 , S,, 1 (x) =0, Sn,2(x)=(I_x)2 . (2.18)

Furthermore, one has

k _ ( n + l ) x ( ( n + l ) x k) mk,,(x ) 1 - x mk'"(x)=S"'l(x)+2 ~ \ -1--~

k=o 2

< 2 (n + 1) x/(1 - x) < 2 (n + 1) x z/q) (x).

Therefore

B l < x -2 S,,2(x)+ ~ k ( n + l ) x mk,,(x ) +(n+l)/~o(x) k=o 1 - x

< 4 (n + 1)/(p (x),

which proves (2.16) and thus the theorem.

In connection with the endpoint weight ~0 let us define the following spaces which wig be used for proving the inverse part of Theorem 1.1 in the nonoptimal cases 0 < e < 2 (cf. [4], p. 700):

Co.'= { f e C[0, 1]; f ( 0 )= f (1 ) =0},

C~:={ f s Co; (p-~/2feC[O, 1]} (0<e<2) ,

2. _ {g e C O; g" e C(0 , 1), q~! - ~/z g,, e C [0, 1 ] } , C~,-- Ilfll~, = 1I~o-~/2fll ( f e Ca),

Igl=,2==ll@-=/2g"ll (geC~).

Note that ql" I1~ defines a norm on C= and l" I~, z a seminorm on C 2, respectively. Let us also observe that C 2 is not always a subspace of C= (for instance, the function g ( t ) : = ( 1 - t ) l o g ( l - t ) belongs to C 2 but not to C 0.

Lemma 2.6. For g e C~, n e N , k>__O one has

4n 2 ( k + l ~-- (1 --~/2) IAk,,gt<=(n+k)(n+k + 2) tg]~,2~ p \ ~ + ~ - ~ ) . (2.19)

Proof. Corresponding to the proof of (2.14) there holds

IAk,.gl<=[g[~,a(n+k+l ) ( n + k + 2 ) ~~ 2 + ( n + k ) cp(u)l_~/2 .

Thus we have to prove the estimate

-i } duds 2 ( t - x l ) 2 I . - j - (te{Xo, X2}), (2.20)


For small values of x 1, t (<1/2) this is an immediate consequence of (cf. [3], Lemma 2.2)

i i ~ duds ( t - -x ) 2 < (x~(O, 1); t,/~e[o, 1]). ~ [ u ~ - ~ ) ] ~ = [ x ( 1 - x ) ] ~

As ~o(u) is decreasing for u > x o > 1/3, there follows

I __< (�89 (t - x 1) 2 m a x (p (u ) - (1 - , /2) g

<~ (t-xl)2/2q~(ya)!- ' /2 for t<=Xl' =[(t-xl )2/2~o(t) 1-'/2 t >=xl.

Thus in view of

k + 2 (1 i .]2>( 1 12 , e(xz) /~o(xO=~f -~) >x n + k + 2 ] =

we obtain (2.20), and thus (2.19).

The most important properties of the operators M, with respect to the spaces C,, C 2 are summarized in the following theorem (constants A may have different values at each occurrence).

Theorem 2.7. Let n >__2, f ~ C,. Then M n f e C~c~ C 2 with

I[M, fII~<=A I[f[[=, (2.21)

IM, f[~,z<=A(n+ l) Ilfll~. (2.22)

Furthermore, for n > 3, g ~ C 2 one has M , g ~ C~ with

IM, gl~, 2 < A [gl~, 2. (2.23)

Proof. Whereas (2.21) is an immediate consequence of (2.7), we have by (2.9), (2.16)

LLky '2 [(M,f)"(x)l<Hfl[, ~, q~ \ n + k ] Irk'"(x)lmk'"(x)

k = 0

<IINH, k~o~O ~ k=o

Thus (2.22) follows from

B2:= q) ~ Irk,,(x)[mk,,(x)<7(n+l ) (n>2). (2.24) k = O

202

To prove (2.24) we may write

rk, . (x) = gk,, (x)/x2 _ (n + 1)/cp (x)

with ( ( n + l ) x ~ ( ( n + l ) x

gk,.(x):= k i - x ! k 1 - x

M. Becker and R.J. Nessel

1) For fixed x, n the quantity gk,~(x) is always non-negative except for k o such that (n + 1) x/(1 - x) < k o < (n + 1) x/(1 - x) + 1, where

~ 1 - x ! i - x ] ] = 4

This yields

B 2 < ~o n-~k ' Igk~ mk~ k = O

+ (2 (n + 1)#p (x)) M, qo (x)

< [(M, cp)"(x)l + (1/2 + 2 (n + 1)/qo (x)) M, cp (x).

In view of qLcp/[ =4/27, [IcP"II ~4 and (2.7), (2.11) this implies for n > 2

Bz < 8 + c p ( x ) + 4 ( n + l ) < 7 ( n + l).

For the proof of (2.23) we use Lemma 2.6. By (2.10), (2.13) one has for n> 3

( k + l ]-.-~/2~ [(M.g)"(x)l<81gl~,2 ~ (o \n~k+l} mk'"-z(X)

k = O

m~ ._ 2(x) "~-~,/2

While the sum can be estimated by 12/~0(x) (cf. (2.8)), it follows for the first term that

( 1 ) - 1 ( n + l ) 3 x ( l _ x ) , + l < 1

q~ ~ mo,._2(~)= n~o(x) =~o(~--5'

since x ( 1 - x ) "+1 attains its maximal value at x = 1/(n+2) which implies

(n+ 1) 3 x ( l _ x ) , + l < ( 1 + 1 ) z (1 1 \,+2 16 . 2 . v . . Vee < 1

This proves (2.23) and thus the theorem.

3. Proof of the Equivalence Theorem

The proof of the direct theorem follows along standard lines using the Jackson- type inequality (2.12) and the Steklov means.

A Global Approximation Theorem for Meyer-K~Snig and Zeller Operators 203

Theorem 3.1. For f a C[0, 1] there holds

IM, f(x)-f(x)l<=2co2(f , 1/~o(x)/(n+l)) (n>3, xe [0 , 1]). (3.1)

In particular, i f f a Lip2 c~ for some c~ e (0, 2], then

[M n f (x) - f ( x ) l < A [~o(x)/(n + 1)-I ~/2, (3.2)

the constant A being independent of n > 3, x ~ [0, 1].

Proof. Let us recall the Steklov means

h/2

fh(x):=h-2 ~ f ( x + s + t ) d s d t --hi2

which satisfy (cf. [5, p. 38, 77])

Ilf--fhll <�89 c02 (f, h), l[ fh"[[ = < h - 2 c%(f, h). (3.3)

Note that for x = 0 , 1 the assertion is trivial. Since IIM~NII < Ilfli, for f e C[0, iI, h > 0 one has by (2.12), (3.3)

[M n f (x) - f (x ) ] __< IMn [f--fh ] (X)I + IM, fh(x)--fh (X)I q-I fh (X)--f(x)l

<2 [I f - fh[I + Nfh"ll q~(x)/(n + 1)

< co 2 (f, h) [ 1 + (p (x)/(n + 1) h2],

so that (3.1) follows upon setting h=l/q)(x)/(n+ 1).

The proof of the inverse theorem in the saturation case a =2 may be based upon an idea of Grundmann [8] (who gave the proof for the Bernstein polynomials).

Theorem 3.2. Let f ~ C[0, 1] satisfy

IMnf (x ) - f (x ) l<A~o(x) / (n+l) (n>3, x e [0, 1]). (3.4)

Then f~L ip2 2.

Proof. By Lemma 2.1 we obtain from (3.4) that

[M~f (x) - f ( x ) l < A q)n(x)= A(Mn(t2; x ) - x2).

Hence

Ax2+_-f(x)<M~(At2+__f(t); x) (n=>3, x~[0 , 1]).

This implies that A x 2 + f ( x ) are convex functions (which may be proved by a parabola argument using the Voronovskaja-type relation (2.5)). Hence

A 2 [A t 2 4- f ( t ) ] (x) > 0,

so that

tA~f(x)I <= A A~ [t 2] (x) = (9 (h=),

proving f ~ Lip2 2.


To complete the proof of Theorem 1.1 there remains the inverse part in the nonoptimal cases 0 < c~ < 2.

Theorem 3.3. For 0<c~<2 let f e C[0, 1] satisfy

IM, f(x)-f(x)l<=A[q~(x)/(n+l)] ~/2 (n>3, x~[0, 1]). (3.5)

Then f ~ Lip2 c~.

Remark 3.4. We shall use the "intermediate space method" as developed by Berens and Lorentz [4]. Since by Theorems2.4, 2.5 and (3.3) one has the Bernstein-type inequality

n + l -2 [(Mnf)"(x)[<Ac%(f, 3 ) [ ~ + 6 ] , (3.6)

one may try to apply the so-called elementary method (cf. [2; 4]). In view of the fact, however, that (p has a double zero at x = 1, the integral (2.6) does not exist for/~= 1 (and x = 1-h) . To circumvent this difficulty one may hope to prove an estimate like

I(M,f)"(x)l<A(n+l)2co2(f, 1/(n + 1)) ( x ~ l - ) . (3.7)

But this seems to be hard to deduce (if at all, cf. (2.10)). On the other hand, the intermediate space method, which is based upon mapping properties of the operators M, with respect to the spaces C~, C~ (cf. Thm. 2.7), has the advantage of using the integral (2.6) only for values/? < 1. Let us mention that this method may also be used to prove inverse theorems for Sz~sz-Mirakjan, Baskakov, and Favard operators (see [1]).

Let us introduce the K-functional for f ~ C o

K~(t, f )=K( t , f; C~, C~)

�9 "=inf{llLll~+tlf2[~,2; f = f ~ +f2, f~ ~ C~, f2 e C2},

and for 0 </~__< 2 the spaces

(C~, 2 C~)p, = { f ~ C o; K,(t, f ) = (9(t~/2), t--+O + }.

Our interest in these spaces arises from (cf. [4])

Lemma 3.5. For 0<e__<2 one has (C~, C~)~cLip2 c~.

Proof. For fixed h ~ (0, 1/8), x ~ [h, 1 - h], and any f ~ C a one has

[A2 f (x)[ < [f (x +h)[ +2 If(x)[ + [f (x -h)[ <4 [I f [I~m(x, h) ~/2, (3.8)

where re(x, h)..=max {cp(x+h), (p(x)}. Using Lemma 2.2, for any g~C~


hi2 hi2 ds dt Id g(x)l<= lg"(x+s+Ola at <lgk2 Jf

--h/2 -hi2 (#(X + S + t) I-e~2

N A h 2 [g[=, 2 re(x, h) -(1-=/2). (3.9)

Let f e (C~, 2 C~)~. For all representations f = f l +f2, f l �9 Co, f2 e C 2 the estimates (3.8), (3.9) deliver

IA2f(x)l < [A~fl (x)] + ]A2f2(x) l

< A m (x, h) ~/2 [II f t lI~ + (h2/m(x, h)) If2 l~, 2].

Hence the definition of (C=, C2)= implies

I A 2 f (x) l < A m (x, h) ~/2 K~ (h 2/m (x, h), f )

<= A rrl(x, h) =/2 [h 2/m (x, h)] =/2 = A h =,

proving f e LiP2 e.

Proof of Theorem 3.3. In view of Lemma 3.5 it suffices to show that (3.5) implies f s ( C ~ , 2 C~)~, which in turn will foIlow from (cf. [-3; 4~)

K~(t,f) < A [6 ~/2 + (t/a) K~(a,f)]. (3.10)

To prove (3.10) we first note that (3.5) means

]lM, f - f l l~<=A(n+ l) -~/2 (n>3). (3.11)

Furthermore, by Theorem 2.7 one has for f = f t +f2, fa ~ C~, f2 ~ C 2,

[M, fl=,2 < [M, f t 1~,2 + IM= f21=,2

__<.4 [-(n+ 1)Ilfx lf + IA I ,A-

Thus we obtain

]M, fl~,2<=A(n+ l)K~(1/(n+ l) , f) (n>3). (3.12)

Since f = ( f - M , f ) + M , f represents a decomposition into ( f - - M , f ) E C = (cf. (3.11)) and M, f e C~ (cf. Thm. 2.7), the estimates (3.11), (3.12) yield that

K~(t,f) < ]lM, f - f l[, + t IM, fl~,2

__< A [(n + 1)-~/2 + t(n + 1) K~ (1/(n + 1),f)].

With (n+ 1) - j __<cS<2(n+ 1) -1 this implies (3.10), so that the proof is complete.

References

1. Becker, M.: Umkehrsiitze ftir positive lineare Operatoren. Dissertation, Aachen 1977 2. Becker, M.: An elementary proof of the inverse theorem for Bernstein polynomials. Aequationes

Math. (to appear) 3. Becker, M., Nessel, R.J.: Inverse results via smoothing. In: Constructive Function Theory.

Proceedings of an International Conference (Blagoevgrad 1977). Sofia: (to appear)


4. Berens, H., Lorentz, G.G.: Inverse theorems for Bernstein polynomials. Indiana Univ. Math. J. 21, 693 708 (1972)

5. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation, Vol. I: One-Dimensional Theory. Basel: Birkh~iuser and New York: Academic Press 1971

6. Cheney, E.W., Sharma, A.: Bernstein power series. Canad. J. Math. 16, 241-252 (1964) 7. DeVore, R.A.: The Approximation of Continuous Functions by Positive Linear Operators.

Lecture Notes in Mathematics 293. Berlin-Heidelberg-New York: Springer 1972 8. Grundmann, A.: Personal communication to Dr. E. Stark. July 1975 9. Knoop, H.B., Pottinger, P.: Ein Satz vom Korovkin-Typ fiir Ck-R~iume. Math. Z. 148, 23-32

(1976) 10. Lorentz, G.G.: Inequalities and the saturation classes of Bernstein polynomials. In: On Approxi-

mation Theory. Proceedings of a Conference (Oberwolfach 1963), pp. 200-207. Editors P.L. Butzer-J . Korevaar. Basel: Birkhfiuser 1964

11. Lupas, A.: Some properties of the linear positive operators (I). Mathematica (Cluj) 9, 77-83 (1967) 12. Lupas, A., Miiller, M.W.: Approximation properties of the M,-operators. Aequationes Math. 5,

19 37 (1970) 13. Meyer-K6nig, W., Zeller, K.: Bernsteinsche Potenzreihen. Studia Math. 19, 8944 (1960) 14. Mtiller, M.W.: Die Folge der Gammaoperatoren. Dissertation, Stuttgart 1967 15. Rathore, R.K.S.: Lipschitz-Nikolskii constants and asymptotic simultaneous approximation of

the Mn-operators. Aequationes Math. (to appear) 16. Schurer, F., Steutel, F.W.: On the degree of approximation of functions in C~[0,1] by the

operators of Meyer-K6nig and Zeller. J. Math. Anal. Appl. (to appear) 17. Sikkema, P.C.: On the asymptotic approximation with operators of Meyer-KiSnig and Zeller.

Indag. Math. 32, 428-440 (1970) 18. Suzuki, Y., Watanabe, S.: Approximation of functions by generalized Meyer-K6nig and Zeller

operators. Bull. Yamagata Univ. Natur. Sci. 7, 123 127 (1969)

Received December 23, 1977

A Global Approximation Theorem for Meyer-König and Zeller operators

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Transcript of A Global Approximation Theorem for Meyer-König and Zeller operators