Direct and Inverse Estimations for A Generalization of Positive Linear
Operators
ABDULLAH ALTIN and OGÜN DOĞRU
Department of Mathematics
Ankara University, Faculty of Science
06100 Tandoğan, Ankara
TURKEY
Abstract: - In this study, we give quantitative estimations and inverse theorems for a generalization of positive
linear operators, which includes some well-known polynomials and operators.
Key-Words: - Positive linear operators, the Bohman-Korovkin theorem, the Gadjiev-Ibragimov operators, first
and second order modulus of continuities and Lipschitz classes.
1 Introduction
Let n  N and a be positive real numbers,
be the sequence of functions
in C[0, a] such that  n (0)  0,  n (t )  0, for each
t  [0, a]. Let also  n  be a sequence of positive

1
numbers such that lim n  1, lim
 0.
2
n n
n n  (0)
n
 n (t ) and  n (t )
Assume that a sequence of functions of three
variables K n ( x, t , u ) ( x, t  [0, a],    u  )
satisfies the following conditions :
10 . Each function of this family is an entire
analytic function with respect to u for fixed x,
t  [0, a] ,
2 0 . K n ( x,0,0)  1 for any x  [0, a] and for all
n N ,





30. (1)   K n ( x, t , u ) u u1   0 ,
t 0 

u


4 0.

   1

K
(
x
,
t
,
u
)
K
(
x
,
t
,
u
)
u u1   nx

n
m

n
 1
t 0
u 
 u

where m  n N  0 ,
u u1
t 0
5 0.

K n ( x, t , u ) u   n  n ( t ) 
t 0
x
.
 nn n (0) K m  n ( x, t , u ) u  n n (t )
t 0
Consider the sequence of positive linear
operators;
 

 
( Ln f )( x)   f ( 2
)    K n ( x, t , u ) 
n  n (0)  u
 0








u   n  n (t )  (1)
t 0

[ n n (0)]
!
where f  C ([0, a], x 2 ) and C ([0, a ], x 2 ) be the
space of functions defined on the entire line,
continuous in the interval [a, b] and increasing to
infinity not more rapidly than x 2 .
By multiple application of property 4 0 , operator
(1) can be reduced to the form


n(n  m)(n  (  1)m)
( Ln f )( x)   f ( 2
)
!
n  n (0)
 0
 [ n n (0)] K n  m ( x,0,  n n (0)) x
.
These operators are a generalization of some
well-known operators and they are called as
Gadjiev-Ibragimov operators [12].
Some approximation properties of these
operators and their some generalizations are
investigated by Radatz and Wood [17]. (see also
[10]).
n
ux 

Remarks. By choosing K n ( x, t , u )  1 
 ,
 1 t 
1
n  n, n (0)  , we have h(n)  n  1 and the
n
operators defined by (1) are transformed into
classical Bernstein polynomials.
For
 n  n,  n (0) 
b
lim n  0), we
n  n
polynomials.
obtain
1
nbn
( lim bn  ,
n 
Bernstein-Chlodovsky
By choosing K n ( x, t , u )  e  n(t ux) ,  n  n,
 n (0) 
If
1
, we get Szasz-Mirakjan operators.
n
K n ( x, t , u )  (1  t  ux )  n ,
 n  n,
1
then we obtain Baskakov operators (see
n
for details [1]).
Therefore these operators include a large
spectrum of some well-known positive linear
operators.
In [12], Gadjiev and Ibragimov proved the
following theorem:
Theorem A. The sequence of linear positive
n (0) 
operators defined by (1) with conditions 1  5
converges
uniformly
to
the
function
0
0
f  C ([0, a], x ) in [0, a].
In the proof of Theorem A, for using the wellknown Bohman-Korovkin theorem [6,14] (see also
[2] ), the authors obtained
 x
( Ln1)( x)  1, ( Lnt )( x)  n and
n
(2)

x

x
m

n
1
( Lnt 2 )( x)  n ( n
 2
).
n
n
n
n  n (0)
2
In this study, we will give the sufficient conditions
to obtain inverse estimates for the general operators
defined in (1). Therefore, main purpose of this note
is to obtain some direct and inverse results for the
operators (1).
We denote n,k ( x) : ( Ln (t  x)k )( x), the k-th order
central moment of the operator (1) for abbreviation.
From the equalities in (2), we have

mn


1
 n, 2 ( x)  ( ( n ) 2
 2 n  1) x 2  n 2
x
n
n
n
n n  n (0)
(3)
for the operators (1).
Now, Using the some standard methods, we will
give the proof of the direct estimates:
Lemma 2.1. If f  C[0, a] then we have
( Ln f (.)  f ( x) )( x)  2( f ; n,2 ( x) ) .
(4)
Proof. By using the well-known technique of
Popovichu [16], we have
 1

( Ln f (.)  f ( x) )( x)  ( f ; )1 
n,2 ( x) .
 

By choosing   n,2 ( x) , we obtain the
quantitative estimate in (4).
Since n,2 ( x)  0 for n   , the estimate in (4)
gives the rate of approximation for the operators in
(1).
Now, we compute this rate of convergence by
means of the second order modulus of continuity.
To obtain this result we will benefit the following
Theorem given by Gonska in [13]:
Theorem B. If L : C ( K )  B( K ) is a positive
linear operator then for f  C ( K ), x  K and
each   0 the following holds


3

( Lf )( x)  f ( x)   ( L  1)   n,2 ( x) max   2 , I ( K )  2 
2



 2 ( f ; )  2  n,1 ( x) max  1, I ( K ) 1 ( f ; )
  n,0 ( x)  1

f  ( f ; )
(5)
2 Direct Results
where n, k ( x) denote the k-th order central
Let us recall that the first and second order
modulus of continuities and Lipschitz classes:
moment of operator L and B(K) denote the Banach
space of bounded and real-valued functions on K,
and I(K) denotes the length of the interval K.
Lemma 2.2. For the operators in (1), if
( f ; )  sup f (t )  f ( x) ; t , x  [0, a], t  x  ,
 f ( x  h)  2 f ( x)  f ( x  h) ;
2 ( f ; )  sup 
( x  h)  [0, a], h  



,


LipM ()  f : f (t )  f ( x)  M t  x , 0    1 ,
 f : f ( x  h)  2 f ( x )  f ( x  h)  M h  , 


*
LipM ()  
.
a
x

[
h
,
a

h
],
0

h



2
f  C[0, a] and
 n, 2 ( x )
2
a
large n then we have
( Ln f )( x)  f ( x)  42 ( f ; n )
where n 
 n, 2 ( x)
2
.
for sufficiently
(6)
Proof. Since n 
large n, we get
 n, 2 ( x)
 a for sufficiently
2


max n 2 , a  2  n 2 .
(7)
If we use n,1( x)  0 , n,0 ( x)  1  0 and (7) in
(5), the proof is completed.
Lemma 2.3. If f  LipM () then the sequence of
( Ln ) satisfies
( Ln f )( x)  f ( x)  M ( n, 2 ( x))

2,
(0    1, 0  x  a).
(8)
Proof. Because of monotonicity of positive linear
operator ( Ln ) , if f  LipM () then we have

( Ln f )( x)  f ( x)  ( Ln f (.)  f ( x) )( x)  ( Ln M 2 .  x )( x).
By Hölder’s inequality for
2
2
1 1
p , q
, ( p, q  1,   1),

2
p q
we obtain (8).
Lemma 2.4. If
n
and
f
 n, 2 ( x )
2
the
sequence
of
( Ln ) satisfies
( Ln f )( x)  f ( x)  4M1 ( n, 2 ( x))

2,
(0    1, 0  x  a).
(9)
Proof. If f  Lip*M1 () then we have
2 ( f , n )  M1n / 2.
 n, 2 ( x)
2
proof is completed.
decreasing in [0,a] according to x,
(ii) n, 2 ( x) is decreasing in n and there is a
positive number c satisfying the
n 1,2 ( x)  c n,2 ( x) for n0  n ,
 K n  m ( x,0,  n n (0)) x .
3 Inverse Results
The equivalence relation
 x(1  x) 
( Bn f )( x)  f ( x)  K 

 n 
Note that all the conditions above are satisfied
for
Bernstein
and
Bernstein-Chlodovsky
polynomials, Szasz-Mirakjan operators and
Baskakov operators.
In this section, we will use the notation
n(n  m)(n  (  1)m)
Pn, ( n ,  n (0), m, x) 
!
 [ n n (0)] K n  m ( x,0,  n n (0)) x 
(10)
and using Lemma 2.2, the

2
 2 ( f ; )  O( )
was proved by Lorentz [15] and Berens and
Lorentz [5] for some  . Later, different proofs
have been given by De Vore [7], Becker [4], Totik
[18], Ditzian and Ivanov [9], Felten [11] for
Bernstein polynomials.
Also Ditzian [8] and Becker [3] obtained
inverse estimation for Szasz-Mirakjan and
Baskakov operators.
inequality
(iii)
d
1

( K n  m ( x,0,  n n (0)) x ) 
( 2
 x)
dx
n,2 ( x) n  n (0)
 a for sufficiently large
 Lip*M1 () then
Choosing n 
These results are starting point of investigations
of the relation between the order of approximation
of some positive linear operators and the
smoothness of the functions approximated.
In this section, we will give some direct and
inverse theorems on the approximation of the
operators in (1).
The theorem, giving us the necessary conditions
for a function to be in the saturation class is called
as inverse theorems. Note that, in general, inverse
theorems are difficult to prove.
We assume the following conditions for the
operators in (1):
(i)
n, 2 ( x) is concave or increasing or
for abbreviation.
Now, we can give the inverse results in the light
of the conditions (i - iii).
Lemma 3.1. If the operators (1) satisfies the
inequality (8) then f  Lip *M () .
Proof. In this proof we will use the similar
technique of Berens-Lorentz theorem in [5]. From
(2) and (iii), we have

d
1

( Ln f )( x) 
)  f ( x))
 (f( 2
dx
 n, 2 ( x)   0
n  n (0)
(

n 2 n (0)
(11)
 x) Pn, ( n ,  n (0), m, x) .
By the basic properties of modulus of continuity
and Cauchy-Schwartz inequality, we have

 1

( Ln f ) / ( x)  ( f ; ) 
x
 2

(
x
)
 n, 2   0 n  n (0)
 1

 1 
x
2
  n  n (0)


 Pn, ( n ,  n (0), m, x) (12)



1
1
 ( f ; ) 
 .
  n, 2 ( x)  
 n, 2 ( x)
x
x
/
/
 ( Ln f ) (t ) dt   ( Ln f ) (t ) dt


 1 
1
1
 y  x ( f ; ) max 
,
 


(
x
)

(
y
)

  
n
,
2
n
,
2


for x  y .
By using the last inequality in the formula
f ( y )  f ( x)  f ( y )  ( Ln f )( y )  f ( x)  ( Ln f )( x)
y
Proof. Because of Lemma 3.1, if we have (9) then
we get f  Lip * () .
4 M1
Thus we can write
Choosing M 2  8M1* , the proof is obvious.
Theorem 3.3. For the operators (1) satisfying the
conditions (i)-(iii), we have
( Ln f )( x)  f ( x)  C1 ( n, 2 ( x))
  ( Ln f ) (t ) dt ,
x

2,
(0    1, 0  x  a) iff f  LipC2 ().
Theorem 3.4. For the operators (1) satisfying the
conditions (i-iii) and
/
 n, 2 ( x)
2
 a for sufficiently
large n, we have
from (4), we get
f ( y )  f ( x)  2M1 ( n, 2 ( x))

2
 y  x ( f ; )


 1 
1
1
 max 
,
  .


(
x
)

(
y
)

  
n
,
2
n
,
2


Because of (ii), we can choose
n,2 (t )    n 1,2 (t )  c n,2 (t )
for all t  [0, a] , then we have
 c  1
f ( y)  f ( x)  2M1  y  x ( f ; ) 
.
  
If we take supremum over y  x  h , we get
h


( f ; h)  M   ( f ; )






where M  max 2M1, c  1 .
Assume that
inequality (9) then f  Lip*M 2 () .
By Lemma 2.3, Lemma 2.4, Lemma 3.1 and
Lemma 3.2, we can give the following theorems
including equivalence relations:
we can write
y
If the operators (1) satisfies the
f ( x  )  2 f ( x)  f ( x  )  8M1* .
Thus by (i) and (12), and concavity of
y
Lemma 3.2.
A  1 such that 2M   A1 .
Choosing h  hn  A n and   hn 1 , since
hn  hn 1 , with the similar calculation to the
Berens-Lorentz Theorem in [5], we have
( f ; h)  M *h
where M *  2M A2 . Taking h  y  x in the last
inequality, the proof is completed.
( Ln f )( x)  f ( x)  C3 ( n, 2 ( x))

2,
*
(0    1, 0  x  a) iff f  LipC4 ().
References:
[1] F. Altomare and M. Campiti, Korovkin Type
Approximation Theory and Its Applications, de
Gruyter Series Studies in Mathematics,
Vol.17, Walter de Gruyter, Berlin-New York,
1994.
[2] V.A. Baskakov, On a Construction of
Converging Sequences of Linear Positive
Operators, Studies of Modern Problems of
Constructive Theory of Functions, Fizmatgiz,
Moskow, 1961, pp.314 - 318. (Russian)
MR#1491.
[3] M. Becker, Global approximation theorems for
Szasz-Mirakjan and Baskakov operators in
polynomial weight spaces, Indiana Univ.
Math. J., Vol. 27, No. 1, 1978, pp.127-142.
[4] M. Becker, An elementary proof of the inverse
theorem
for
Bernstein
polynomials,
Aequationes Math., Vol. 19, 1979, pp.145150.
[5] H. Berens and G. G. Lorentz; Inverse
theorems for Bernstein polynomials, Indiana
Univ. Math. J., Vol. 21, No. 8, 1972, pp.693708.
[6] H. Bohman, On approximation of continuous
and analytic functions, Arkif für Math., Vol. 2,
No. 3, 1951, pp.43-56.
[7] R. A. DeVore, The Approximation of
Continuous Functions by Positive Linear
Operators, Lecture Notes in Mathematics,
Springer-Verlag, 293, Berlin, 1972.
[8] Z. Ditzian, On global inverse theorems of
Szasz and Baskakov Operators, Can. J. Math.,
Vol. 31, No.2, 1979, pp.255-263.
[9] Z. Ditzian and K. Ivanov; Bernstein-type
operators and their derivatives, J. Approx. Th.,
Vol. 56, No.1, 1989, pp.72-90.
[10] O. Doğru, On a certain family of linear
positive operators, Tr. J. of Math., Vol. 21,
No. 4, 1997, pp.387-399.
[11] M. Felten; Direct and inverse estimates for
Bernstein polynomials, Constr. Approx., Vol.
14, 1989, pp.459-468.
[12] A.D. Gadjiev and I.I. Ibragimov, On a
Sequence of Linear Positive Operators, Soviet
Math. Dokl., Vol. 11, 1970, pp.1092-1095.
[13] H.H. Gonska, Quantitative Korovkin type
theorems on simultaneous approximation,
Math. Z., Vol. 186, 1984, pp.419-433.
[14] Korovkin, P. P., Linear Operators and
Approximation Theory, Delhi, 1960.
[15] Lorentz, G. G., Inequalities and the saturation
classes of Bernstein polynomials, in P.L.
Butzer & J. Korevaar, eds., On Approximation
Theory, Proc. Oberwolfach, ISNM 5,
Birkhäuser, Basel, 1964, pp.200-207.
[16] T. Popoviciu; Sur l'approximation des
fonctions
convexes
d'ordre
supérieur,
Mathematica (Cluj), Vol. 10, 1934, pp.49-54.
[17] P. Radatz and B. Wood, Approximating
Derivatives of Functions Unbounded on The
Positive Axis with Linear Operators, Rev.
Roum. Math. Pures et Appl., Bucarest, Vol.
23, No 5, 1978, pp.771 - 781.
[18] V. Totik, An interpolation theorem and its
applications to positive operators, Pacific J.
Math., Vol. 111, 1984, pp.447-481.