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SOME RESULTS CONCERNING MASTROIANNI OPERATORS BY POWER SERIES METHOD
EMRE TA¸S
Abstract. In this paper, we consider power series method which is also mem-ber of the class of all continuous summability methods. We study a Korovkin type approximation theorem for the Mastroianni operators with the use of power series method which includes Abel and Borel methods. We also give some estimates in terms of the modulus of continuity and the second modulus of smoothness.
1. Introduction
In the development of the theory of approximation by positive linear operators, the Korovkin theory has big importance. The classical Korovkin type theorems pro-vide conditions for whether a given sequence of positive linear operators converges to the identity operator in the space of continuous functions on a compact interval [3], [11]. This theory has closely connections with real analysis, functional analysis and summability theory. In approximation theory, in order to correct the lack of convergence summability methods are used since it is well known that they provide a nonconvergent sequence to converge [4], [7], [10]. Also Holhos [9] has given a char-acterization of the functions which is uniformly approximated by Bernstein-Stancu operators.
In this paper, using power series method we give an approximation theorem and quantitative estimates by the Mastroianni operators [13] which contain many well known operators, such as Bernstein polynomials, Baskakov operators and S´zasz-Favard-Mirakjan operators.
First of all we recall some basic de…nitions and notations used in the paper. Let (pj) be real sequence with p1 > 0 and p2; p3; p4::: 0, and such that the
corresponding power series p(t) :=P1j=1pjtj 1 has radius of convergence R with
Received by the editors: Feb. 08, 2016, Accepted: March 22, 2016. 2010 Mathematics Subject Classi…cation. 41A25, 40G10, 41A36.
Key words and phrases. Power series method, Korovkin type approximation theorem, Mas-troianni operators, quantitative estimates.
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0 < R 1. If, for all t 2 (0; R), lim t!R 1 p(t) 1 X j=1 xjpjtj 1= L
then we say that x = (xj) is convergent in the sense of power series method [12],
[16]. Note that the power series method is regular if and only if lim
t!R
pjtj 1
p(t) = 0; f or each j 2 N (1.1)
hold [5]. Throughout the paper we assume that power series method is regular. Let for each j 2 N, j : R+ ! R be an in…nitely di¤erentiable function on
R+:= [0; 1) for which the following conditions hold:
(i) j(0) = 1
(ii) ( 1)m (m)j (x) 0 for every x 2 R+ and m 2 N0
(iii) for every m 2 N0, there exists a positive integer q(j; m) 2 N and a function j;m: R+! R such that j;m(0) = O(jm) as j ! 1 (i+m) j (x) = ( 1)m (i) q(j;m)(x) j;m(x) (1.2)
for every x 2 R+, i 2 N0 and also
lim t!R 1 p(t) 1 X j=1 pjtj 1j q(j; m) = limt!R 1 p(t) 1 X j=1 pjtj 1 j;m(0) jm = 1: (1.3)
In [8] the following facts have been obtained for each m 2 N0 and x 2 R+
(a) j;m(x) 0 f or every j 2 N; (b) (m)j (0) = O(j m
) as j ! 1: (1.4) Lemma 1. The following statements are satis…ed for each m 2 N0,
(i) lim
t!R
[m]
t = 1,
(ii) for each v 2 N, lim t!R 1 p(t) 1 X j=1 ( 1)mpjt j 1 (m) j (0) jm+v = limt!R 1 p(t) 1 X j=1 pjtj 1 j;m(0) jm+v = 0; where [m]t :=p(t)1 P1j=1pjt j 1( 1)m (m) j (0) jm .
Proof. If we choose i = 0, x = 0 in (1.2) and use (i), then we get
(m) j (0) jm = ( 1) m q(j;m)(0) j;m(0) jm = ( 1) m j;m(0) jm :
Hence we obtain 1 p(t) 1 X j=1 ( 1)mpjt j 1 (m) j (0) jm = 1 p(t) 1 X j=1 pjtj 1 j;m(0) jm : (1.5)
Using (1.3) and (1.5), we obtain (i). Since, for each v 2 N, the sequence (1
jv) is null sequence, for a given " > 0, there
exists a positive integer j0= j0("; v) such that
0 1 p(t) 1 X j=1 pjtj 1( 1)m (m)j (0) jm+v 1 p(t) j0 X j=1 pjtj 1( 1)m (m)j (0) jm +" 1 p(t) 1 X j=j0+1 pjtj 1( 1)m (m)j (0) jm :
Using the last inequality and also Lemma 1-(i) and (1.4)-(b), the next inequality is obtained for some M > 0
0 lim t!R 1 p(t) 1 X j=1 pjtj 1( 1)m mj (0) jm+v M limt !R 1 p(t) j0 X j=1 pjtj 1+ "
which implies (ii) by (1.1).
Now let q 2 N0 and consider the following space
Eq := ff 2 C(R+) : lim
x!1
f (x)
1 + xq existsg:
This space is endowed with the norm k:k ; kfk := supx 01+xf (x)q is a Banach space.
The classical Mastroianni operators are given as follows Mj(f; x) = 1 X m=0 ( 1)mf (m j )x m (m) j (x) m!
and map Eq(R+) into C(R+). In case of j(x) = e jx, q(j; m) = j and j;m= jm,
we obtain S´zasz-Favard-Mirakjan operators also if j(x) = (1+x) j, q(j; m) = j+m
and j;m= j(j + 1):::(j + m 1)(1 + x) mthen we obtain Baskakov operators ([1],
[14]).
In this paper we investigate the following operators Mt(f; x) = 1 p(t) 1 X j=1 pjtj 1Mj(f; x):
These operators can be written as Mt(f; x) = 1 p(t) 1 X j=1 pjtj 1 1 X m=0 ( 1)mf (m j )x m (m) j (x) m! :
For the classical Mastroianni operators, we know that Mj(1; x) = 1; Mj(y; x) = 0 j(0) j x; Mj(y2; x) = 00 j(0) j2 x 2 0 j(0) j2 x; Mj((y x)2; x) = ( 00 j(0) j2 + 2 0 j(0) j + 1)x 2 0 j(0) j2 x:
We claim that Mt(eq; x) is well de…ned for the functions eq(y) = yq; q = 0; 1; 2; :::.
It is known from Lemma 3 of [15] that Mj(eq; x) = 1 jq q X v=1 j;v(0) (q; v)xv;
where j;v(0) is the same as in (iii) and (q; v); v = 1; 2; :::q are Stirling numbers
of the second kind. Then we get Mt(eq; x) = 1 p(t) q X v=1 (q; v)xv 1 X j=1 pjtj 1 j;v(0) jq = 1 p(t)x q 1 X j=1 pjtj 1 j;q(0) jq + 1 p(t) q 1 X v=1 (q; v)xv 1 X j=1 pjtj 1 j;v(0) jq :
The last inequality implies that 0 Mt(eq; x) 1 p(t)x qX1 j=1 pjtj 1 j;q(0) jq + 1 p(t) q 1 X v=1 j (q; v)j xv 1 X j=1 pjtj 1 j;v(0) jq
and this proves our claim. So for all f 2 Eq , Mt(f; x) are well de…ned.
Theorem 1. Let q be a positive integer such that q 2: For every f 2 Eq, x 2 R+,
we have
lim
t!R Mt(f; x) = f (x) (1.6)
uniformly on every compact subsets of R+ or equivalently fMj(f; x)gj2N is power
series summable to f (x) uniformly on every compact subsets of R+.
Proof. We follow the similar procedure in the Korovkin type approximation theory. We use the approximation theorems by Altomare-Campiti [2]. Thus, it is enough to prove (1.6) holds for three test functions e0, e1, eq. It is easy to observe that
Mt(e0; x) = 1 p(t) 1 X j=1 pjtj 1fMj(e0; x)g = [0]t :
Hence from the regularity of the power series method we see that lim
t!R Mt(e0; x) = e0(x):
We also see that
Mt(e1; x) = 1 p(t) 1 X j=1 pjtj 1fMj(e1; x)g = x [1]t :
Hence, by Lemma 1-(i), we obtain that lim
t!R Mt(e1; x) = e1(x) = x:
Finally we can write Mt(eq; x) = 1 p(t)x qX1 j=1 pjtj 1 j;q(0) jq + 1 p(t) q 1 X v=1 (q; v)xv 1 X j=1 pjtj 1 j;v(0) jq
and by (1.3) and Lemma1-(ii) we have lim
t!R Mt(eq; x) = x
q = e
q(x):
The pointwise approximation in (1.6) with respect to x becomes uniform on every compact subsets of R+ [1] . This completes the proof.
The next result concerns with the pointwise order of approximation in Theorem 1.
Lemma 2. For every x 2 R+, we have
Mt( 2x; x) 2t(x)
where x(y) := y x and
t(x) := v u u t(x2+ x) maxf [0] t 2 [1] t + [2] t ; 1 p(t) 1 X j=1 pjtj 1 [ 0j(0)] j2 g:
Proof. By simple calculation, we obtain Mt( 2x; x) = ( [0] t 2 [1] t + [2] t )x2+ x 1 p(t) 1 X j=1 pjtj 1 [ 0j(0)] j2
which completes the proof.
By w(f; ) = supjy xj jf(y) f (x)j; > 0; x; y 2 R+ we denote the usual
modulus of continuity of a function f 2 Cb(R+), the space of all continuous and
bounded functions on R+: Now we will estimate the rate of convergence in terms
Theorem 2. Let f 2 Cb(R+), x 2 R+: Then we have
jMt(f; x) f (x)j jf(x)jj [0]t 1j + w(f; t(x))( [0]t +
q
[0]
t ):
Proof. Following Theorem 5.1.2 of [2], we obtain for any > 0, that jMt(f; x) f (x)j jf(x)jjMt(e0; x) 1j + w(f; )(Mt(e0; x))
+w(f; ) pMt(e0; x)
p
Mt( 2x; x):
Then from Lemma 2, taking := t(x), we get
jMt(f; x) f (x)j jf(x)jj [0]t 1j + w(f; t(x)) [0]t +
w(f; )q [0] t 2t(x):
The following result gives an estimation on the space of di¤erentiable functions. Theorem 3. Let f 2 Eq(R+); (q 2 N0) for which f is di¤ erentiable on R+ and
f02 C
b(R+) and for every x 2 R+, we have
jMt(f; x) f (x)j jf(x)jj [0]t 1j+jxf0(x)jj [1] t [0] t j+w(f0; t(x)) t(x)( q [0] t +1):
Proof. Following Theorem 5.1.2 of [2], we obtain for any > 0, that jMt(f; x) f (x)j jf(x)jjMt(e0; x) 1j + w(f0; ) p Mt( 2x; x) p Mt(e0; x) +1 pMt( 2x; x) + jf0(x)jjMt( x; x)j:
By using Lemma 2, we obtain that
jMt(f; x) f (x)j jf(x)jj [0]t 1j + jxf0(x)j [1] t [0] t j +w (f0; t(x)) q 2 t(x) q [0] t + 1 q 2 t(x) :
Now, we consider the approximation property. It is said to be that a function f 2 Eq(R+); (q 2 N0), satis…es the locally Lipschitz condition on a subset U of R+
provided that
jf(x) f (y)j cfjx yj ; (x; y) 2 (R+; U ); (1.7)
holds for some positive constant cf depending on and f , where 2 (0; 1]: The
next result gives an estimation on the class of functions satisfying locally Lipschitz condition.
Theorem 4. For every x 2 R+ the following estimate
jMt(f; x) f (x)j cff( [0]t ) (2 ) 2 t(x) + 2 [0] t d (x; U )g + jf(x)jj [0] t 1j
holds for any function f 2 Eq(R+); (q 2 N0) satisfying the locally Lipschitz
con-dition on a subset U of R+ as in (1.7), where [0]t and t(x) are given before and
d(x; U ) denotes the distance x from U; i.e.,
d(x; U ) := inf fjx yj : y 2 Ug:
Proof. Let x 2 R+. By the de…nition of d(x; U ), there exists a point x02 U, such
that d(x; U ) = jx x0j: Then using the fact that jf(x) f (y)j jf(y) f (x0)j +
jf(x) f (x0)j for any y 2 R+, it follows from the positivity and linearity of the
operators that
jMt(f; x) f (x)j Mt(jf(y) f (x)j; x) + jf(x)jjMt(e0; x) 1j
Mt(jf(y) f (x0)j; x) + jf(x) f (x0)jMt(e0; x)
+ jf(x)jjMt(e0; x) 1j:
By (1.7), we can write that
jMt(f; x) f (x)j cfMt(jy x0j ; x) + cfjx x0jtMt(e0; x) + jf(x)jjMt(e0; x) 1j cfMt(jy xj ; x) + 2cfjx x0jtMt(e0; x) + jf(x)jjMt(e0; x) 1j = cf 1 p(t) 1 X j=1 pjtj 1 1 X m=0 ( 1)mjm j xj xm (m) j (x) m! + 2cfjx x0jtMt(e0; x) + jf(x)jjMt(e0; x) 1j:
Now, using the Hölder’s inequality with the Hölder conjugates 2 and 22 , one can see that jMt(f; x) f (x)j cf( [0]t ) (2 ) 2 f 1 p(t) 1 X j=1 pjtj 1( 1 X m=0 ( 1)m(m j x) 2x m (m) j (x) m! )g 2 + 2cfjx x0jtMt(e0; x) + jf(x)jjMt(e0; x) 1j = cff( [0]t ) (2 ) 2 t Mt2( 2x; x) + 2 [0] t jx x0j g + jf(x)jj [0]t 1j cff( [0]t ) (2 ) 2 t(x) + 2 [0] t jx x0j g + jf(x)jj [0]t 1j:
Therefore, the proof is completed.
Let w2(f; ); > 0 denote the second modulus of smoothness of a function
f 2 Cb(R+): Then we get the following theorem.
Theorem 5. For every f 2 Cb(R+) and x 2 R+ we have
jMt(f; x) f (x)j C( [0]t + 1)fw2(f;
p
t(x)) + t(x)kfkg;
where C is a positive constant and
t(x) := maxfj [0]t 1j; jx( [1] t [0] t )j; 2 t(x) 2 g:
Proof. Let g 2 C2
b(R+): Then, by the Taylor’s formula, one can write that
g(y) = g(x) + (y x)g0(x) +1 2g
002; y 2 R +
where lies between y and x. So,
Mt(g; x) = g(x)Mt(e0; x)) + g0(x)Mt( x; x) +
1 2Mt(
2
xg00( ); x)
which implies that
jMt(g; x) g(x)j kgkj [0]t 1j + kg0kjx( [1] t [0] t )j +kg 00k 2 Mt( 2 x; x) kgkj [0]t 1j + kg0kjx( [1] t [0] t )j +kg 00k 2 2 t(x) t(x)(kgk + kg0k + kg00k) = t(x)kgkC2 b(R+) where kgkC2
b(R+)= (kgk + kg0k + kg00k): Then it is easy to see that
jMt(f; x) f (x)j jMt(f g; x)j + jMt(g; x) g(x)j + jf(x) g(x)j:
By the de…nition of the operators we can write that
jMt(f; x) f (x)j kf gk( [0]t + 1) + t(x)kgkC2 b(R+)
( [0]t + 1)(kf gk + kgkC2
b(R+) t(x))
and also by taking in…mum over g 2 C2
b(R+) we obtain that jMt(f; x) f (x)j ( [0]t + 1)K(f; t(x)) where K(f; ) := inf g2C2 b(R+) fkf gk + kgkC2 b(R+) g
known as the Peetre’s K-functional. Now, using the fact that K(f; ) C(w2(f;
p
) + kfk minf1; g) for some positive constant C independent of ; f [6], we get
jMt(f; x) f (x)j C( [0]t + 1)fw2(f;
p
t(x)) + t(x)kfkg
2. Concluding Remarks In the case of R = 1, p (t) = 1
1 t and for j 1, pj = 1 the power series method coincides with Abel method which is a sequence-to-function transformation.
In the case of R = 1, p (t) = et and for j 1, p
j =
1
(j 1)! the power series method coincides with Borel method.
We can therefore give all of the theorems of this paper for Abel and Borel con-vergences.
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Current address : Department of Mathematics, Ahi Evran University, K¬r¸sehir, Turkey E-mail address : [email protected]
