Volume 10 (2009), Issue 3, Article 61, 8 pp.
APPROXIMATION BY MODIFIED SZÁSZ-MIRAKYAN OPERATORS
L. REMPULSKA AND S. GRACZYK
I NSTITUTE OF M ATHEMATICS
P OZNAN U NIVERSITY OF T ECHNOLOGY
UL . P IOTROWO 3 A , 60-965 P OZNAN
P OLAND
lucyna.rempulska@put.poznan.pl
szymon.graczyk@pisop.org.pl
Received 23 December, 2008; accepted 28 May, 2009
Communicated by I. Gavrea
A BSTRACT. We introduce the modified Szász-Mirakyan operators Sn;r related to the Borel
methods Br of summability of sequences. We give theorems on approximation properties of
these operators in the polynomial weight spaces.
Key words and phrases: Szász-Mirakyan operator, Polynomial weight space, Order of approximation, Voronovskaya type
theorem.
2000 Mathematics Subject Classification. 41A36, 41A25.
1. I NTRODUCTION
The approximation of functions by Szász-Mirakyan operators
∞
X
(nx)k
k
−nx
(1.1)
Sn (f ; x) := e
f
, x ∈ R0 , n ∈ N,
k!
n
k=0
(R0 = [0, ∞), N = {1, 2, ...}) has been examined in many papers and monographs (e.g. [11],
[1], [2], [4], [5]).
The above operators were modified by several authors (e.g. [3], [6], [9], [10], [12]) which
showed that new operators have similar or better approximation properties than Sn . M. Becker
in the paper [1] studied approximation problems for the operators Sn in the polynomial weight
space Cp , p ∈ N0 = N ∪ {0}, connected with the weight function wp ,
(1.2)
w0 (x) := 1,
wp (x) := (1 + xp )−1
if
p ∈ N,
for x ∈ R0 . The Cp is the set of all functions f : R0 → R (R = (−∞, ∞)) for which f wp is
uniformly continuous and bounded on R0 and the norm is defined by
(1.3)
kf kp ≡ kf (·)kp := sup wp (x)|f (x)|.
x∈R0
003-09
2
L. R EMPULSKA AND S. G RACZYK
The space Cpm , m ∈ N, p ∈ N0 , of m-times differentiable functions f ∈ Cp with derivatives
f (k) ∈ Cp , 1 ≤ k ≤ m, and the norm (1.3) was considered also in [1].
In [1] it was proved that Sn is a positive linear operator acting from the space Cp to Cp for
every p ∈ N0 . Moreover, for a fixed p ∈ N0 there exist Mk (p) = const. > 0, k = 1, 2,
depending only on p such that for every f ∈ Cp there hold the inequalities:
(1.4)
kSn (f )kp ≤ M1 (p)kf kp
n ∈ N,
for
and
(1.5)
wp (x) |Sn (f ; x) − f (x)| ≤ M2 (p)ω2
r x
, x ∈ R0 , n ∈ N,
f ; Cp ;
n
where ω2 (f ; Cp ; ·) is the second modulus of continuity of f .
In this paper we introduce the following modified Szász-Mirakyan operators
∞
(1.6)
X (nx)rk
1
Sn;r (f ; x) :=
f
Ar (nx) k=0 (rk)!
rk
n
,
x ∈ R0 , n ∈ N,
for f ∈ Cp and every fixed r ∈ N, where
(1.7)
∞
X
trk
Ar (t) :=
(rk)!
k=0
for
t ∈ R0 .
Clearly A1 (t) = et , A2 (t) = cosh t ≡ 12 (et + e−t ) and Sn;1 (f ; x) ≡ Sn (f ; x) for f ∈ Cp ,
x ∈ R0 and n ∈ N. (The operators Sn;2 were investigated in [9] for functions belonging to
exponential weight spaces.)
We mention that the definition of Sn;r is related to the Borel method of summability of sequences. It is well known ([7]) that a sequence (an )∞
0 , an ∈ R, is summable to g by the Borel
P∞ xrk
method Br , r ∈ N, if the series k=0 (rk)! ak is convergent on R and
lim re−x
x→∞
∞
X
xrk
ak = g.
(rk)!
k=0
In Section 2 we shall give some elementary properties of Sn;r . The approximation theorems
will be given in Section 3.
2. AUXILIARY R ESULTS
It is known ([1]) that for ek (x) = xk , k = 0, 1, 2, there holds: Sn (e0 ; x) = 1, Sn (e1 ; x) = x
and Sn (e2 ; x) = x2 + nx , which imply that
(2.1)
x
Sn (e1 (t) − e1 (x))2 ; x =
n
for
x ∈ R0 , n ∈ N.
Pq
k
Moreover, for every fixed q ∈ N, there exists a polynomial Pq (x) =
k=0 ak x with real
coefficients ak , aq 6= 0, depending only on q such that
(2.2)
Sn (e1 (t) − e1 (x))2q ; x ≤ Pq (x)n−q for x ∈ R0 , n ∈ N.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 61, 8 pp.
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A PPROXIMATION BY M ODIFIED S ZÁSZ -M IRAKYAN O PERATORS
3
From (1.1) – (1.4), (1.6) and (1.7) we deduce that Sn;r is a positive linear operator well
defined on every space Cp , p ∈ N0 , and
Sn;r (e0 ; x) = 1,
(2.3)
x A0r (nx)
,
n Ar (nx)
x A0 (nx)
x2 A00 (nx)
(2.5)
+ 2 r
,
Sn;r (e2 ; x) = 2 r
n Ar (nx) n Ar (nx)
for x ∈ R0 and n, r ∈ N, and
(2.4)
Sn;r (e1 ; x) =
(2.6)
Sn;r (f ; 0) = f (0)
for
f ∈ Cp , n, r ∈ N.
Here we derive a simpler formula for Ar .
Lemma 2.1. Let r ∈ N be a fixed number. Then Ar defined by (1.7) can be rewritten in the
form: A1 (t) = et , A2 (t) = cosh t,
"
#
m−1
X
1
kπ
kπ
(2.7)
A2m (t) =
cosh t +
exp t cos
cos t sin
,
m
m
m
k=1
for 2 ≤ m ∈ N, and
(2.8)
"
#
m
X
1
2kπ
2kπ
t
A2m+1 (t) =
e +2
exp t cos
cos t sin
,
2m + 1
2m + 1
2m + 1
k=1
for m ∈ N and t ∈ R0 .
Proof. The formulas for A1 and A2 are obvious by (1.7). For r ≥ 3 and t ∈ R0 we have
∞
∞
∞
X
X
X
trk
trk+1
trk+r−1
t
e =
+
+ ··· +
(rk)! k=0 (rk + 1)!
(rk + r − 1)!
k=0
k=0
which by (1.7) can be written in the form
Z t
Z t Z v1
t
e = Ar (t) +
Ar (u) du +
Ar (u) du dv1
0
0
0
Z tZ
+ ··· +
0
v1
Z
...
0
vr−2
Ar (u) du dvr−2 . . . dv1 .
0
By (r − 1)-times differentiation we get the equality
et = A(r−1)
(t) + A(r−2)
(t) + · · · + A0r (t) + Ar (t) for
r
r
t ∈ R0 ,
which shows that y = Ar (t) is the solution of the differential equation
(2.9)
y (r−1) + y (r−2) + · · · + y 0 + y = et
satisfying the initial conditions
(2.10)
y(0) = 1,
y 0 (0) = y 00 (0) = · · · = y (r−2) (0) = 0.
Using now the Laplace transformation
Z
L[y(t)] = Y (s) :=
∞
y(t)e−st dt,
s = x + iy,
0
we have by (2.10)
L y (k) (t) = sk Y (s) − sk−1
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 61, 8 pp.
for
k = 1, . . . , r − 1,
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4
L. R EMPULSKA AND S. G RACZYK
and consequently we get from (2.7)
sr−1 + sr−2 + · · · + s + 1 Y (s) =
1
+ sr−2 + sr−3 + · · · + s + 1,
s−1
and
Y (s) =
(2.11)
sr−1
.
sr − 1
By the inverse Laplace transformation we get
r−1 s
−1
(2.12)
y(t) = L
sr − 1
for
t ∈ R0 ,
and this L−1 transform can be calculated by the residues of Y .
P (s)
It is known that the inverse transform of a rational function Q(s)
with the simple poles sk can
be written as follows
X
X∗∗ P (sk )esk t
∗ P (sk )esk t
−1 P (s)
(2.13)
L
=
+
2re
,
0 (s )
0 (s )
Q(s)
Q
Q
k
k
sk
sk
P∗
P∗∗
where
denotes the sum for all real sk and
denotes the sum for all complex sk = xk +iyk
with a positive yk .
√
The function Y defined by (2.11) has the simple poles sk = r 1 = e2kπi/r for k = 0, 1, . . . , r−
1. From this and (2.12) and (2.13) for r = 2m, 2 ≤ m ∈ N, we get
!
X∗∗
1 X ∗ sk t
y(t) =
e + 2re
esk t
2m s
sk
k
"
#
m−1
X
1
kπ
kπ
=
cosht +
exp t cos
cos t sin
.
m
m
m
k=1
This shows that the formula (2.7) is proved.
Analogously by (2.12) and (2.13) we obtain (2.8).
From (2.7) and (2.8) we have that
A3 (t) =
1
3
et + 2e−t/2 cos
√ !!
3
t
,
2
1
A4 (t) = (cosht + cos t),
2
1
t
A6 (t) =
cosht + 2cosh cos
3
2
√ !!
3
t
,
2
for
t ∈ R0 .
Applying the formula (1.7) and Lemma 2.1, we immediately obtain the following:
Lemma 2.2. For every fixed r ∈ N there exists a positive constant M3 (r) depending only on r
such that
enx
(2.14)
1≤
≤ M3 (r) for x ∈ R0 , n ∈ N.
Ar (nx)
Lemma 2.3. Let r ∈ N. Then for e1 (x) = x there holds
(2.15)
lim nSn;r (e1 (t) − e1 (x); x) = 0
n→∞
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A PPROXIMATION BY M ODIFIED S ZÁSZ -M IRAKYAN O PERATORS
5
and
lim nSn;r (e1 (t) − e1 (x))2 ; x = x,
n→∞
at every x ∈ R0 . Moreover, we have
(2.16)
Sn;r (e1 (t) − e1 (x))2q ; x ≤ M3 (r)Sn (e1 (t) − e1 (x))2q ; x
for x ∈ R0 , n ∈ N and every fixed q ∈ N.
Proof. The inequality (2.16) is obvious by (1.1), (1.6) and (2.14).
We shall prove only (2.15) for r = 2m, m ∈ N.
If r = 2 then A2 (t) = cosh t and by (2.4) we have
sinh nx
Sn;2 (e1 (t) − e1 (x); x) = x
−1
cosh nx
−2x
= 2nx
for x ∈ R0 , n ∈ N,
e +1
which implies (2.15).
If r = 2m with 2 ≤ m ∈ N, then by (2.4), (2.7) and (2.14) we get
1 0
x
|Sn;2m (e1 (t) − e1 (x); x)| =
A (nx) − A2m (nx)
A2m (nx) n 2m
=
x
mA2m (nx)
sinh nx − cosh nx
m−1
X
kπ
kπ
kπ
cos
cos nx sin
+
exp nx cos
m
m
m
k=1
kπ
kπ
kπ
− sin
sin nx sin
− cos nx sin
m
m
m
"
#
m−1
X
x −2nx
2 kπ
≤ M3 (2m)
e
+3
exp −2nx sin
m
m
k=1
and from this we immediately obtain (2.15).
From (1.6), (1.1) – (1.4) and (2.14) the following lemma results.
Lemma 2.4. The operator Sn;r , n, r ∈ N, is linear and positive, and acts from the space Cp to
Cp for every p ∈ N0 . For f ∈ Cp
kSn;r (f )kp ≤ kf kp kSn;r (1/wp )kp
≤ M3 (r)kf kp · kSn (1/wp )kp ≤ M4 (p, r)kf kp
f or
n, r, ∈ N,
where M4 (p, r) = M1 (p)M3 (r) and M1 (p), M3 (r) are positive constants given in (1.4) and
(2.14).
3. T HEOREMS
First we shall prove two theorems on the order of approximation of f ∈ Cp by Sn;r , r ≥ 2.
Theorem 3.1. Let p ∈ N0 and 2 ≤ r ∈ N be fixed numbers. Then there exists M5 (p, r) =
const. > 0 (depending only on p and r) such that for every f ∈ Cp1 there holds the inequality
r
x
0
(3.1)
wp (x) |Sn;r (f ; x) − f (x)| ≤ M5 (p, r)kf kp
,
n
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6
L. R EMPULSKA AND S. G RACZYK
for x ∈ R0 and n ∈ N.
Proof. Let f ∈ Cp1 . Then by (1.6), (1.7) and (2.14) it follows that
|Sn;r (f ; x) − f (x)| ≤ Sn;r (|f (t) − f (x)| ; x)
≤ M3 (r)Sn (|f (t) − f (x)|; x)
for x ∈ R0 , n ∈ N,
and for t, x ∈ R0
Z
|f (t) − f (x)| =
t
0
0
f (u) du ≤ kf kp
x
1
1
+
wp (t) wp (x)
|t − x|.
Using now the operator Sn , (1.1) – (1.4) and (2.1), we get
|t − x|
0
wp (x)Sn (|f (t) − f (x)|; x) ≤ kf kp wp (x)Sn
; x + Sn (|t − x|; x)
wp (t)
o
1/2 n
1/2
≤ kf 0 kp Sn (t − x)2 ; x
2kSn (1/w2p )k2p + 1
r
p
x
0
≤ 2 M1 (2p) + 1 kf kp
for x ∈ R0 , n ∈ N.
n
Combining the above, we obtain the estimation (3.1).
Theorem 3.2. Let p ∈ N0 and 2 ≤ r ∈ N be fixed. Then there exists M6 (p, r) = const. > 0
(depending only on p and r) such that for every f ∈ Cp , x ∈ R0 and n ∈ N there holds
r x
(3.2)
wp (x) |Sn;r (f ; x) − f (x)| ≤ M6 (p, r)ω1 f ; Cp ;
,
n
where ω1 (f ; Cp ; ·) is the modulus of continuity of f ∈ Cp , i.e.
(3.3)
ω1 (f ; Cp ; t) := sup k∆u f (·)kp
for
t ≥ 0,
0≤u≤t
and ∆u f (x) = f (x + u) − f (x).
Proof. The inequality (3.2) for x = 0 follows by (1.2), (2.6) and (3.3).
Let f ∈ Cp and x > 0. We use the Steklov function fh ,
Z
1 h
fh (x) :=
f (x + t) dt for x ∈ R0 , h > 0.
h 0
This fh belongs to the space Cp1 and by (3.3) it follows that
(3.4)
kf − fh kp ≤ ω1 (f ; Cp ; h)
and
(3.5)
kfh0 kp ≤ h−1 ω1 (f ; Cp ; h),
for
h > 0.
By the above properties of fh and (2.3) we can write
Sn;r f (t); x − f (x) ≤ |Sn;r (f (t) − fh (t); x)| + Sn;r fh (t); x − fh (x) + |fh (x) − f (x)| ,
for n ∈ N and h > 0. Next, by Lemma 2.4 and (3.4) we get
wp (x) Sn;r f (t) − fh (t); x ≤ M4 (p, r)kf − fh kp ≤ M4 (p, r)ω1 (f ; Cp ; h).
In view of Theorem 3.1 and (3.5) we have
r
r
x
x
−1
0
≤ M5 (p, r)h
ω1 (f ; Cp ; h).
wp (x) Sn;r fh ; x − fh (x) ≤ M5 (p, r)kfh kp
n
n
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A PPROXIMATION BY M ODIFIED S ZÁSZ -M IRAKYAN O PERATORS
7
Consequently,
(3.6)
−1
wp (x) |Sn;r (f ; x) − f (x)| ≤ ω1 (f ; Cp ; h) M4 (p, r) + M5 (p, r)h
for x > 0, n ∈ N and h > 0. Putting h =
desired estimation (3.2).
p
r
x
+1 ,
n
x/n in (3.6) for given x and n, we obtain the
Theorem 3.2 implies the following:
Corollary 3.3. If f ∈ Cp , p ∈ N0 , and 2 ≤ r ∈ N, then
lim Sn;r (f ; x) = f (x) at every
n→∞
x ∈ R0 .
This convergence is uniform on every interval [x1 , x2 ], x1 ≥ 0.
The Voronovskaya type theorem given in [1] for the operators Sn can be extended to Sn;r
with r ≥ 2.
Theorem 3.4. Suppose that f ∈ Cp2 , p ∈ N0 , and 2 ≤ r ∈ N. Then
lim n (Sn;r (f ; x) − f (x)) =
(3.7)
n→∞
x 00
f (x)
2
at every x ∈ R0 .
Proof. The statement (3.7) for x = 0 is obvious by (2.6). Choosing x > 0, we can write the
Taylor formula for f ∈ Cp2 :
1
f (t) = f (x) + f 0 (x) + f 00 (x)(t − x)2 + ϕ(t, x)(t − x)2
2
for
t ∈ R0 ,
where ϕ(t) ≡ ϕ(t, x) is a function belonging to Cp and lim ϕ(t) = ϕ(x) = 0.
t→x
Using now the operator Sn;r and (2.3), we get
1
Sn;r (f (t); x) = f (x)+f 0 (x)Sn;r (t−x; x)+ f 00 (x)Sn;r ((t−x)2 ; x)+Sn;r ϕ(t)(t − x)2 ; x ,
2
for n ∈ N, which by Lemma 2.3 implies that
x
(3.8)
lim n (Sn;r (f (t); x) − f (x)) = f 00 (x) + lim nSn;r ϕ(t)(t − x)2 ; x .
n→∞
n→∞
2
It is clear that
1/2
(3.9)
Sn;r ϕ(t)(t − x)2 ; x ≤ Sn;r (ϕ2 (t); x)Sn;r ((t − x)4 ; x)
,
and by Corollary 3.3
(3.10)
lim Sn;r (ϕ2 (t); x) = ϕ2 (x) = 0.
n→∞
∞
Moreover, by (2.16) and (2.2) we deduce that the sequence (n2 Sn;r ((t − x)4 ; x))1 is bounded
at every fixed x ∈ R0 . From this and (3.9) and (3.10) we get
lim nSn;r ϕ(t)(t − x)2 ; x = 0
n→∞
which with (3.8) yields the statement (3.7).
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 61, 8 pp.
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L. R EMPULSKA AND S. G RACZYK
4. R EMARKS
Remark 1. We observe that the estimation (1.5) for the operators Sn is better than (3.2) obtained
for Sn;r with r ≥ 2. It is generated by formulas (2.3) – (2.5) and Lemma 2.1 which show that
the operators Sn;r , r ≥ 2, preserve only the function e0 (x) = 1. The operators Sn preserve the
function ek (x) = xk , k = 0, 1.
Remark 2. In the paper [2], the approximation properties of the Szász-Mirakyan operators Sn
in the exponential weight spaces Cq∗ , q > 0, with the weight function vq (x) = e−qx , x ∈ R0
were examined. Obviously the operators Sn;r , r ≥ 2, can be investigated also in these spaces.
Remark 3. G. Kirov in [8] defined the new Bernstein polynomials for m-times differentiable
functions and showed that these operators have better approximation properties than classical
Bernstein polynomials.
The Kirov idea was applied to the operators Sn in [10].
We mention that the Kirov method can be extended to the operators Sn;r with r ≥ 2, i.e. for
functions f ∈ Cpm , m ∈ N, p ∈ N0 , and a fixed 2 ≤ r ∈ N we can consider the operators
j
∞
m
X
1
rk
(nx)rk X f (j) rk
∗
n
Sn;r (f ; x) :=
−x ,
j!
n
Ar (nx) k=0 (rk)! j=0
for x ∈ R0 and n ∈ N.
∗
In [10] it was proved that the Sn;1
have better approximation properties for f ∈ Cpm , m ≥ 2,
than Sn;1 .
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J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 61, 8 pp.
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