statistical fuzzy approximation to fuzzy differentiable functions by

Hacettepe Journal of Mathematics and Statistics
Volume 39 (4) (2010), 497 – 514
STATISTICAL FUZZY APPROXIMATION TO
FUZZY DIFFERENTIABLE FUNCTIONS BY
FUZZY LINEAR OPERATORS
Oktay Duman∗
Received 10 : 09 : 2009 : Accepted 18 : 03 : 2010
Abstract
In this paper, we obtain fuzzy approximations to fuzzy differentiable
functions by means of fuzzy linear operators whose positivity condition
and classical limits fail. In order to get more powerful results than
the classical approach we investigate the effects of matrix summability
methods on the fuzzy approximation. So, we mainly use the notion of
A-statistical convergence from summability theory instead of the usual
convergence.
Keywords: Korovkin theorem, Statistical convergence, Fuzzy approximation, Fuzzy
continuity, Fuzzy differentiable function.
2000 AMS Classification: 26 E 50, 41 A 25, 41 A 36.
1. Introduction
The classical Korovkin theory is mainly based on two conditions: positivity of linear operators and existence of their classical limits (see [1, 28]). So far, the former
has been weakened by considering monotonicity and convexity of the functions being approximated (see, for instance, [2, 8, 9, 26]). Furthermore, by using the notion of statistical convergence, some Korovkin–type approximation theorems have been
obtained when the classical limit of a sequence of positive linear operators fails (see
[6, 10, 11, 12, 14, 15, 16, 17, 25]). Their fuzzy analogs have also been introduced in
[5, 13, 21]. In this study, we obtain various fuzzy approximation results without these
two conditions as mentioned above. We use the notions of statistical convergence of
sequences of fuzzy numbers and fuzzy differentiability of fuzzy real valued functions to
get our theorems. When proving our results, we use not only classical techniques from
This research was supported by the Scientific and Technological Research Council of Turkey
(TUBITAK); Project No: 108T229.
∗
TOBB Economics and Technology University, Faculty of Arts and Sciences, Department of
Mathematics, 06530 Söğütözü, Ankara, Turkey. E-mail: [email protected]
498
O. Duman
approximation theory but also new methods from summability theory and fuzzy logic
theory.
We first recall some basic concepts used in the present paper.
The (asymptotic) density of a subset K of N, the set of all natural numbers, is defined
by
δ(K) := lim
j
# {n ≤ j : n ∈ K}
,
j
provided the limit exists, where the symbol # {B} denotes the cardinality of the set B.
Then, we say that a sequence (xn )n∈N is statistically convergent to a number L (see [18])
if, for every ε > 0, the set {n ∈ N : |xn − L| ≥ ε} has density zero, i.e.,
δ ({n ∈ N : |xn − L| ≥ ε}) = lim
j
# {n ≤ j : |xn − L| ≥ ε}
= 0.
j
Now let A = (ajn ) be an infinite non-negative regular summability matrix. Recall that A
is said to be regular if limj (Ax)j = L whenever
P limj xj = L, where Ax := ((Ax)j ) denotes
the A-transform of x given by (Ax)j = ∞
n=1 ajn xn , provided the series converges for
each j (see, for instance, [7]). Then, the A-density of a subset K is defined by
X
δA (K) := lim
ajn
j
n∈K
provided the limit exists. Observe that if we take A = C1 = (cjn ), the Cesáro matrix of
order one, defined by

 1 , if 1 ≤ n ≤ j,
cjn := j
0, otherwise,
then C1 -density coincides with (asymptotic) density. With the help of A-density, Freedman and Sember [19] introduced the notion of A-statistical convergence, which is a more
general method of statistical convergence. Recall that the sequence (xn )n∈N is said to
be A-statistically convergent to L if, for every ε > 0, δA {n ∈ N : |xn − L| ≥ ε} = 0; or
equivalently
X
lim
ajn = 0.
j
n: |xn −L|≥ε
This limit is denoted by stA -limn xn = L. It is not hard to see that if we take A = C1 , then
C1 -statistical convergence coincides with the statistical convergence mentioned above. If
A is replaced by the identity matrix, then we get the ordinary convergence of number
sequences. We also note that if A = (ajn ) is any non-negative regular summability
matrix for which limj maxn {ajn } = 0, then A-statistical convergence is stronger than
convergence (see [27]). Actually, every convergent sequence is A-statistically convergent
to the same value for any non-negative regular matrix A, but its converse is not always
true. Some other results regarding statistical and A-statistical convergence may be found
in the papers [20, 30].
As usual, a fuzzy number is a function µ : R → [0, 1], which is normal, convex, upper
semi-continuous and the closure of the set supp(µ) is compact, where supp(µ) := {x ∈
R : µ(x) > 0}. We denote the set of all fuzzy numbers by RF . Let
[µ]0 := {x ∈ R : µ(x) > 0} and [µ]r := {x ∈ R : µ(x) ≥ r}, (0 < r ≤ 1).
Then, it is well-known [22] that, for each r ∈ [0, 1], the set [µ]r is a closed and bounded
interval of R. For any u, v ∈ RF and λ ∈ R, it is possible to define uniquely the sum
Statistical Fuzzy Approximation
499
u ⊕ v and the product λ ⊙ v as follows:
[u ⊕ v]r = [u]r + [v]r and [λ ⊙ u]r = λ[u]r , (0 ≤ r ≤ 1).
(r)
(r)
(r)
Now denote the interval [u]r by [u(r) , u+ ], where u(r) ≤ u+ and u(r) , u+
r ∈ [0, 1]. Then, for u, v ∈ RF , define
(r)
∈ R for
(r)
u v ⇐⇒ u(r) ≤ v (r) and u+ ≤ v+ for all 0 ≤ r ≤ 1.
Define also the following metric D : RF × RF → R+ by
o
n
(r)
(r) D(u, v) = sup max u(r) − v (r) , u+ − v+ .
r∈[0,1]
In this case, it is known [32] that (RF , D) is a complete metric space on RF with the
properties
D(u ⊕ w, v ⊕ w) = D(u, v) for all u, v, w ∈ RF ,
D(λ ⊙ u, λ ⊙ v) = |λ| D(u, v) for all u, v ∈ RF and λ ∈ R,
D(u ⊕ v, w ⊕ z) ≤ D(u, w) + D(v, z) for all u, v, w, z ∈ RF .
Let f, g : [a, b] ⊂ R → RF be fuzzy valued functions. Then, the distance between f and
g on [a, b] is given by
n
(r)
o
(r)
D∗ (f, g) = sup sup max f (r) (x) − g (r) (x), f+ (x) − g+ (x) .
x∈[a,b] r∈[0,1]
Now let (µn )n∈N be a fuzzy number valued sequence. Then, Nuray and Savaş [31] introduced the fuzzy analog of statistical convergence by using the fuzzy metric D instead of
the classical absolute value in the above definition. So, by a similar idea, one can obtain
the following definition of A-statistical convergence of fuzzy valued sequences. We say
that a sequence (µn )n∈N of fuzzy numbers is A-statistically convergent to µ ∈ RF , which
is denoted by stA -limn D(µn , µ) = 0, if for every ε > 0, δA ({n ∈ N : D(µn , µ) ≥ ε}) = 0,
i.e.,
X
lim
ajn = 0
j
n:D(µn ,µ)≥ε
holds. Of course, the case of A = C1 immediately reduces to the statistical convergence
of fuzzy valued sequences. Also, replacing A with the identity matrix, we get the classical
fuzzy convergence introduced by Matloka [29].
2. Statistical Fuzzy Approximation
A fuzzy valued function f : [a, b] → RF is said to be fuzzy continuous at x0 ∈ [a, b]
provided that, whenever xn → x0 , then D (f (xn ), f (x0 )) → 0 as n → ∞. Also, we say
that f is fuzzy continuous on [a, b] if it is fuzzy continuous at every point x ∈ [a, b]. The
set of all fuzzy continuous functions on [a, b] is denoted by CF [a, b] (see, for instance,
[3, 4]). Notice that CF [a, b] is only a cone, not a vector space.
Following [24, 33], a fuzzy valued function f : [a, b] → RF is said to satisfy the condition
(H) on [a, b] if, for any x, y ∈ [a, b] satisfying x ≤ y, there exists u ∈ RF such that
f (y) = f (x) + u.
In this case, we call u the H-difference (or, Henstock-difference) of f (y) and f (x), and
denote it by f (y) − f (x). For brevity, throughout this paper, when we use the “−”
operation of fuzzy numbers, we always assume that the condition (H) is satisfied.
500
O. Duman
Assume that f : [a, b] → RF is a fuzzy valued function. Then f is called fuzzy
differentiable at x ∈ (a, b) if there exists a f ′ (x) ∈ RF such that the following limits
lim
h→0+
f (x + h) − f (x)
f (x) − f (x − h)
, lim
h
h
h→0+
exist and are equal to f ′ (x). Also, if x = a or x = b, then we use the following
f ′ (a) := lim
h→0+
f (a + h) − f (a)
f (b) − f (b − h)
and f ′ (b) := lim
h
h
h→0+
provided that the above limits exist. Observe that the limits are taken in the metric
space (RF , D). If f is (fuzzy) differentiable at every point x ∈ [a, b], then we say that
f is (fuzzy) differentiable on [a, b] with derivative f ′ (see [32]). It follows from [23] that
a function f : [a, b] → RF is fuzzy differentiable at x ∈ [a, b], and f ′ (x) is the fuzzy
derivative, if and only if for every ε > 0 there exists a δ > 0 such that, for any interval
[x1 , x2 ] ⊂ (x − δ, x + δ) we have
f (x2 ) − f (x1 ) ′
, f (x) < ε.
D
x2 − x1
Similarly, we can define higher order fuzzy derivatives. Also, by CFm [a, b], (m ∈ N), we
mean the set of all fuzzy valued functions from [a, b] into RF that are m-times continuously
differentiable in the fuzzy sense. However, in this paper, we only discuss the cases
m = 0, 1, 2.
Using these definitions, Kaleva [24] proved the following result.
2.1. Lemma. [24] Let f : [a, b] ⊂ R → RF be fuzzy differentiable, and x ∈ [a, b],
0 ≤ r ≤ 1. Then, clearly
[f (x)]r = [(f (x))(r) , (f (x))(r)
+ ] ⊂ R.
Then (f (x))(r)
± is differentiable and
(r) ′
[f (x)]r = ((f (x))(r) )′ , ((f (x))+ )′ ,
i.e.,
′
(r)
(r)
(f ′ )± = f±
for any r ∈ [0, 1].
Also, for higher order fuzzy derivatives, Anastassiou [4] gave the similar result:
(r)
2.2. Lemma. [4] Let k ∈ N and f ∈ CFk [a, b]. Then, we have f± ∈ C k [a, b] (for any
r ∈ [0, 1]) and
h
i
(i)
(r) (i)
[f (i) (x)]r = (f (x))(r)
, (f (x))+
for i = 0, 1, . . . , k, and, in particular, we get
(r)
(r) (i)
f (i) ± = f±
for any r ∈ [0, 1] and i = 0, 1, . . . , k.
In this paper, for simplicity, we use the unit interval [0, 1] instead of [a, b]. Now let
L : CF [0, 1] → CF [0, 1] be an operator. Then L is said to be fuzzy linear if, for every
λ1 , λ2 ≥ 0, f1 , f2 ∈ CF [0, 1], and x ∈ [0, 1],
L (λ1 ⊙ f1 ⊕ λ2 ⊙ f2 ; x) = λ1 ⊙ L(f1 ; x) ⊕ λ2 ⊙ L(f2 ; x)
holds. Let k be a non-negative integer. As usual, by C k [0, 1] we denote the space of
all k-times continuously differentiable functions (in the usual sense) on [0, 1], endowed
Statistical Fuzzy Approximation
501
with the sup-norm k · k. Then, throughout the paper, we consider the following function
spaces:
A := f ∈ C 2 [0, 1] : f ≥ 0 ,
B := f ∈ C 2 [0, 1] : f ′′ ≥ 0 ,
C := f ∈ C 2 [0, 1] : f ′′ ≤ 0 ,
D := f ∈ C 1 [0, 1] : f ≥ 0 ,
E := f ∈ C 1 [0, 1] : f ′ ≥ 0 ,
F := {f ∈ C[0, 1] : f ≥ 0} .
We also consider the test functions
ei (y) = y i (i = 0, 1, 2, . . .) for every y ∈ [0, 1].
Then we have following results.
2.3. Theorem. Let A = (ajn ) be a non-negative regular summability matrix, and
{Ln }n∈N a sequence of fuzzy linear operators from CF2 [0, 1] onto itself. Assume that there
exists a corresponding sequence {L̃n }n∈N of linear operators from C 2 [0, 1] onto itself for
which the following conditions hold:
(r)
(r)
(2.1)
Ln (f ; x)± = L̃n f± ; x (for x ∈ [0, 1], r ∈ [0, 1], n ∈ N)
and
(2.2)
If
(2.3)
n
o
δA n ∈ N : L̃n (A ∩ B) ⊂ A = 1.
stA - lim L̃n (ei ) − ei = 0 for i = 0, 1, 2,
n
then we have
(2.4)
stA - lim D∗ (Ln (f ), f ) = 0 for all f ∈ CF2 [0, 1].
n
Proof. Let x ∈ [0, 1] be fixed, and f ∈ CF2 [0, 1] and r ∈ [0, 1]. Then we may write that,
for every ε > 0 there exists a δ > 0 such that, for every y ∈ [0, 1],
D (f (y), f (x)) < ε
holds. So, for every y, r ∈ [0, 1] and for any β ≥ 1, we have
(r)
(2.5)
(r)
where M±
and
(r)
2M± β
2M± β
(r)
(r)
ϕx (y) ≤ f± (y) − f± (x) ≤ ε +
ϕx (y),
δ2
δ2
(r) = f± and ϕx (y) = (y − x)2 . Then, by (2.5), we obtain that
−ε −
(r)
gβ
(r)
±
(y) :=
2M± β
(r)
(r)
ϕx (y) + ε − f± (y) + f± (x) ≥ 0
δ2
(r)
2M± β
(r)
(r)
ϕx (y) + ε + f± (y) − f± (x) ≥ 0
δ2
±
(r)
(r)
hold for all y ∈ [0, 1]. So, the functions gβ
and hβ
belong to A. On the other
(r)
hβ
(y) :=
±
hand, it is clear that, for all y ∈ [0, 1],
(r)
4M± β h (r) i′′
(r)
gβ
(y) =
− f±
(y)
δ2
±
±
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O. Duman
and
(r)
4M± β h (r) i′′
+ f±
(y).
δ2
±
If we choose the number β such that
 h
i′′ 
2
(r)

 f±
δ 
(2.6)
β ≥ max 1,
,
(r)


4M±


(r)
hβ
′′
(y) =
(r)
(r)
we observe that (2.5) holds for such β’s and also the functions gβ
and hβ
±
±
′′
′′
(r)
(r)
belong to B because of gβ
(y) ≥ 0 and hβ
(y) ≥ 0 for all y ∈ [0, 1]. So, we
±
±
(r)
(r)
have gβ
, hβ
∈ A ∩ B under the condition (2.6). Let
±
±
K := {n ∈ N : L̃n (A ∩ B) ⊂ A}.
By (2.2), it is clear that δA {K} = 1, and so
(2.7)
δA {N \ K} = 0.
Then, we may write that
(r)
(r)
L̃n gβ
; x ≥ 0 and L̃n hβ
; x ≥ 0 for any n ∈ K.
±
±
Now using the fact that ϕx ∈ A ∩ B, and considering the linearity of L̃n , we obtain, for
every n ∈ K, that
(r)
and
2M± β
(r)
(r)
L̃n (ϕx ; x) + εL̃n (e0 ; x) − L̃n f± ; x + f± (x)L̃n (e0 ; x) ≥ 0
δ2
(r)
2M± β
(r)
(r)
L̃n (ϕx ; x) + εL̃n (e0 ; x) + L̃n f± ; x − f± (x)L̃n (e0 ; x) ≥ 0,
δ2
or equivalently
(r)
−
2M± β
(r)
L̃n ϕx ; x − εL̃n (e0 ; x) + f± (x) L̃n (e0 ; x) − e0
δ2
(r)
(r)
≤ L̃n f± ; x − f± (x)
(r)
≤
Then, we have
2M± β
(r)
L̃n (ϕx ; x) + εL̃n (e0 ; x) + f± (x) L̃n (e0 ; x) − e0 .
2
δ
(r)
L̃n (f (r) ; x) − f (r) (x) ≤ ε + 2M± β L̃n ϕx ; x
±
±
δ2
(r) + ε + f± (x) L̃n (e0 ; x) − e0 for every n ∈ K. The last inequality gives that, for every ε > 0 and n ∈ K,
(r)
(r)
(r)
L̃n (f±
) − f± ≤ ε + (ε + M± )L̃n (e0 ) − e0 +
(r)
(r)
2M± β L̃n (e2 ) − e2 + 4M± β L̃n (e1 ) − e1 2
2
δ
δ
(r)
2M± β L̃n (e0 ) − e0 .
+
δ2
Statistical Fuzzy Approximation
503
Hence, we get, for any n ∈ K, that
(2.8)
2
X
(r)
(r)
(r)
L̃n (f±
L̃n (ek ) − ek ,
) − f± ≤ ε + C± (ε)
k=0
(r) 4M± β
. Now it follows from (2.1) that
δ2
D∗ Ln (f ), f = sup D Ln (f ; x), f (x)
(r)
(r)
where C± (ε) := max ε + M± +
(r)
2M± β
,
δ2
x∈[0,1]
= sup
n
sup max L̃n f (r) ; x − f (r) (x),
x∈[0,1] r∈[0,1]
n
= sup max L̃n
r∈[0,1]
o
(r)
(r)
L̃n f+
; x − f+ (x)
o
(r) (r)
f (r) − f (r) , L̃n f+ − f+ .
Combining the above equality with (2.8), we have, for any n ∈ K,
(2.9)
D∗ (Ln (f ), f ) ≤ ε + C(ε)
2
X
L̃n (ek ) − ek ,
k=0
where C(ε) = sup max
r∈[0,1]
′
n
(r)
(r)
C− (ε), C+ (ε)
o
, Now, for a given ε′ > 0, choose an ε > 0
such that ε < ε , and define the following sets:
F := n ∈ N : D∗ (Ln (f ), f ) ≥ ε′ ,
n
ε′ − ε o
Fk := n ∈ N : L̃n (ek ) − ek ≥
, k = 0, 1, 2.
3C(ε)
Then, it follows from (2.9) that
F ∩K ⊂
2
[
(Fk ∩ K),
k=0
which yields, for every j ∈ N, that
X
2 2 X
X
X
X
ajn .
(2.10)
ajn ≤
ajn ≤
n∈F ∩K
k=0
n∈Fk ∩K
k=0
n∈Fk
Now, taking the limit as j → ∞ on both-sides of (2.10), and using (2.3), we immediately
see that
X
(2.11) lim
ajn = 0.
j
n∈F ∩K
Furthermore, since
X
X
ajn =
ajn +
n∈F ∩K
n∈F
≤
X
n∈F ∩K
ajn +
X
ajn
n∈F ∩(N\K)
X
ajn
n∈(N\K)
holds for every j ∈ N, taking again the limit as j → ∞ in the last inequality, and using
(2.7), (2.11), we obtain
X
lim
ajn = 0,
j
n∈F
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O. Duman
which means that
stA - lim D∗ (Ln (f ), f ) = 0.
n
The theorem is proved.
2.4. Theorem. Let A = (ajn ) be a non-negative regular summability matrix, and
{Ln }n∈N a sequence of fuzzy linear operators from CF2 [0, 1] onto itself. Assume that there
exists a corresponding sequence {L̃n }n∈N of linear operators from C 2 [0, 1] onto itself for
which the following conditions hold:
h
i′′
h
i′′
(r) (2.12)
{Ln (f )}(r)
(x) = L̃n f±
(x) for x ∈ [0, 1], r ∈ [0, 1], n ∈ N
±
and
(2.13)
If
(2.14)
δA n ∈ N : L̃n (A ∩ C) ⊂ C = 1.
stA - lim [L̃n (ei )]′′ − e′′i = 0 for i = 0, 1, 2, 3, 4,
n
then, we have
(2.15)
stA - lim D∗ [Ln (f )]′′ , f ′′ = 0 for all f ∈ CF2 [0, 1].
n
Proof. We should remark that the derivatives in (2.12) and (2.14) are in the usual sense
while the derivatives in (2.15) are in the fuzzy sense. Now let f ∈ CF2 [0, 1], x ∈ [0, 1] and
r ∈ [0, 1] be fixed. By Lemmas 2.1 and 2.2, and as in the proof of Theorem 2.3, we can
write that, for every ε > 0, there exists a δ > 0 such that
(r)
(r)
(2.16)
−ε +
(r) ′′
(r) ′′
2U± β ′′
2U± β ′′
σx (y) ≤ f±
(y) − f±
(x) ≤ ε −
σx (y)
2
δ
δ2
(y − x)4
(r)
holds for all y ∈ [0, 1] and for any β ≥ 1, where σx (y) = −
+ 1 and U± =
12
(r) ′′ f
. Then, define the following functions on [0, 1]:
±
(r) ′′
(r)
f
(x) 2
2U± β
ε 2
(r)
(r)
uβ
(y) :=
σx (y) + f± (y) − y − ±
y ,
δ2
2
2
±
and
(r)
vβ
(r) ′′
(r)
2U± β
f±
(x) 2
ε 2
(r)
σx (y) − f± (y) − y +
y .
(y) :=
δ2
2
2
±
It follows from (2.16) that
′′
′′
(r)
(r)
uβ
(y) ≤ 0 and vβ
(y) ≤ 0 for all y ∈ [0, 1],
±
±
which implies that the functions
11
for all y ∈ [0, 1]. Then
12
(r)
±f± (y) − ε2 y 2 ±
(r)
h
(r)
uβ
i
(r) ′′
f±
2U± σx (y)
2
(x)
±
(r)
and vβ
belong to C. Observe that σx (y) ≥
y2 δ2
±
≤
(r)
(r)
M± + U± + ε δ 2
(r)
U±
Statistical Fuzzy Approximation
(r)
505
h
i (r) (r) ′′ = f± and U±(r) = f±
as stated before.
holds for all y ∈ [0, 1], where M±
Now, if we choose β such that
(
)
(r)
(r)
(M± + U± + ε)δ 2
(2.17) β ≥ max 1,
,
(r)
U±
then inequality (2.16) holds for such β’s and
(r)
(r)
uβ
(y) ≥ 0 and vβ
(y) ≥ 0 for all y ∈ [0, 1].
±
±
(r)
(r)
(r)
Hence, we also get uβ
, vβ
∈ A, which gives that the functions uβ
and
±
±
±
(r)
vβ
belong to A ∩ C under the condition (2.17). Now let
±
K := {n ∈ N : L̃n (A ∩ C) ⊂ C}.
Then, by (2.13), we have
(2.18)
δA (N \ K) = 0.
Also we get, for every n ∈ K,
h
i′′
h
i′′
(r) (r) L̃n vβ ± ≤ 0 and L̃n vβ ± ≤ 0.
Then, we obtain, for every n ∈ K, that
(r) ′′
(r)
i′′ h
i′′ ε
f
(x)
2U± β h
(r)
′′
L̃n (σx ) + L̃n (f± ) − [L̃n (e2 )] − ±
[L̃n (e2 )]′′ ≤ 0
δ2
2
2
and
(r) ′′
(r)
h
i′′ ε
f±
(x)
2U± β
(r)
′′
′′
[L̃n (σx )] − L̃n (f± ) − [L̃n (e2 )] +
[L̃n (e2 )]′′ ≤ 0.
δ2
2
2
These inequalities yield that
(r) ′′
(r)
(r) ′′
2U± β
f±
(x)
ε
′′
′′
[
L̃
(σ
)]
(x)
−
[
L̃
(e
)]
(x)
+
[L̃n (e2 )]′′ (x) − f±
(x)
n
x
n
2
2
δ
2
2
h
i′′
h
i′′
(r)
(r)
≤ L̃n (f± ) (x) − f±
(x)
(r)
≤−
2U± β
ε
[L̃n (σx )]′′ (x) + [L̃n (e2 )]′′ (x)
δ2
2
(r) ′′
(r) ′′
f
(x)
+ ±
[L̃n (e2 )]′′ (x) − f±
(x).
2
Observe now that [L̃n (σx )]′′ ≤ 0 on [0, 1] for every n ∈ K because of σx ∈ A ∩ C. Using
this, the last inequality gives, for every n ∈ K, that
(r)
(r) ′′ ′′
′′ 2U β ε (r) ′′
(x) ≤ − ±2
L̃n (σx ) (x) + L̃n (e2 ) (x)
L̃n (f± ) (x) − f±
δ
2
(r) ′′ f±
′′
(x) +
L̃n (e2 ) (x) − 2 ,
2
and hence
(r) ′′ ′′
ε + f±
(x) (r) ′′
′′
′′
L̃n (f± ) (x) − f (x) ≤ ε +
L̃n (e2 ) (x) − e2 (x)
2
(2.19)
(r)
′′
2U± β +
L̃n (−σx ) (x).
δ2
506
O. Duman
Now we compute the quantity [Ln (−σx )]′′ in inequality (2.19). Observe that
h (y − x)4
i′′
′′
L̃n (−σx ) (x) = L̃n
−1
(x)
12
′′
′′
′′
x
x2 1
L̃n (e4 ) (x) −
L̃n (e3 ) (x) +
L̃n (e2 ) (x)
=
12
3
2
x4
′′
′′
x3 −
L̃n (e1 ) (x) +
− 1 L̃n (e0 ) (x)
3
12
o x n
o
′′
′′
1 n
=
L̃n (e4 ) (x) − e′′4 (x) −
L̃n (e3 ) (x) − e′′3 (x)
12
3
o x3 n o
′′
′′
x2 n +
L̃n (e2 ) (x) − e′′2 (x) −
L̃n (e1 ) (x) − e′′1 (x)
2
3
x4
n
o
′′
+
−1
L̃n (e0 ) (x) − e′′0 (x) .
12
Combining this with (2.19), for every ε > 0 and n ∈ K, we have
(r) ′′ (r)
(r) ′′ ε + f±
(x)
U β± x2
(r) ′′
+ ± 2
(x) ≤ ε +
L̃n (f± ) (x) − f±
2
δ
′′
× Ln (e2 ) (x) − e′′2 (x)
(r)
′′
U± β ′′
L̃
(e
)
(x)
−
e
(x)
n
4
4
6δ 2
(r)
′′
2U± βx ′′
+
L̃n (e3 ) (x) − e3 (x)
2
3δ
(r)
′′
2U± βx3 ′′
L̃
(e
)
(x)
−
e
(x)
+
n
1
1
3δ 2
(r) ′′
2U± β
x4 ′′
+
1−
L̃n (e0 ) (x) − e0 (x) .
2
3δ
12
Therefore, we obtain, for every ε > 0 and n ∈ K, that
4 X
(r) ′′ ′′
(r) ′′
(r)
′′ (2.20) L̃n (f± ) − f±
≤ ε + E± (ε)
L̃n (ek ) − ek ,
+
k=0
(r)
where E± (ε) :=
ε+
(r)
U±
+
(r)
U± β
δ2
(r) ′′ (r)
and U± = f±
, as stated before. On the other
2
hand, by (2.12), since
′′
′′
D∗ Ln (f ) , f ′′ = sup D Ln (f ) (x), f ′′ (x)
x∈[0,1]
n ′′ (r) ′′
(x) − f (r) (x) ,
L̃n f
x∈[0,1] r∈[0,1]
(r) ′′ o
(r) ′′
(x) − f+
(x)
L̃n f+
n ′′ (r) ′′ = sup max L̃n f (r)
− f
,
= sup
sup max
r∈[0,1]
o
(r) ′′ (r) ′′
− f+
L̃n f+
,
we may write from (2.20) that
(2.21)
4 X
′′
′′
′′ D∗ Ln (f ) , f ′′ ≤ ε + E(ε)
L̃n (ek ) − ek k=0
Statistical Fuzzy Approximation
507
(r)
(r)
holds for all n ∈ K, where E(ε) = sup max E− (ε), E+ (ε) . Now, for a given ε′ > 0,
r∈[0,1]
choose an ε such that 0 < ε < ε′ , and consider the following sets:
n
o
′′
G := n ∈ N : D∗ Ln (f ) , f ′′ ≥ ε′ ,
′′
ε′ − ε
Gk := n ∈ N : L̃n (ek ) − e′′k ≥
, k = 0, 1, 2, 3, 4.
5E(ε)
In this case, by (2.21),
G∩K ⊂
4
[
(Gk ∩ K),
k=0
which yields, for every j ∈ N, that
X
4 4 X
X
X
X
(2.22)
ajn ≤
ajn ≤
ajn .
n∈G∩K
k=0
n∈Gk ∩K
k=0
n∈Gk
Letting j → ∞ on both sides of (2.22), and using (2.14), we immediately see that
X
(2.23) lim
ajn = 0.
j
n∈G∩K
Furthermore, if we use the inequality
X
X
X
ajn =
ajn +
n∈G
n∈G∩K
≤
X
n∈G∩K
ajn
n∈G∩(N\K)
ajn +
X
ajn
n∈(N\K)
and take the limit as j → ∞, then it follows from (2.18) and (2.23) that
X
lim
ajn = 0.
j
n∈G
Thus, we get
′′
stA - lim D∗ Ln (f ) , f ′′ = 0.
n
The theorem is proved.
2.5. Theorem. Let A = (ajn ) be a non-negative regular summability matrix and {Ln }n∈N
a sequence of fuzzy linear operators from CF1 [0, 1] onto itself. Assume that there exists
a corresponding sequence {L̃n }n∈N of linear operators from C 1 [0, 1] onto itself for which
the following conditions hold:
(r) ′
(r) ′
(2.24)
Ln (f ) ± (x) = L̃n f±
(x) (for x ∈ [0, 1], r ∈ [0, 1], n ∈ N)
and
(2.25)
If
(2.26)
δA n ∈ N : L̃n (D ∩ E) ⊂ E = 1.
′
stA - lim L̃n (ei ) − e′i = 0 for i = 0, 1, 2, 3,
n
then we have
(2.27)
′ stA - lim D∗ Ln (f ) , f ′ = 0 for all f ∈ C 1 [0, 1].
n
508
O. Duman
Proof. Let f ∈ CF1 [0, 1], x ∈ [0, 1] and r ∈ [0, 1] be fixed. Then, for every ε > 0, there
exists a positive number δ such that
(r)
(2.28)
(r)
(r) ′
(r) ′
2V± β ′
2V β
wx (y) ≤ f± (y) − f± (x) ≤ ε + ±2 wx′ (y)
δ2
δ
−ε −
holds for all y ∈ [0, 1] and for any β ≥ 1, where wx (y) :=
(r) ′ f± , Now considering the functions defined by
(y − x)3
(r)
+ 1 and V± :=
3
(r)
(r) θβ
and
(y) :=
±
(r) ′
2V± β
(r)
wx (y) − f± (y) + εy + y f± (x)
2
δ
(r)
(r) ′
2V± β
(r)
wx (y) + f± (y) + εy − y f± (x),
2
δ
(r) (r) we can easily check that θβ ± and λβ ± belong to E for any β ≥ 1. Also, observe
2
that wx (y) ≥ for all y ∈ [0, 1]. Then
3
(r) ′
(r)
(r)
(r)
± f± (y) − εy ± f± (x)y δ 2
(M± + V± + ε)δ 2
≤
(r)
(r)
2V± wx (y)
V±
(r) λβ
(y) :=
±
(r)
(r)
holds for all y ∈ [0, 1], where M± = kf± k, as stated before. Now, if we choose a number
β such that
(
)
(r)
(r)
(M± + V± + ε)δ 2
(2.29) β ≥ max 1,
,
(r)
V±
then inequality (2.28) holds for such β’s and
(r) (r) θβ ± (y) ≥ 0 and λβ ± (y) ≥ 0 for all y ∈ [0, 1],
(r) (r) (r) (r) which yields that θβ ± , λβ ± ∈ D. Thus, we get θβ ± , λβ ± ∈ D ∩ E for any
β satisfying (2.29). Let
K := {n ∈ N : L̃n (D ∩ E) ⊂ E}.
Then, by (2.25), we have
(2.30)
δA {N \ K} = 0.
Also we get, for every n ∈ K,
(r) ′
(r) ′
L̃n θβ ± ≥ 0 and L̃n λβ ± ≥ 0.
Hence we obtain, for every n ∈ K, that
(r)
and
′ (r) ′ ′
2V± β
(r) ′
[L̃n (wx )]′ − L̃n f±
+ ε L̃n (e1 ) + f± (x) L̃n (e1 ) ≥ 0
δ2
(r)
′ (r) ′ ′
2V± β
(r) ′
[L̃n (wx )]′ + L̃n (f± ) + ε L̃n (e1 ) − f± (x) L̃n (e1 ) ≥ 0.
δ2
Statistical Fuzzy Approximation
509
Then, we may write that
−
(r)
′
(r) ′ ′
(r) ′
2V± β
[L̃n (wx )]′ (x) − ε L̃n (e1 ) (x) + f± (x) L̃n (e1 ) (x) − f± (x)
2
δ
(r) ′
(r) ′
≤ L̃n f±
(x) − f± (x)
≤
(r)
′
′
2V± β L̃n (wx ) (x) + ε L̃n (e1 ) (x)
δ2
(r) ′ ′
(r) ′
+ f± (x) L̃n (e1 ) (x) − f± (x),
and hence
(r) ′ ′
(r) ′ (r) ′
′
L̃n (f± ) (x) − f± (x) ≤ ε + ε + f± (x) L̃n (e1 ) (x) − e1 (x)
(2.31)
(r)
′
2V β + ±2
L̃n (wx ) (x)
δ
holds for every n ∈ K because of the fact that the function wx belongs to D ∩ E. Since
i′
h (y − x)3
′
+ 1 (x)
L̃n (wx ) (x) = L̃n
3
′
′
′
1
= L̃n (e3 ) (x) − x L̃n (e2 ) (x) + x2 L̃n (e1 ) (x)
3
′
x3 + 1−
L̃n (e0 ) (x)
3
′
′
1 =
L̃n (e3 ) (x) − e′3 (x) − x L̃n (e2 ) (x) − e′2 (x)
3
′
+ x2 L̃n (e1 ) (x) − e′1 (x)
′
x3 + 1−
L̃n (e0 ) (x) − e′0 (x) ,
3
it follows from (2.31) that
(r) ′ (r) ′
L̃n (f± ) (x) − f± (x) ≤ ε +
!
(r)
2
(r) ′ 2V± β± x
ε + f± (x) +
δ2
′
× L̃n (e1 ) (x) − e′1 (x)
(r)
′
2V± β ′
L̃
(e
)
(x)
−
e
(x)
n
3
3
3δ 2
(r)
′
2V βx ′
+ ±2
L̃n (e2 ) (x) − e2 (x)
δ
(r)
′
2V β x 3 ′
+ ±2
1−
L̃n (e0 ) (x) − e0 (x) .
δ
3
+
Thus, we deduce from the last inequality that
(2.32)
3 X
(r) ′ ′
(r) ′
(r)
′
L̃n (f± ) − f± ≤ ε + F± (ε)
L̃n (ek ) − ek k=0
(r)
(r)
(r)
holds for any n ∈ K, where F± (ε) := ε + V± +
since
2V± β±
. On the other hand, by (2.24),
δ2
510
O. Duman
′ ′
D∗ Ln (f ) , f ′ = sup D Ln (f ) (x), f ′ (x)
x∈[0,1]
= sup
sup max
x∈[0,1] r∈[0,1]
= sup max
r∈[0,1]
n ′ (r) ′
(x) − f (r) (x) ,
L̃n f
n L̃n
(r) ′ o
(r) ′
(x) − f+ (x)
L̃n f+
′ ′ f (r) − f (r) ,
o
(r) ′ (r) ′
− f+ ,
L̃n f+
we may write from (2.32) that
(2.33)
3 X
′ ′
′
D∗ Ln (f ) , f ′ ≤ ε + F (ε)
L̃n (ek ) − ek k=0
(r)
(r)
holds for all n ∈ K, where F (ε) = sup max F− (ε), F+ (ε) . Now, for a given ε′ > 0,
r∈[0,1]
choose an ε such that 0 < ε < ε′ , and consider the following sets:
′ J := n ∈ N : D∗ Ln (f ) , f ′ ≥ ε′ ,
′
ε′ − ε
, k = 0, 1, 2, 3.
Jk := n ∈ N : L̃n (ek ) − e′k ≥
4G(ε)
In this case, by (2.33),
J ∩K ⊂
3
[
(Jk ∩ K),
k=0
which yields, for every j ∈ N, that
(2.34)
X
ajn ≤
n∈J ∩K
3
X
k=0
X
ajn
n∈Jk ∩K
!
≤
3
X
k=0
X
n∈Jk
ajn
!
Letting j → ∞ on both sides of (2.34), and also using (2.26), we immediately see that
X
(2.35) lim
ajn = 0.
j
n∈J ∩K
Now, using the fact that
X
X
ajn =
ajn +
n∈J ∩K
n∈J
≤
X
n∈J ∩K
ajn +
X
ajn
n∈J ∩(N\K)
X
ajn ,
n∈(N\K)
and taking the limit as j → ∞, then it follows from (2.30) and (2.35) that
X
lim
ajn = 0.
j
n∈J
Thus, we get
′ stA - lim D∗ Ln (f ) , f ′ = 0,
n
whence the result.
Statistical Fuzzy Approximation
511
2.6. Theorem. Let A = (ajn ) be a non-negative regular summability matrix and {Ln }n∈N
a sequence of fuzzy linear operators from CF [0, 1] onto itself. Assume that there exists a
corresponding sequence {L̃n }n∈N of linear operators from C[0, 1] onto itself for which the
following conditions hold:
(r)
(r)
(2.36) Ln (f ; x)± = L̃n f± ; x for x ∈ [0, 1], r ∈ [0, 1], n ∈ N
and
If
δA n ∈ N : L̃n F ⊂ F = 1.
(2.38)
stA - lim kL̃n (ei ) − ei k = 0 for i = 0, 1, 2,
(2.37)
n
then we have
(2.39)
′ stA - lim D∗ Ln (f ) , f ′ for all f ∈ C[0, 1].
n
Proof. If L̃n (F) ⊂ F holds for every n ∈ N instead of (2.37), i.e., the corresponding
sequence {L̃n }n∈N consists of positive linear operators on C[0, 1], then the proof follows
immediately from [5, Theorem 2.1]. However, our previous proofs show that even when
the weaker condition (2.37) holds, the property (2.38) implies (2.39).
3. Concluding Remarks
If we replace the non-negative regular matrix A by the identity matrix, then our
Theorems 2.3-2.6 reduce to the following results which are also new in the literature,
except for the last one.
3.1. Corollary. Let {Ln }n∈N be a sequence of fuzzy linear operators from CF2 [0, 1] onto
itself. Assume that there exists a corresponding sequence {L̃n }n∈N of linear operators
from C 2 [0, 1] onto itself for which the following conditions hold:
(r)
(r)
Ln (f ; x)± = L̃n f± ; x for x ∈ [0, 1], r ∈ [0, 1], n ∈ N
and
L̃n (A ∩ B) ⊂ A for n ∈ N).
If the sequence {L̃n (ei )}, (i = 0, 1, 2), is uniformly convergent to ei on [0, 1] then, for all
f ∈ CF2 [0, 1],
lim D∗ Ln (f ), f = 0.
n
3.2. Corollary. Let {Ln }n∈N be a sequence of fuzzy linear operators from CF2 [0, 1] onto
itself. Assume that there exists a corresponding sequence {L̃n }n∈N of linear operators
from C 2 [0, 1] onto itself for which the following conditions hold:
(r) ′′
(r) ′′
{Ln (f )}± (x) = L̃n f±
(x) for x ∈ [0, 1], r ∈ [0, 1], n ∈ N
and
L̃n (A ∩ C) ⊂ C for n ∈ N .
′′ If the sequence L̃n (ei )
, (i = 0, 1, 2, 3, 4), is uniformly convergent to e′′i on [0, 1] then,
2
for all f ∈ CF [0, 1],
′′
lim D∗ Ln (f ) , f ′′ = 0.
n
512
O. Duman
3.3. Corollary. Let {Ln }n∈N be a sequence of fuzzy linear operators from CF1 [0, 1] onto
itself. Assume that there exists a corresponding sequence {L̃n }n∈N of linear operators
from C 1 [0, 1] onto itself for which the following conditions hold:
(r) ′
(r) ′
Ln (f ) ± (x) = L̃n f±
(x) for x ∈ [0, 1], r ∈ [0, 1], n ∈ N
and
L̃n (D ∩ E) ⊂ E for n ∈ N .
′ If the sequence L̃n (ei ) , (i = 0, 1, 2, 3), is uniformly convergent to e′i on [0, 1] then,
for all f ∈ CF1 [0, 1],
′ lim D∗ Ln (f ) , f ′ = 0.
n
3.4. Corollary. [3] Let {Ln }n∈N be a sequence of fuzzy linear operators from CF [0, 1]
onto itself. Assume that there exists a corresponding sequence {L̃n }n∈N of positive linear
operators from C[0, 1] onto itself for which the following condition holds:
(r)
(r)
Ln (f ; x)± = L̃n f± ; x for x ∈ [0, 1], r ∈ [0, 1], n ∈ N).
If the sequence {L̃n (ei )} (i = 0, 1, 2) is uniformly convergent to ei on [0, 1] then, for all
f ∈ CF [0, 1],
lim D∗ Ln (f ), f = 0.
n
Finally, we give an application satisfying all the conditions of our Theorem 2.3.
3.5. Application. We first take A = C1 , the Cesáro matrix. Then, we know that C1 statistical convergence reduces to the concept of statistical convergence; in this case, we
use the notation st- lim instead of stC1 - lim. Also, we have δC1 ≡ δ. Assume now that
{Ln } is a sequence of linear operators on CF2 [0, 1] whose corresponding sequence {L̃n } of
linear operators on C 2 [0, 1] is given by
(
−x2 ,
if n = m2 m ∈ N ,
(r)
(3.1)
L̃n (f± ; x) =
(r)
Bn (f± ; x), if n 6= m2 ,
where x ∈ [0, 1], r ∈ [0, 1], n ∈ N, and {Bn } denotes the sequence of classical Bernstein
polynomials on C[0, 1]. Then, observe that
δ n ∈ N : L̃n A ∩ B ⊂ A = δ n 6= m2 : m ∈ N
= 1.
Also we have, for each i = 0, 1, 2,
st- lim L̃n (ei ) − ei = 0.
n
Then, it follows from Theorem 2.3 that, for all f ∈ CF2 [0, 1],
st- lim D∗ Ln (f ), f = 0.
n
However, for the function e0 = 1, since
(
−x2 if n = m2 m ∈ N
L̃n (e0 ; x) :=
1
otherwise,
we get, for all x ∈ [0, 1], that the sequence {L̃n (e0 ; x)} is non-convergent in the ordinary
sense. Furthermore,
we see that infinitely many terms of L̃n do not satisfy the inclusion
L̃n A ∩ B ⊂ A. Hence, this example clearly shows that our statistical fuzzy approximation theorems obtained in this paper have a wider range of applications than the classical
fuzzy approximation results.
Statistical Fuzzy Approximation
513
3.6. Remark. In this paper, we focus on fuzzy approximation only for the one dimensional case. However, by simple manipulations, all our results are also valid for multi
dimensional cases. However, we omit their details.
Acknowledgement. The author would like to thank the referee for carefully reading
the manuscript.
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