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Stability of some positive linear operators on compact disk
Acta Mathematica Scientia 2015,35B(6):1492–1500 http://actams.wipm.ac.cn
STABILITY OF SOME POSITIVE LINEAR OPERATORS ON COMPACT DISK∗ M. MURSALEEN Khursheed J. ANSARI
Asif KHAN
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail : [email protected]; [email protected]; [email protected] Abstract Recently, Popa and Ra¸sa [27, 28] have shown the (in)stability of some classical operators defined on [0, 1] and found best constant when the positive linear operators are stable in the sense of Hyers-Ulam. In this paper we show Hyers-Ulam (in)stability of complex Bernstein-Schurer operators, complex Kantrovich-Schurer operators and Lorentz operators on compact disk. In the case when the operator is stable in the sense of Hyers and Ulam, we find the infimum of Hyers-Ulam stability constants for respective operators. Key words
Hyers-Ulam stability; Bernstein-Schurer operators; Kantrovich-Schurer operators; Lorentz operators; stability constants
2010 MR Subject Classification
1
39B82; 41A35; 41A44
Introduction
The equation of homomorphism is stable if every “approximate” solution can be approximated by a solution of this equation. The problem of stability of a functional equation was formulated by S.M. Ulam [35] in a conference at Wisconsin University, Madison in 1940: “Given a metric group (G, ., ρ), a number ε > 0 and a mapping f : G → G which satisfies the inequality ρ(f (xy), f (x)f (y)) < ε for all x, y ∈ G, does there exist a homomorphism a of G and a constant k > 0, depending only on G, such that ρ(a(x), f (x)) ≤ kε for all x ∈ G?” If the answer is affirmative the equation a(xy) = a(x)a(y) of the homomorphism is called stable; see [10, 17]. The first answer to Ulam’s problem was given by D.H. Hyers [16] in 1941 for the Cauchy functional equation in Banach spaces, more precisely he proved: “Let X, Y be Banach spaces, ε a nonnegative number, f : X → Y a function satisfying kf (x + y) − f (x) − f (y)k ≤ ε for all x, y ∈ X, then there exists a unique additive function with the property kf (x) − a(x)k ≤ ε for all x ∈ X.” Due to the question of Ulam and the result of Hyers this type of stability is called today HyersUlam stability of functional equations. A similar problem was formulated and solved earlier by G. P´olya and G. Szeg¨ o in [25] for functions defined on the set of positive integers. After Hyers result a large amount of literature was devoted to study Hyers-Ulam stability for various equations. A new type of stability for functional equations was introduced by T. Aoki [2] and Th.M. Rassias [29] by replacing ε in the Hyers theorem with a function depending on x and y, such that the Cauchy difference can be unbounded. The results of Aoki and Rassias have been ∗ Received
May 8, 2014; revised October 2, 2014.
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complemented later in the papers [12] and [7]. Moreover, a lot of useful recent information on that type of stability can be found in [6]. The Hyers-Ulam stability of linear operators was considered for the first time in the papers by Miura, Takahasi et al. (see [14, 15, 21]). Similar type of results are obtained in [34] for weighted composition operators on C(X), where X is a compact Hausdorff space. A result on the stability of a linear composition operator of the second order was given by J. Brzdek and S.M. Jung in [9]. Recently, Popa and Ra¸sa obtained [26] a result on Hyers-Ulam stability of the BernsteinSchnabl operators using a new approach to the Fr´echet functional equation, and in [27, 28], they have shown the (in)stability of some classical operators defined on [0, 1] and found the best constant for the positive linear operators in the sense of Hyers-Ulam. For other results on the Hyers-Ulam stability of functional equations one can refer to [22, 23]. Motivated by their work, in this paper, we show the (in)stability of some complex positive linear operators on compact disk in the sense of Hyers-Ulam. We find the infimum of the HyersUlam stability constants for complex Bernstein-Schurer operators and complex KantrovichSchurer operators on compact disk. Further we show that Lorentz polynomials are not stable in the sense of Hyers-Ulam on a compact disk. Issues considered in this paper are strictly connected with the problems of stability of the equation of fixed point investigated in [30]. Also, some related results have been obtained in [4, 5, 24, 31, 32, 36] and [8].
2
The Hyers-Ulam Stability Property of Operators
In this section, we recall some basic definitions and results on Hyers-Ulam stability property which form the background of our main results. Definition 2.1 (see [34]) Let A and B be normed spaces and T a mapping from A into B. We say that T has the Hyers-Ulam stability property (briefly, T is HU-stable) if there exists a constant K such that: (i) for any g ∈ T (A), ε > 0 and f ∈ A with kT f − gk ≤ ε, there exists an f0 ∈ A such that T f0 = g and kf − f0 k ≤ Kε. The number K is called a HUS constant of T , and the infimum of all HUS constants of T is denoted by KT . Generally, KT is not a HUS constant of T (see [14] and [15]). Let now T be a bounded linear operator with the kernel denoted by N (T ) and the range denoted by R(T ). Consider the one-to-one operator Te from the quotient space A/N (T ) into B: Te(f + N (T )) = T f, f ∈ A,
and the inverse operator Te−1 : R(T ) → A/N (T ).
Theorem 2.2 (see [34]) Let A and B be Banach spaces and T : A → B be a bounded linear operator. Then the following statements are equivalent: (a) T is HU-stable; (b) R(T ) is closed; (c) Te−1 is bounded.
Moreover, if one of the conditions (a), (b), (c) is satisfied, then KT = kTe−1 k.
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Remark 2.3 (1) Condition (i) of Definition 2.1 expresses the Hyers-Ulam stability of the equation T f = g, where g ∈ R(T ) is given and f ∈ A is unknown. (2) If T : A → B is a bounded linear operator, then (i) is equivalent to: (ii) for any f ∈ A with kT f k ≤ 1 there exists an f0 ∈ N (T ) such that kf − f0 k ≤ K (see [13]). So, in what follows, we shall study the HU-stability of a bounded linear operator T : A → B by checking the existence of a constant K for which (ii) is satisfied, or equivalently, by checking the boundedness of Te−1 .
The main results used in our approach for obtaining, in some concrete cases, the explicit value of KT are the formula given above and a result by Lubinsky and Ziegler [19] concerning coefficient bounds in the Lorentz representation of a polynomial. Let P ∈ Πn , where Πn is the set of all polynomials of degree at most n with real coefficients. A Lorentz representation of a polynomial P (x), is a representation of the form P (x) =
n X
ck xk (1 − x)n−k ,
(2.1)
k=0
where ck ∈ R, k = 0, 1, · · · , n. While it is not unique in general − for example n X n k 1= x (1 − x)n−k , any n ≧ 0, k k=0
− it becomes unique if we insist in (2.1) that n equals the degree of P . Note that, in fact, it is a representation in Bernstein-B´ezier basis. Let Tn denote the usual n-th degree Chebyshev polynomial of the first kind. From the expressions given in ([11], p.34), one readily derives the representation: Tn (2x − 1) =
n X
dn,k xk (1 − x)n−k (−1)n−k ,
(2.2)
k=0
where dn,k :=
min{k,n−k}
n 2j
X j=0
It is proved in [27] that dn,k = Therefore Tn (2x − 1) =
n − 2j j 4 , k = 0, 1, 2, · · · , n. k−j
(2.3)
2n , k = 0, 1, · · · , n. 2k
n X 2n (−1)n−k xk (1 − x)n−k . 2k k=0
Theorem 2.4 (Lubinsky and Ziegler [19]) Let P (x) have the representation (2.1), and let 0 ≤ k ≤ n. Let dn,k be defined by (2.3). Then |ck | ≤ dn,k kP k∞ with equality if and only if P (x) is a constant multiple of Tn (2x − 1), where kP k∞ = ess sup |P (x)| = inf K : |P (x)| ≤ K a.e. on [0, 1] .
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As in [19], we observe that kP k∞ ≦ max
0≦k≦n
|ck |/
n . k
(2.4)
Let A be the Banach space and M the closed subspace of A, then by A/M , we denote the quotient space with the usual norm kf + M k = inf kf + hk. h∈M
(2.5)
For more details, one can refer to [14]. Let C[0, 1] be the space of all continuous, real-valued functions defined on [0, 1], and CB [0, +∞) the space of all continuous, bounded, real-valued functions on [0, +∞). Endowed with the supremum norm, they are Banach spaces. Popa and Ra¸sa have shown the Hyers-Ulam stability of the following operators: (i) Bernstein operators [27] For each integer n ≥ 1, the sequence of classical Bernstein operators Bn : C[0, 1] → C[0, 1] is defined by (see [1]) n X n k k Bn f (x) = x (1 − x)n−k f , f ∈ C[0, 1], n ≥ 1. k n k=0
They are stable in the Hyers-Ulam sense and the best Hyers-Ulam stability constant is given by 2n KBn = , n ∈ N. 2[ n2 ] (ii) Sz´asz-Mirakjan operators [27] The nth Sz´asz-Mirakjan operator Ln : Cb [0, +∞) → Cb [0, +∞) defined by (see [1], pp. 338) j ∞ X j n j −nx Ln f (x) = e f x , x ∈ [0, +∞) n j! j=0 is not HU-stable for each n ≥ 1. (iii) Beta operators [27] For each n ≥ 1, the Beta operator Bn : C[0, 1] → C[0, 1] defined by [20] R 1 nx t (1 − t)n(1−x) f (t)dt Ln f (x) := 0R 1 nx n(1−x) dt 0 t (1 − t)
is not stable in the sense of Hyers-Ulam. (iv) Stancu operators [28] Let C[0, 1] be the linear space of all continuous functions f : [0, 1] → R, endowed with the supremum norm denoted by k.k, and a, b real numbers, 0 ≤ a ≤ b. The Stancu operator [33] Sn : C[0, 1] → Πn is defined by n X k+a n k Sn f (x) = f x (1 − x)n−k , n+b k k=0 f ∈ C[0, 1]. It is HU-stable and the infimum of the Hyers-Ulam constant is KSn = 2[2nn ] / [ nn ] , 2 2 for each n ≥ 1.
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(v) Kantorovich operators [28] Let X = {f : [0, 1] → R : where f is bounded and Riemann integrable} be endowed with the supremum norm denoted by k.k. The Kantorovich operators defined by n Z k+1 X n+1 n k f (t)dt x (1 − x)n−k , Kn f (x) = (n + 1) k k n+1 k=0
f ∈ X, x ∈ [0, 1] are HU-stable and the best HUS constant is KSn =
3
2n 2[ n 2]
/
n [n 2]
.
Main Results
In this section, we show the Hyers-Ulam stability of some other operators. Let DR denote the compact disk having radius R, i.e., DR = {z ∈ C : |z| ≤ R}. (i) Bernstein-Schurer Operators Let XDR = {f : DR → C | f is analytic in DR } be the collection of all analytic functions endowed with the supremum norm denoted by k.k for f ∈ XDR . The supremum norm is not over the whole space (as it is usually understood) and that the dimension of Πn+m is over reals. The complex Bernstein-Schurer operator Sn,m : XDR → Πn+m is defined by (see [3]) n+m X n + m k k n+m−k Sn,m (f )(z) = z (1 − z) f , z ∈ C, f ∈ XDR . k n k=0
We have N (Sn,m ) = {f ∈ XDR : f ( nk ) = 0, 0 ≤ k ≤ n + m}, which is a closed subspace of XDR , and R(Sn,m ) = Πn+m . The operator Sen,m : XDR /N (Sn,m ) → Πn+m is bijective, −1 Sen,m : Πn+m → XDR /N (Sn,m ) is bounded since dimΠn+m = 2(n + m + 1) over R. So according to Theorem 2.2 the operator Sn,m is Hyers-Ulam stable. As usual, the norm of −1 Sen,p : Πn+p → XDR /N (Sn,p ) is defined by −1 −1 kSen,p k = sup kSen,p (q)k = sup
inf
kqk≤1 h∈N (Sn,p )
kqk≤1
kfq + hk (by (2.5)).
Theorem 3.1 For n ≥ 1, the Hyers-Ulam stability best constant (by Definition 2.1 and Theorem 2.2) is given by n+m 2(n + m) −1 e KSn,m = kSn,m k = / n+m . 2[ n+m [ 2 ] 2 ] Proof
Let p(z) ∈ Πn+m and kpk ≤ 1. The Lorentz representation of p(z) is given by p(z) =
n+m X
ck (p)z k (1 − z)n+m−k , |z| ≤ R.
k=0
Consider the constant function fp ∈ XDR defined by k ck (p) = n+m , 0 ≤ k ≤ n + m. fp n k
−1 Then Sn,m fp = p and Sen,m (p) = fp + N (Sn,m ) (see [28]). Clearly n+m inf kfp + hk = kfp k = max |ck (p)|/ (by (2.4)). 0≤k≤n+m h∈N (Sn,m ) k
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Hence using the above equality, we have −1 −1 kSen,m k = sup kSen,m (p)k = sup
inf
kpk≤1 h∈N (Sn,m )
kpk≤1
kfp + hk
n+m k kpk≤1 0≤k≤n+m n+m ≤ sup max dn+m,k .kpk/ (using Theorem (2.4)) k kpk≤1 0≤k≤n+m n+m = max dn+m,k / . 0≤k≤n+m k = sup
max
|ck (p)|/
(3.1)
(3.2)
On the other hand, let r(z) = Tn (2z − 1), |z| ≤ R. Then krk = 1 and |ck (r)| = dn+m,k , 0 ≤ k ≤ n + m, according to Theorem 2.4. Consequently by (3.1), we have n+m n+m −1 kSen,m k ≥ max |ck (r)|/ = max dn+m,k / (3.3) 0≤k≤n+m 0≤k≤n+m k k and so by (3.2) and (3.3), we obtain −1 kSen,m k=
Let
max
0≤k≤n+m
ak = Then
The inequality
dn+m,k n+m =
2(n+m) 2k n+m k
k
max
0≤k≤n+m
2(n+m) 2k n+m k
.
, 0 ≤ k ≤ n + m.
ak+1 2n + 2m − 2k − 1 = , 0 ≤ k ≤ n + m. ak 2k + 1 ak+1 ak
≥ 1 is satisfied if and only if k ≤ [ n+m−1 ], therefore 2 a n+m , n + m is even; [ 2 ] max ak = a[ n+m−1 ]+1 = 2 0≤k≤n+m a n+m , n + m is odd. [
2
]+1
Since a[ n+m ]+1 = a[ n+m ] if n + m is an odd number, we conclude that 2 2 n+m 2(n + m) −1 e / n+m . KSn,m = kSn,m k = [ 2 ] 2[ n+m 2 ] This completes the proof of the theorem.
(ii) Kantrovich-Schurer Operators Let XDR = {f : DR → C is analytic in DR } be the collection of all analytic functions endowed with the supremum norm denoted by k.k. The complex Kantrovich-Schurer operator [3] Kn,m : XDR → Πn+m is defined by k+1 Z n+1 n+m X n + m Kn,m (f )(z) = (n + m + 1) z k (1 − z)n+m−k f (t)dt, z ∈ C, f ∈ XDR . k k n+1 k=0
We have N (Kn,m ) = {f ∈ XDR : f (t) = 0, t ∈ DR }. The operators Kn,m are Hyers-Ulam stable since their ranges are finite dimensional spaces.
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Theorem 3.2 For n ≥ 1
Proof
−1 KKn,m = kTen,m k=
(n + 1) (n + m +
2(n+m) 2[ n+m 2 ] . 1) [n+m n+m 2 ]
Let p(z) ∈ Πn+m , kpk ≤ 1, and its Lorentz representation n+m X
p(z) =
ck (p)z k (1 − z)n+m−k , |z| ≤ R.
k=0
Consider the constant function fp ∈ XDR defined by fp (t) =
(n + 1)ck (p) , 0 ≤ k ≤ n + m, t ∈ DR . (n + m + 1) n+m k
−1 e n,m Then Kn,m fp = p and K (p) = fp + N (Kn,m ). −1 e As usual, the norm of Kn,m : Πn+m → XDR /N (Kn,m ) is defined by −1 −1 e n,m e n,m kK k = sup kK (p)k = sup
Clearly
inf
h∈N (Kn,m )
inf
kpk≤1 h∈N (Kn,m )
kpk≤1
kfp + hk = kfp k =
max
0≤k≤n+m
Therefore e −1 k = sup kK n,m
max
kpk≤1 0≤k≤n+m
≤ sup
max
kpk≤1 0≤k≤n+m
=
max
0≤k≤n+m
kfp + hk
(n + 1)|ck (p)| (n + m + 1) n+m k
(n + 1)|ck (p)| (n + m + 1) n+m k (n + 1)kpkdn+m,k (n + m + 1) n+m k
(by (2.5)).
(by (2.4)).
(3.4) (using Theorem (2.4))
(n + 1)dn+m,k . (n + m + 1) n+m k
(3.5)
On the other hand, let r(z) = Tn (2z − 1), |z| ≤ R. Then krk = 1 and |ck (r)| = dn+m,k , 0 ≤ k ≤ n + m, according to Theorem 2.4. Consequently by (3.4), we have −1 e n,m kK k≥
max
0≤k≤n+m
(n + 1)dn+m,k (n + 1)|ck (r)| = max 0≤k≤n+m (n + m + 1) n+m (n + m + 1) n+m k k
and so by (3.5) and (3.6), we can conclude
Let
−1 e n,m kK k
(n + 1) 2(n+m) (n + 1)dn+m,k 2k = max . = max 0≤k≤n+m (n + m + 1) n+m 0≤k≤n+m (n + m + 1) n+m k k ak =
Then
The inequality
(n + m
2(n+m) 2k , + 1) n+m k
(n + 1)
0 ≤ k ≤ n + m.
ak+1 2n + 2m − 2k − 1 = , 0 ≤ k ≤ n + m. ak 2k + 1 ak+1 ak
≥ 1 is satisfied if and only if k ≤ [ n+m−1 ]. Therefore 2 a n+m , n + m is even; [ 2 ] max ak = a[ n+m−1 ]+1 = 2 0≤k≤n+m a n+m , n + m is odd. [
2
]+1
(3.6)
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Since a[ n+m ]+1 = a[ n+m ] if n + m is an odd number, we conclude that 2 2 2(n+m) (n + 1) 2[ n+m ] −1 2 e n,m k = . KKn,m = kL (n + m + 1) [n+m n+m ]
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2
This completes the proof of the theorem.
(iii) Lorentz Operators The complex Lorentz polynomial [13] attached to any analytic function f in a domain containing the origin is given by n k X n z f (k) (0), n ∈ N. Ln (f )(z) = n k k=0
Suppose that f : DR → C is analytic in DR , i.e., f (z) =
∞ P
ck z k , for all z ∈ DR := {z ∈
k=0
C; |z| < R, R > 1}.
Theorem 3.3 For each n ≥ 1, the Lorentz polynomial on compact disk is not stable in the sense of Hyers and Ulam. Proof To prove this theorem, we use the approach used in [27, Theorem 4.1]. Let us denote ej (z) = z j , then from Lorentz operators we can easily obtain that Ln (e0 )(z) = 1, Ln (e1 )(z) = e1 (z); and for all j, n ∈ N, j ≥ 2, we have j n z Ln (ej )(z) = j! j , 1 ≤ R1 < R j n 2 j−1 1 j = z 1− 1− ··· 1 − . n n n Also an easy computation shows that
Ln (f )(z) =
∞ X
cj Ln (ej )(z), ∀ |z| ≤ R1 ,
j=0
and Ln (e0 )(z) = 1, Ln (e1 )(z) = e1 (z). It follows that for each j ≥ 2, (1 − n1 )(1 − n2 ) · · · (1 − j−1 n ) is an eigen value of Ln . It can be easily seen that Ln is injective. Therefore (1− 1 )(1− 21)···(1− j−1 ) is an eigen value n
of L−1 n . Since lim
j→∞
1 (1 −
1 n )(1
−
2 n ) · · · (1
−
j−1 n )
n
n
nj = +∞, j→∞ (n − 1)(n − 2) · · · (n − j + 1)
= lim
we conclude that L−1 n is unbounded and so Ln is not HU-stable. This completes the proof of the theorem.
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STABILITY OF SOME POSITIVE LINEAR OPERATORS ON COMPACT DISK∗ M. MURSALEEN Khursheed J. ANSARI
Asif KHAN
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail : [email protected]; [email protected]; [email protected] Abstract Recently, Popa and Ra¸sa [27, 28] have shown the (in)stability of some classical operators defined on [0, 1] and found best constant when the positive linear operators are stable in the sense of Hyers-Ulam. In this paper we show Hyers-Ulam (in)stability of complex Bernstein-Schurer operators, complex Kantrovich-Schurer operators and Lorentz operators on compact disk. In the case when the operator is stable in the sense of Hyers and Ulam, we find the infimum of Hyers-Ulam stability constants for respective operators. Key words
Hyers-Ulam stability; Bernstein-Schurer operators; Kantrovich-Schurer operators; Lorentz operators; stability constants
2010 MR Subject Classification
1
39B82; 41A35; 41A44
Introduction
The equation of homomorphism is stable if every “approximate” solution can be approximated by a solution of this equation. The problem of stability of a functional equation was formulated by S.M. Ulam [35] in a conference at Wisconsin University, Madison in 1940: “Given a metric group (G, ., ρ), a number ε > 0 and a mapping f : G → G which satisfies the inequality ρ(f (xy), f (x)f (y)) < ε for all x, y ∈ G, does there exist a homomorphism a of G and a constant k > 0, depending only on G, such that ρ(a(x), f (x)) ≤ kε for all x ∈ G?” If the answer is affirmative the equation a(xy) = a(x)a(y) of the homomorphism is called stable; see [10, 17]. The first answer to Ulam’s problem was given by D.H. Hyers [16] in 1941 for the Cauchy functional equation in Banach spaces, more precisely he proved: “Let X, Y be Banach spaces, ε a nonnegative number, f : X → Y a function satisfying kf (x + y) − f (x) − f (y)k ≤ ε for all x, y ∈ X, then there exists a unique additive function with the property kf (x) − a(x)k ≤ ε for all x ∈ X.” Due to the question of Ulam and the result of Hyers this type of stability is called today HyersUlam stability of functional equations. A similar problem was formulated and solved earlier by G. P´olya and G. Szeg¨ o in [25] for functions defined on the set of positive integers. After Hyers result a large amount of literature was devoted to study Hyers-Ulam stability for various equations. A new type of stability for functional equations was introduced by T. Aoki [2] and Th.M. Rassias [29] by replacing ε in the Hyers theorem with a function depending on x and y, such that the Cauchy difference can be unbounded. The results of Aoki and Rassias have been ∗ Received
May 8, 2014; revised October 2, 2014.
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complemented later in the papers [12] and [7]. Moreover, a lot of useful recent information on that type of stability can be found in [6]. The Hyers-Ulam stability of linear operators was considered for the first time in the papers by Miura, Takahasi et al. (see [14, 15, 21]). Similar type of results are obtained in [34] for weighted composition operators on C(X), where X is a compact Hausdorff space. A result on the stability of a linear composition operator of the second order was given by J. Brzdek and S.M. Jung in [9]. Recently, Popa and Ra¸sa obtained [26] a result on Hyers-Ulam stability of the BernsteinSchnabl operators using a new approach to the Fr´echet functional equation, and in [27, 28], they have shown the (in)stability of some classical operators defined on [0, 1] and found the best constant for the positive linear operators in the sense of Hyers-Ulam. For other results on the Hyers-Ulam stability of functional equations one can refer to [22, 23]. Motivated by their work, in this paper, we show the (in)stability of some complex positive linear operators on compact disk in the sense of Hyers-Ulam. We find the infimum of the HyersUlam stability constants for complex Bernstein-Schurer operators and complex KantrovichSchurer operators on compact disk. Further we show that Lorentz polynomials are not stable in the sense of Hyers-Ulam on a compact disk. Issues considered in this paper are strictly connected with the problems of stability of the equation of fixed point investigated in [30]. Also, some related results have been obtained in [4, 5, 24, 31, 32, 36] and [8].
2
The Hyers-Ulam Stability Property of Operators
In this section, we recall some basic definitions and results on Hyers-Ulam stability property which form the background of our main results. Definition 2.1 (see [34]) Let A and B be normed spaces and T a mapping from A into B. We say that T has the Hyers-Ulam stability property (briefly, T is HU-stable) if there exists a constant K such that: (i) for any g ∈ T (A), ε > 0 and f ∈ A with kT f − gk ≤ ε, there exists an f0 ∈ A such that T f0 = g and kf − f0 k ≤ Kε. The number K is called a HUS constant of T , and the infimum of all HUS constants of T is denoted by KT . Generally, KT is not a HUS constant of T (see [14] and [15]). Let now T be a bounded linear operator with the kernel denoted by N (T ) and the range denoted by R(T ). Consider the one-to-one operator Te from the quotient space A/N (T ) into B: Te(f + N (T )) = T f, f ∈ A,
and the inverse operator Te−1 : R(T ) → A/N (T ).
Theorem 2.2 (see [34]) Let A and B be Banach spaces and T : A → B be a bounded linear operator. Then the following statements are equivalent: (a) T is HU-stable; (b) R(T ) is closed; (c) Te−1 is bounded.
Moreover, if one of the conditions (a), (b), (c) is satisfied, then KT = kTe−1 k.
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Remark 2.3 (1) Condition (i) of Definition 2.1 expresses the Hyers-Ulam stability of the equation T f = g, where g ∈ R(T ) is given and f ∈ A is unknown. (2) If T : A → B is a bounded linear operator, then (i) is equivalent to: (ii) for any f ∈ A with kT f k ≤ 1 there exists an f0 ∈ N (T ) such that kf − f0 k ≤ K (see [13]). So, in what follows, we shall study the HU-stability of a bounded linear operator T : A → B by checking the existence of a constant K for which (ii) is satisfied, or equivalently, by checking the boundedness of Te−1 .
The main results used in our approach for obtaining, in some concrete cases, the explicit value of KT are the formula given above and a result by Lubinsky and Ziegler [19] concerning coefficient bounds in the Lorentz representation of a polynomial. Let P ∈ Πn , where Πn is the set of all polynomials of degree at most n with real coefficients. A Lorentz representation of a polynomial P (x), is a representation of the form P (x) =
n X
ck xk (1 − x)n−k ,
(2.1)
k=0
where ck ∈ R, k = 0, 1, · · · , n. While it is not unique in general − for example n X n k 1= x (1 − x)n−k , any n ≧ 0, k k=0
− it becomes unique if we insist in (2.1) that n equals the degree of P . Note that, in fact, it is a representation in Bernstein-B´ezier basis. Let Tn denote the usual n-th degree Chebyshev polynomial of the first kind. From the expressions given in ([11], p.34), one readily derives the representation: Tn (2x − 1) =
n X
dn,k xk (1 − x)n−k (−1)n−k ,
(2.2)
k=0
where dn,k :=
min{k,n−k}
n 2j
X j=0
It is proved in [27] that dn,k = Therefore Tn (2x − 1) =
n − 2j j 4 , k = 0, 1, 2, · · · , n. k−j
(2.3)
2n , k = 0, 1, · · · , n. 2k
n X 2n (−1)n−k xk (1 − x)n−k . 2k k=0
Theorem 2.4 (Lubinsky and Ziegler [19]) Let P (x) have the representation (2.1), and let 0 ≤ k ≤ n. Let dn,k be defined by (2.3). Then |ck | ≤ dn,k kP k∞ with equality if and only if P (x) is a constant multiple of Tn (2x − 1), where kP k∞ = ess sup |P (x)| = inf K : |P (x)| ≤ K a.e. on [0, 1] .
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As in [19], we observe that kP k∞ ≦ max
0≦k≦n
|ck |/
n . k
(2.4)
Let A be the Banach space and M the closed subspace of A, then by A/M , we denote the quotient space with the usual norm kf + M k = inf kf + hk. h∈M
(2.5)
For more details, one can refer to [14]. Let C[0, 1] be the space of all continuous, real-valued functions defined on [0, 1], and CB [0, +∞) the space of all continuous, bounded, real-valued functions on [0, +∞). Endowed with the supremum norm, they are Banach spaces. Popa and Ra¸sa have shown the Hyers-Ulam stability of the following operators: (i) Bernstein operators [27] For each integer n ≥ 1, the sequence of classical Bernstein operators Bn : C[0, 1] → C[0, 1] is defined by (see [1]) n X n k k Bn f (x) = x (1 − x)n−k f , f ∈ C[0, 1], n ≥ 1. k n k=0
They are stable in the Hyers-Ulam sense and the best Hyers-Ulam stability constant is given by 2n KBn = , n ∈ N. 2[ n2 ] (ii) Sz´asz-Mirakjan operators [27] The nth Sz´asz-Mirakjan operator Ln : Cb [0, +∞) → Cb [0, +∞) defined by (see [1], pp. 338) j ∞ X j n j −nx Ln f (x) = e f x , x ∈ [0, +∞) n j! j=0 is not HU-stable for each n ≥ 1. (iii) Beta operators [27] For each n ≥ 1, the Beta operator Bn : C[0, 1] → C[0, 1] defined by [20] R 1 nx t (1 − t)n(1−x) f (t)dt Ln f (x) := 0R 1 nx n(1−x) dt 0 t (1 − t)
is not stable in the sense of Hyers-Ulam. (iv) Stancu operators [28] Let C[0, 1] be the linear space of all continuous functions f : [0, 1] → R, endowed with the supremum norm denoted by k.k, and a, b real numbers, 0 ≤ a ≤ b. The Stancu operator [33] Sn : C[0, 1] → Πn is defined by n X k+a n k Sn f (x) = f x (1 − x)n−k , n+b k k=0 f ∈ C[0, 1]. It is HU-stable and the infimum of the Hyers-Ulam constant is KSn = 2[2nn ] / [ nn ] , 2 2 for each n ≥ 1.
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(v) Kantorovich operators [28] Let X = {f : [0, 1] → R : where f is bounded and Riemann integrable} be endowed with the supremum norm denoted by k.k. The Kantorovich operators defined by n Z k+1 X n+1 n k f (t)dt x (1 − x)n−k , Kn f (x) = (n + 1) k k n+1 k=0
f ∈ X, x ∈ [0, 1] are HU-stable and the best HUS constant is KSn =
3
2n 2[ n 2]
/
n [n 2]
.
Main Results
In this section, we show the Hyers-Ulam stability of some other operators. Let DR denote the compact disk having radius R, i.e., DR = {z ∈ C : |z| ≤ R}. (i) Bernstein-Schurer Operators Let XDR = {f : DR → C | f is analytic in DR } be the collection of all analytic functions endowed with the supremum norm denoted by k.k for f ∈ XDR . The supremum norm is not over the whole space (as it is usually understood) and that the dimension of Πn+m is over reals. The complex Bernstein-Schurer operator Sn,m : XDR → Πn+m is defined by (see [3]) n+m X n + m k k n+m−k Sn,m (f )(z) = z (1 − z) f , z ∈ C, f ∈ XDR . k n k=0
We have N (Sn,m ) = {f ∈ XDR : f ( nk ) = 0, 0 ≤ k ≤ n + m}, which is a closed subspace of XDR , and R(Sn,m ) = Πn+m . The operator Sen,m : XDR /N (Sn,m ) → Πn+m is bijective, −1 Sen,m : Πn+m → XDR /N (Sn,m ) is bounded since dimΠn+m = 2(n + m + 1) over R. So according to Theorem 2.2 the operator Sn,m is Hyers-Ulam stable. As usual, the norm of −1 Sen,p : Πn+p → XDR /N (Sn,p ) is defined by −1 −1 kSen,p k = sup kSen,p (q)k = sup
inf
kqk≤1 h∈N (Sn,p )
kqk≤1
kfq + hk (by (2.5)).
Theorem 3.1 For n ≥ 1, the Hyers-Ulam stability best constant (by Definition 2.1 and Theorem 2.2) is given by n+m 2(n + m) −1 e KSn,m = kSn,m k = / n+m . 2[ n+m [ 2 ] 2 ] Proof
Let p(z) ∈ Πn+m and kpk ≤ 1. The Lorentz representation of p(z) is given by p(z) =
n+m X
ck (p)z k (1 − z)n+m−k , |z| ≤ R.
k=0
Consider the constant function fp ∈ XDR defined by k ck (p) = n+m , 0 ≤ k ≤ n + m. fp n k
−1 Then Sn,m fp = p and Sen,m (p) = fp + N (Sn,m ) (see [28]). Clearly n+m inf kfp + hk = kfp k = max |ck (p)|/ (by (2.4)). 0≤k≤n+m h∈N (Sn,m ) k
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Hence using the above equality, we have −1 −1 kSen,m k = sup kSen,m (p)k = sup
inf
kpk≤1 h∈N (Sn,m )
kpk≤1
kfp + hk
n+m k kpk≤1 0≤k≤n+m n+m ≤ sup max dn+m,k .kpk/ (using Theorem (2.4)) k kpk≤1 0≤k≤n+m n+m = max dn+m,k / . 0≤k≤n+m k = sup
max
|ck (p)|/
(3.1)
(3.2)
On the other hand, let r(z) = Tn (2z − 1), |z| ≤ R. Then krk = 1 and |ck (r)| = dn+m,k , 0 ≤ k ≤ n + m, according to Theorem 2.4. Consequently by (3.1), we have n+m n+m −1 kSen,m k ≥ max |ck (r)|/ = max dn+m,k / (3.3) 0≤k≤n+m 0≤k≤n+m k k and so by (3.2) and (3.3), we obtain −1 kSen,m k=
Let
max
0≤k≤n+m
ak = Then
The inequality
dn+m,k n+m =
2(n+m) 2k n+m k
k
max
0≤k≤n+m
2(n+m) 2k n+m k
.
, 0 ≤ k ≤ n + m.
ak+1 2n + 2m − 2k − 1 = , 0 ≤ k ≤ n + m. ak 2k + 1 ak+1 ak
≥ 1 is satisfied if and only if k ≤ [ n+m−1 ], therefore 2 a n+m , n + m is even; [ 2 ] max ak = a[ n+m−1 ]+1 = 2 0≤k≤n+m a n+m , n + m is odd. [
2
]+1
Since a[ n+m ]+1 = a[ n+m ] if n + m is an odd number, we conclude that 2 2 n+m 2(n + m) −1 e / n+m . KSn,m = kSn,m k = [ 2 ] 2[ n+m 2 ] This completes the proof of the theorem.
(ii) Kantrovich-Schurer Operators Let XDR = {f : DR → C is analytic in DR } be the collection of all analytic functions endowed with the supremum norm denoted by k.k. The complex Kantrovich-Schurer operator [3] Kn,m : XDR → Πn+m is defined by k+1 Z n+1 n+m X n + m Kn,m (f )(z) = (n + m + 1) z k (1 − z)n+m−k f (t)dt, z ∈ C, f ∈ XDR . k k n+1 k=0
We have N (Kn,m ) = {f ∈ XDR : f (t) = 0, t ∈ DR }. The operators Kn,m are Hyers-Ulam stable since their ranges are finite dimensional spaces.
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Theorem 3.2 For n ≥ 1
Proof
−1 KKn,m = kTen,m k=
(n + 1) (n + m +
2(n+m) 2[ n+m 2 ] . 1) [n+m n+m 2 ]
Let p(z) ∈ Πn+m , kpk ≤ 1, and its Lorentz representation n+m X
p(z) =
ck (p)z k (1 − z)n+m−k , |z| ≤ R.
k=0
Consider the constant function fp ∈ XDR defined by fp (t) =
(n + 1)ck (p) , 0 ≤ k ≤ n + m, t ∈ DR . (n + m + 1) n+m k
−1 e n,m Then Kn,m fp = p and K (p) = fp + N (Kn,m ). −1 e As usual, the norm of Kn,m : Πn+m → XDR /N (Kn,m ) is defined by −1 −1 e n,m e n,m kK k = sup kK (p)k = sup
Clearly
inf
h∈N (Kn,m )
inf
kpk≤1 h∈N (Kn,m )
kpk≤1
kfp + hk = kfp k =
max
0≤k≤n+m
Therefore e −1 k = sup kK n,m
max
kpk≤1 0≤k≤n+m
≤ sup
max
kpk≤1 0≤k≤n+m
=
max
0≤k≤n+m
kfp + hk
(n + 1)|ck (p)| (n + m + 1) n+m k
(n + 1)|ck (p)| (n + m + 1) n+m k (n + 1)kpkdn+m,k (n + m + 1) n+m k
(by (2.5)).
(by (2.4)).
(3.4) (using Theorem (2.4))
(n + 1)dn+m,k . (n + m + 1) n+m k
(3.5)
On the other hand, let r(z) = Tn (2z − 1), |z| ≤ R. Then krk = 1 and |ck (r)| = dn+m,k , 0 ≤ k ≤ n + m, according to Theorem 2.4. Consequently by (3.4), we have −1 e n,m kK k≥
max
0≤k≤n+m
(n + 1)dn+m,k (n + 1)|ck (r)| = max 0≤k≤n+m (n + m + 1) n+m (n + m + 1) n+m k k
and so by (3.5) and (3.6), we can conclude
Let
−1 e n,m kK k
(n + 1) 2(n+m) (n + 1)dn+m,k 2k = max . = max 0≤k≤n+m (n + m + 1) n+m 0≤k≤n+m (n + m + 1) n+m k k ak =
Then
The inequality
(n + m
2(n+m) 2k , + 1) n+m k
(n + 1)
0 ≤ k ≤ n + m.
ak+1 2n + 2m − 2k − 1 = , 0 ≤ k ≤ n + m. ak 2k + 1 ak+1 ak
≥ 1 is satisfied if and only if k ≤ [ n+m−1 ]. Therefore 2 a n+m , n + m is even; [ 2 ] max ak = a[ n+m−1 ]+1 = 2 0≤k≤n+m a n+m , n + m is odd. [
2
]+1
(3.6)
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Since a[ n+m ]+1 = a[ n+m ] if n + m is an odd number, we conclude that 2 2 2(n+m) (n + 1) 2[ n+m ] −1 2 e n,m k = . KKn,m = kL (n + m + 1) [n+m n+m ]
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2
This completes the proof of the theorem.
(iii) Lorentz Operators The complex Lorentz polynomial [13] attached to any analytic function f in a domain containing the origin is given by n k X n z f (k) (0), n ∈ N. Ln (f )(z) = n k k=0
Suppose that f : DR → C is analytic in DR , i.e., f (z) =
∞ P
ck z k , for all z ∈ DR := {z ∈
k=0
C; |z| < R, R > 1}.
Theorem 3.3 For each n ≥ 1, the Lorentz polynomial on compact disk is not stable in the sense of Hyers and Ulam. Proof To prove this theorem, we use the approach used in [27, Theorem 4.1]. Let us denote ej (z) = z j , then from Lorentz operators we can easily obtain that Ln (e0 )(z) = 1, Ln (e1 )(z) = e1 (z); and for all j, n ∈ N, j ≥ 2, we have j n z Ln (ej )(z) = j! j , 1 ≤ R1 < R j n 2 j−1 1 j = z 1− 1− ··· 1 − . n n n Also an easy computation shows that
Ln (f )(z) =
∞ X
cj Ln (ej )(z), ∀ |z| ≤ R1 ,
j=0
and Ln (e0 )(z) = 1, Ln (e1 )(z) = e1 (z). It follows that for each j ≥ 2, (1 − n1 )(1 − n2 ) · · · (1 − j−1 n ) is an eigen value of Ln . It can be easily seen that Ln is injective. Therefore (1− 1 )(1− 21)···(1− j−1 ) is an eigen value n
of L−1 n . Since lim
j→∞
1 (1 −
1 n )(1
−
2 n ) · · · (1
−
j−1 n )
n
n
nj = +∞, j→∞ (n − 1)(n − 2) · · · (n − j + 1)
= lim
we conclude that L−1 n is unbounded and so Ln is not HU-stable. This completes the proof of the theorem.
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