Abstract
Full Text
UDC 517.512.6
MATHEMATICS
Academician of the Academy of Sciences of the Azerbaijan SSR I. I. IBRAGIMOV, A. D. GADZHIEV
ON A SEQUENCE OF LINEAR POSITIVE OPERATORS
In this article a sequence of linear positive operators of a very general kind is constructed, containing as special cases the well-studied operators of S. N. Bernstein, Bernstein—Chlodovsky, G. Mirakyan, V. A. Baskakov, and others. Questions of uniform convergence of these operators in the class of continuous functions (C[0,A]), where (A>0) is a given number, are studied, as well as their properties of convexity or concavity.
I. Consider two sequences of functions ({\varphi_n(t)}) and ({\psi_n(t)}) from the class (C[0,A]) such that
[
\varphi_n(0)=0,\quad \psi_n(0)\ne 0,\quad \psi_n(t)>0\quad (0\le t\le A,\ n=1,2,\ldots),
\tag{1}
]
and let ({a_n}) be a sequence of positive numbers having the properties
[
\lim_{n\to\infty} a_n=\infty,\quad
\lim_{n\to\infty}\frac{\alpha_n}{n}=1,\quad
\lim_{n\to\infty}\frac{1}{n^2\psi_n(0)}=0.
\tag{2}
]
Suppose that the sequence of functions of three variables
({K_n(x,t,u)}) ((x,t\in[0,A],\ -\infty<u<\infty)) satisfies the following conditions:
(1^\circ). Each function of this sequence is an entire analytic function with respect to (u) for fixed (x) and (t) from the interval ([0,A]).
(2^\circ). (K_n(x,0,0)=1) ((n=1,2,\ldots)) for any (x\in[0,A]).
(3^\circ).
[
\left{(-1)^\nu
\left[
\frac{\partial^\nu}{\partial u^\nu}K_n(x,t,u)
\right]_{t=0}^{u=u_1}
\right}\ge 0
\quad *
\quad (\nu,n=1,2,\ldots;\ x\in[0,A]).
]
(4^\circ).
[
-\left.
\frac{\partial^\nu}{\partial u^\nu}K_n(x,t,u)
\right|{\substack{u=u_1\ t=0}}
=
nx\left.
\left[
\frac{\partial^{\nu-1}}{\partial u^{\nu-1}}K(x,t,u)
\right]\right|_{\substack{u=u_1\ t=0}}
\quad
(\nu,n=
]
[
=1,2,\ldots,\ x\in[0,A]),
]
where (m) is a natural number.
Consider the sequence of linear positive operators
[
L_n[f;x]=
\sum_{\nu=0}^{\infty}
f\left(\frac{\nu}{n^2\psi_n(0)}\right)
\left{
\left[
\frac{\partial^\nu}{\partial u^\nu}K_n(x,t,u)
\right]_{t=0}^{u=\alpha_n\psi_n(0)}
\right}
\frac{[-\alpha_n\psi_n(0)]^\nu}{\nu!}.
\tag{3}
]
It is obvious that by applying property (4^\circ) (\nu) times the operator (3) can be brought to the form
[
L_n[f;x]=
\sum_{\nu=0}^{\infty}
f\left(\frac{\nu}{n^2\psi_n(0)}\right)
\frac{n(n+m)\ldots[n+(\nu-1)m]}{\nu!}
\times
]
[
\times
[\alpha_n\psi_n'(0)]^\nu
K_{n+\nu m}(x,0,\alpha_n\psi_n(0))\,x^\nu .
\tag{4}
]
II. We shall show that the operator (L_n[f;x]) contains as a special case a number of known operators.
* This notation means that the derivative with respect to (u) is taken (\nu) times, and then (u=u_1) and (t=0) are set.
- In the case (K_n(x,t,u)=[1-xu/(1+t)]^n), the operator (L_n[f;x]) takes the form
[
L_n^{\langle 1\rangle}[f;x]
=
\sum_{\nu=0}^{n}
f!\left(\frac{\nu}{n^2\psi_n(0)}\right)
\binom{n}{\nu}
[1-x\alpha_n\psi_n(0)]^{\,n-\nu}
(\alpha_n\psi_n(0)x)^\nu .
]
The conditions (1^\circ)—(4^\circ), obviously, are satisfied, and (m=1). For (\alpha_n=n), (\psi_n(0)=1/n), the operator (L_n^{\langle 1\rangle}[f;x]) coincides with the classical polynomial of S. N. Bernstein.
Moreover, putting
[
\alpha_n=n,\qquad
\psi_n(0)=\frac{1}{n b_n}
\left(\lim_{n\to\infty} b_n=\infty,\ \lim_{n\to\infty}\frac{b_n}{n}=0\right),
]
we arrive at the Bernstein–Chlodowsky polynomials
[
L_n^{\langle 2\rangle}[f;x]
=
\sum_{\nu=0}^{n}
f!\left(\frac{\nu b_n}{n}\right)
\binom{n}{\nu}
\left(1-\frac{x}{b_n}\right)^{n-\nu}
\left(\frac{n}{b_n}\right)^\nu .
]
- Let us now put (K_n(x,t,u)=e^{-n(t+xu)}). The conditions (1^\circ)—(4^\circ), as is easily verified, are satisfied, and (m=0). In this case (L_n[f;x]) takes the form
[
L_n^{\langle 3\rangle}[f;x]
=
e^{-nx\alpha_n\psi_n(0)}
\sum_{\nu=0}^{\infty}
f!\left(\frac{\nu}{n^2\psi_n(0)}\right)
\frac{(nx)^\nu}{\nu!}
(\alpha_n\psi_n(0))^\nu .
]
For (\alpha_n=n), (\psi_n(0)=1/n), this operator coincides with the operator of G. M. Mirakyan.
- Let (K_n(x,t,u)=K_n(t+xu)), where (K_n(z)) is an entire analytic function. Simple computations show that in this case (L_n[f;x]) takes the form
[
L_n^{\langle 4\rangle}[f;x]
=
\sum_{\nu=0}^{\infty}
f!\left(\frac{\nu}{n^2\psi_n(0)}\right)
\frac{1}{\nu!}
K_n^{(\nu)}(x\alpha_n\psi_n(0))
[-x\alpha_n\psi_n(0)]^\nu .
]
For (\alpha_n=n), (\psi_n(0)=1/n), we arrive at the operators of V. A. Baskakov ((^1)). If, however, one puts (\alpha_n=n), (\psi_n(0)=1/\alpha_n) and denotes (n^2/\alpha_n=\beta_n), then the operator (L_n^{\langle 4\rangle}[f;x]) turns into another operator of V. A. Baskakov ((^2)).
III. With regard to the convergence of the operators (L_n[f;x]) in the class (C[0,A]), where (A>0) is a given number, the following assertions hold.
Theorem 1. If the function (f(x)) from the class (C[0,A]) is continuous from the right at the point (x=A) and grows at infinity no faster than (x^2), then, uniformly on ([0,A]), we have
[
\lim_{n\to\infty} L_n[f;x]=f(x).
]
Proof. Obviously, it is sufficient to verify the conditions of the well-known theorem of P. P. Korovkin (the growth restrictions on the function, as indicated in ((^2)), do not change its proof).
Writing for the function (K_n(x,t,u)) the Taylor expansion in powers of ((u-u_1)), we put in it (u=\varphi_n(t)), (u_1=\alpha_n\psi_n(t)), where (\varphi_n(t)) and (\psi_n(t)) and the sequence ({\alpha_n}) are defined in (1) and (2). Then putting (t=0), by virtue of condition (2^\circ), we obtain (L_n[1;x]=1).
Next, according to (3^\circ), we have
[
L_n[y;x]
=
\frac{x\alpha_n}{n}
\sum_{\nu=0}^{\infty}
\frac{\partial^\nu}{\partial u^\nu}
K_{n+m}(x,0,\alpha_n\psi_n(0))
\frac{[-\alpha_n\psi_n(0)]^\nu}{\nu!}
=
\frac{x\alpha_n}{n},
]
and, consequently, by virtue of (2), (L_n[y;x] \rightrightarrows x) ((n\to\infty,\ x\in[0,A])). Finally, applying condition (3^\circ) twice, we obtain
[
L_n[y^2;x]=\left(\frac{x\alpha_n}{n}\right)^2\frac{n+m}{n}
\sum_{\nu=2}^{\infty}\frac{\partial^{\nu-2}}{\partial u^{\nu-2}}
K_{n+2m}(x,0,\alpha_n\psi_n(0))
\frac{[-\alpha_n\psi_n(0)]^{\nu-2}}{(\nu-2)!}
+
]
[
+\frac{x\alpha_n}{n}\frac{1}{n^2\psi_n(0)}
=
\left(\frac{x\alpha_n}{n}\right)^2\frac{n+m}{n}
+
\frac{x\alpha_n}{n}\frac{1}{n^2\psi_n(0)};
]
by virtue of (2) it follows from this that, for (x\in[0,A]), (L_n(y^2;x)\rightrightarrows x^2) ((n\to\infty)). The theorem is proved.
A more general result is contained in the following theorem.
Theorem 2. If the function (K_n(x,t,u)), in addition to conditions (1^\circ)—(3^\circ), also satisfies the condition
[
\lim_{n\to\infty}\frac1n\frac{\partial}{\partial u}K_n(x,0,0)=-x,
\qquad
\lim_{n\to\infty}\frac1{n^2}\frac{\partial^2}{\partial u^2}K_n(x,0,0)=x^2
]
uniformly with respect to (x\in[0,A]), then the sequence of operators (L_n[f;x]) converges uniformly on ([0,A]) to the function (f(x)\in C[0,A]), continuous from the right at the point (x=A) and increasing at infinity no faster than (x^2).
The proof can be carried out by the same method as in ((^2)). We note that Theorem 1 can be obtained from Theorem 2.
IV. We now show that the operators (L_n[f;x]) preserve convexity, concavity, and polynomiality of arbitrary order on the interval ([0,A]). As is known ((^3)), a real function (f(x)) is called convex, non-concave, polynomial, non-convex, concave of (k)-th order on ([a,b]), if its divided difference of order ((k+2)), ([x_1,x_2,\ldots,x_{k+2};f]), for any system of ((k+2)) points of the interval ([a,b]), is respectively (>0,\ \ge0,\ =0,\ \le0,\ <0). For functions (F(x)\in C^{(k+1)}[0,A]), as T. Popoviciu showed (see, for example, ((^3))), the conditions of convexity, polynomiality, and concavity of (k)-th order on ([0,A]) mean that on ([0,A]) there hold, respectively, the relations
[
F^{(k+1)}(x)>0,\ =0,\ <0.
]
Let us now suppose that the function (K_n(x,t,u)), besides conditions (1^\circ)—(4^\circ), also satisfies the condition
[
-\frac{\partial}{\partial x}K_n(x,t,u)\bigg|{\substack{u=u_1\ t=0}}
=
nu_1K(x,0,u_1),
\tag{5}
]
where (m) is the natural number defined in (4^\circ).
Differentiating (4) with respect to (x) and taking (5) into account, by induction one can obtain the equality
[
\frac{d^p}{dx^p}L_n[f;x]
=
\frac{\alpha_n^p p!}{n^{2p}}
\sum_{\nu=0}^{\infty}
\left[
\frac{\nu}{n^2\psi_n(0)},\ldots,
\frac{\nu+p}{n^2\psi_n(0)};f
\right]
\times
]
[
\times
K_{n+(\nu-p)m}(x,0,\alpha_n\psi_n(0))
\frac{[\alpha_n\psi_n(0)]^\nu}{\nu!}
\,n(n+m)\cdots[n+(\nu+p-1)m]\,x^\nu,
]
where ([x_1,x_2,\ldots,x_p;f]) is the divided difference of the function (f(x)) at the nodes (x_1=\nu/n^2\psi_n(0),\ldots,x_p=(\nu+p)/n^2\psi_n(0)). With the help of this representation one obtains
Theorem 3. If the function (f(x)) is convex (concave) or polynomial of (k)-th order on the half-axis ([0,\infty)), then the sequence of operators (L_n[f;x]) on the interval ([0,A]) is respectively convex (concave) or polynomial.
In conclusion, we note that the operator (L_n[f;x]) can be represented in the form
[
L_n[f;x]=\sum_{\nu=0}^{\infty}
\frac{\nu!}{n^{2\nu}}\,n(n+m)\ldotsn+(\nu-1)m^\nu \times
]
[
\times\left[0,-\frac{1}{n^2\psi_n^(0)},\ldots,
-\frac{\nu}{n^2\psi_n^(0)};f\right]
K_{n+\nu m}(0,0,a'_n\psi_n^*(l)).
]
From this representation it follows that if (f(x)) is a polynomial of degree (r), then (L_n[f;x]) is its polynomial of degree (\le r), since in this case all divided differences of the function (f(x)) of order greater than (r) are equal to zero.
We note that similar assertions for the operator of V. A. Baskakov ((^1)) were proved in ((^3)).
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
11 V 1970
CITED LITERATURE
(^1) V. A. Baskakov, DAN, 113, 249 (1957).
(^2) V. A. Baskakov, Studies on Contemporary Problems of the Constructive Theory of Functions, Moscow, 1961.
(^3) A. Lupas, Mathematica, 9 (32), I, 77 (1967); 11, 295 (1967).