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1 Iterates of unidimensional operators
Bulletin of the Transilvania
University of BraΕov
Series III:
Mathematics and Computer Science, Vol. 3(65), No. 1 - 2023, xx-xx
https://doi.org/10.31926/but.mif.2023.3.65.1.x
ON THE ITERATES OF UNI- AND MULTIDIMENSIONAL OPERATORS
Abstract
The problem of the convergence of the iterates of operators or of a
sequence of operators is discussed in a general framework related to
the fixed point theory, but with a permanent look towards the theory
of linear approximation operators. The results are obtained for
operators not necessarily linear. Some examples including the class
of approximation operators defined by H. Brass are given.
2000 Mathematics Subject Classification: 41A36, 47H10.
Key words: Bernstein type operator, contraction principle,
Perov theorem.
Dedicated to Professor Radu PΔltΔnea on his 70th
anniversary
1 Iterates of unidimensional operators
1.1 Iterates of a single operator
Any discussion about the iterates of an operator should start with the
notion of a contraction operator and Banachβs contraction principle.
Let be a metric space and let an
operator. One says that is Lipschitz continuous with Lipschitz constant if
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If then is a contraction operator. Observe that if is
Lipschitz continuous with Lipschitz constant then any iterate of where is also Lipschitz continuous
with Lipschitz constant With this remark, we have the
following proposition.
Theorem 1.
Let be such that:
- (i)
-
is a contraction operator with Lipschitz constant
- (ii)
-
has the (unique) fixed point
Then for every the sequence
of the iterates of calculated at is convergent to
Proof.
Clearly for all Then
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Since one has and then as Thus converges to as claimed.
β
This proposition is useful in case that the fixed point is known
for a contraction operator This happens for all many linear
approximation operators leaving invariant the functions and that is with and Then, restricted to a set
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the operator maps into itself and the function
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belongs to and is a fixed point of as can easily be
seen. It remains to be clarified the contraction property of This
aspect in the context of the theory of linear approximation operators was
first highlighted by Rus [15] for Bernsteinβs operators. The contraction
property was later highlighted for other well-known classes of operators
(see, e.g., [1], [2], [3], [6], [7]).
The proposition above is given in the idea of a priori knowledge of the
fixed point. For a contraction operator on a complete metric space, the
existence and uniqueness of the fixed point is guaranteed by Banachβs
contraction principle which in addition gives an estimate of the
approximation error, namely
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(1) |
In case of linear approximation operators, when metric is given by the
uniform norm of the space
this estimate reads as
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(2) |
for all and Thus the error of the approximation of by the -th iterate is expressed by the error with which the arbitrary start function is approximated by the operator This error is known for many
operators beginning with Bernsteinβs ones, in terms of some smoothness
modules (see [13]).
1.2 Iterates of a sequence of operators
Assume now that is an approximation process
in in the sense that
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For such a sequence of operators, one has
Theorem 2.
Assume that every is a contraction operator with a
Lipschitz constant and is the unique
fixed point of all operators Then
- (a)
-
- (b)
-
For each a sequence of iterates converges to if
is such
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(3) |
and some constant
Proof.
(a) Let be any limit point of the sequence Obviously Passing eventually to a
subsequence, we may assume that Choose
two different element From
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letting and using
and we deduce that Since we find Then and the proof of statement (a) is
finished.
(b) Using (1) one has
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Here as while from (3), the front coefficients are uniformly
bounded. As a result as , which is our statement.
β
1.3 Example: Iterates of Brassβs operators
Let and let such that
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Consider the polynomials
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where and the summation is done in relation to all systems of nonnegative
integer numbers for
which
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Define the operator
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(4) |
and consider the class of all operators which are
convex combinations of operators of the form (4) which leave invariant
the functions and Hence
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where the coefficients in finite number, are nonnegative and with the sum equal to one, and
We note that Bernsteinβs operators are of type (4), more exactly
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The class also contains other classical operators such as the
operators of Cheney-Sharma [8], MΓΌhlbach [12] and Stancu [16].
Lemma 1.
Any operator from
is a contraction operator on with the
Lipschitz constant and has the fixed point
Proof.
Indeed, for any since
one has
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Also, for one has and so while if then and so Hence
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Then
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|
The fact that is the fixed point of in
is a simple consequence of the two properties
β
The above lemma guarantees that any operator from the class is
a contraction operator and in has the unique fixed
point Thus Theorem 1 applies for any operator from the
class and we have
Theorem 3.
If then as for every
Additionally, formula (2) is true and
|
|
|
For a subclass of namely that of the so called operators of
Bernstein type, the result given by Theorem 3 was proved by using
completely different techniques, by Albu [4] and myself [14].
Obviously, Theorem 2 also applies to sequences of operators in which are assumed to be
approximation processes. A better result was obtained in [14] for the
subclass of Bernstein type operators by using the extremality property of
the classical Bernstein operators in that subclass, namely the relations
|
|
|
for every nonnegative and convex function
2 Iterates of multidimensional operators
We begin this section with the matrix version of Theorem 1.
Theorem 4.
Let be metric spaces and let where be such that
- (i)
-
For each
|
|
|
(5) |
for all and some nonnegative numbers for which the matrix is convergent to zero in the sense that its power
tends to the zero matrix as
- (ii)
-
has the (unique) fixed point
Then for each the sequence of the iterates of calculated at is convergent in
to
Proof.
Denoting
|
|
|
we may rewrite (5) in the matrix way as
|
|
|
(6) |
Then
|
|
|
and the result follows since as
β
If are complete metric spaces, then inequality (6) showing that is a Perov contraction on implies the
existence and uniqueness of the fixed point of and
moreover that the following estimate holds for all and
|
|
|
(7) |
Example 1. Let where is a
closed convex subset of a Banach space, and
be -Lipschitz continuous, i.e.,
|
|
|
Let be such that for Define the operator by
|
|
|
If the matrix is convergent to zero, then the operator is a
Perov contraction on Consequently, it has a unique fixed point its iterates converge to and estimate (7) holds. In particular, if Β for all and matrix is
convergent to zero as shown in [1]. Indeed, letting one has
and where and the powers of are
dominated by the matrix having all entries equal to (this is
proved by induction). Then
|
|
|
as claimed. In this case, each operator has a unique fixed
point In case that for each one has that
for all then Such a case
occurs if the operators are classical Bernstein
operators (of different degrees), or other operators of Bernstein type, and when the fixed point of is
Theorem 2 extends to multidimensional operators as follows.
Theorem 5.
Let be a sequence of
Perov contraction operators with Lipschitz matrices convergent
to zero, and let is the unique fixed point of all operators Assume that
|
|
|
(8) |
Then
- (a)
-
The element from the diagonal of tend to as
- (b)
-
For each a sequence of iterates converges to if is such that the matrices
|
|
|
are bounded (componentwise).
Proof.
(a) It is known that the elements from the diagonal of a matrix which is
convergent to zero are strictly less than one. Letting one has for For any let be a limit point of the sequence Then Passing eventually to a
subsequence, we may assume that Now we take two element having the components
equal to zero except and which are chosen different. From
|
|
|
looking at the -th component, one has
|
|
|
Here we let and use and to deduce that Since
we find Then and the proof of statement (a) is finished.
(b) Using (7) one has
|
|
|
whence the conclusion of (b) is immediate.
β
The next example gives a scheme of construction of sequences of
multidimensional operators which are approximation processes in the
sense of (8).
Example 2. Let be a
sequence of operators of the type of those from the previous example, that
is
|
|
|
It is easily seen that the sequence
is an approximation process on , i.e., as for every if
the following conditions are satisfied:
- (i)
-
for every the sequence is an approximation process on
- (ii)
-
for every with and for each the sequence is bounded;
- (ii)
-
for every with as
References
-
[1]
Agratini, O., and Precup, R., Iterates of multidimensional
operators via Perov theorem, Carpathian J. Math. 38 (2022),
539-546.
-
[2]
Agratini, O., and Precup, R., Estimates related to the iterates
of positive linear operators and their multidimensional analogues,
submitted.
-
[3]
Agratini, O., and Rus, I.A., Iterates of a class of discrete
operators via contraction principle, Comment. Math. Univ. Carolinae
44 (2003), 555-563.
-
[4]
Albu, M., Asupra convergenΕ£ei iteratelor unor operatori
liniari Εi mΔrginiΕ£i, In βSem. Itin. Ec. Func. Aprox.
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[5]
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[6]
CΔtinaΕ, T., Iterates of a modified Bernstein type
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[7]
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[8]
Cheney, E.W., and Sharma, A., On a generalization of Bernstein
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[9]
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[10]
Karlin, S., and Ziegler, Z., Iteration of positive approximation
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[11]
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[12]
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positiver linearer operatoren, Dissertation Technische
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[13]
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195-235.
-
[14]
Precup, R., ProprietΔΕ£i de alurΔ Εi unele aplicaΕ£ii ale lor, Ph.D. Thesis, BabeΕ-Bolyai University,
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[15]
Rus, I.A., Iterates of Bernstein operators, via contraction
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[16]
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