On the Interpolation in Linear Normed Spaces
using multiple nodes
April 10, 2015.
Dedicated to prof. I. Păvăloiu on the occasion of his 75th anniversary
In the papers
[
2
]
,
[
3
]
,
[
4
]
,
[
6
]
,
[
7
]
we indicated a method of extending the notion of interpolation polynomial to the case of a non-linear mapping
To switch to the case using multiple nodes, case that compulsorily uses the notion of Fréchet differential of the first order as well as of higher orders, we will point out the definition and certain properties of these differentials. On this basis we can present the manner of building an abstract interpolation polynomial with multiple nodes.
MSC. 41A05, 46B70.
Keywords. interpolation in linear normed spaces, multiple nodes.
1 Introduction
The topic of the interpolation of the functions defined between linear spaces or between linear normed spaces has been approached by Păvăloiu, I. in [ 9 ] , [ 10 ] , [ 11 ] Prenter, M. in [ 12 ] , Argyros, I. K. [ 1 ] , Makarov, V. L., Hlobistov, V. V. [ 8 ] and by myself in [ 2 ] , [ 3 ] , [ 4 ] , [ 6 ] , [ 7 ] .
We will recall the elements of the construction of the abstract interpolation polynomial with simple nodes.
Let us consider
We note by
For
For
Let
Let us consider now the bilinear mapping
We will now suppose the following properties:
there exists
the identity element of the semi-group and as well with so that form a group.there exists
and the linear and bijective mappingthere exists the linear mapping
so that
Using the mappings
We consider now the points
and for any
noting that
A first result from
[
2
]
,
[
7
]
shows that the restrictions to
and this mapping can be prolonged through linearity to
If we denote by
where:
and these mappings verify, for any
Because
we will have:
The relation
2 The Fréchet differential of a mapping. Special properties
Let us consider the linear normed spaces
First we will have the following:
The mapping
and:
For the mapping
We can easily prove that there exists at most a mapping
In this way Definition 1 is completed with:
The mapping
Now we have:
Usually there exists a subset
The definition of the previous function allows for the introduction of differentials with higher orders.â–¡
Thus we have:
Besides the data from Definition 1 let us consider a number
If:
there exists
a neighborhood of the point so that for any there exists the differential of the order of the mapping at the point denoted by so the function:is defined;
the function defined by
is a differential (of the first order) at the point then we can say that the mapping admits a differential with the order at the point and in this case:
In the paper [ 5 ] , we have established certain properties of the Fréchet differentials of higher orders, which are relevant for the statements below.
We will recall some of these properties.
I) Let us consider the bilinear and symmetrical mapping
With the aid of this mappings we consider:
We have the following proposition:
If the non-linear mappings
where, for a fixed
For the case
We can notice that the equation
II) It is necessary to generalize the property expressed by Proposition 6.
For this extension let us consider the sequence of mappings
where by
We will now consider the natural numbers
We introduce the set:
and obviously:
where
Let us now consider
For a fixed
For
we choose:
Finally, for a fixed
It is clear that for any
Let us denote by
It is obvious that:
But
Let us consider now the non-linear mappings
which represents a more general case of the mappings
In this way we have the following extension of the Proposition 6.
If the mappings
where for
For the case
III) Let us consider now for a fixed
where
For these mappings we have:
The mappings defined by
IV) Taking into account the mappings
where
We have the following result:
The mappings defined by
V) We will also consider another extension of Leibnitz’ formula concerning the derivative with the
In this way let us consider
Using the considered functions we consider the function:
and for this function we have:
If the mappings
where
For the case where
VI) Let us consider now the mapping
Therefore we can consider the mapping:
We obtain the following result:
If the non-linear mapping
for every
3 The Fréchet differential of certain essential mappings that appear in the interpolation with multiple nodes
We will consider the sequence of mappings
For
It is clear that if
We can thus consider the mapping:
Concerning this mapping we have:
If for any
where:
Taking into account Propositions 7 and 9 we deduce that:
for any
If we choose
after that for
where
so:
Thus:
We now show the equality:
Because the extension from
From the bijectivity of the mappings
The first member of this equality can be written as:
and the second is the value of the mapping
and this can be written as:
consequently the equality
Because for every
consequently the equality
But from the same relation
and:
after that replacing these values in
Because of the relation
where:
So:
consequently:
We remark that for any
this fact being evident, the element
Because the fact that
from where, using the relation
from where through the properties of the bilinear mapping
consequently:
from where:
For the beginning let be
Then in the previous relation we consider, with
and:
consequently:
thus:
so the relation
Let us suppose now that the relation
and for
From the relations
where:
At the same time:
where
Because
where:
and:
Therefore:
from where:
If we denote
and:
so if we denote:
we have:
We have used the identity:
In this way, because
whence we deduce that there exists the mapping
It is clear that:
where we have denoted:
Therefore:
and so the equality
Based on the principle of mathematical induction the equality
Proposition 12 is proved.
4 The construction of an abstract interpolation polynomial with multiple nodes
Let us consider the linear normed spaces
The general interpolation problem has the following setting.
Being given the distinct elements
To provide an answer to this problem let us suppose that the hypotheses from the preliminaries are fulfilled. Then let us consider the mappings that we have introduced by
To simplify the writing we will introduce:
together with the mapping from
If
We denote this inverted mapping as
As an answer to the aforementioned interpolation problem, we have the following theorem:
If for every
where
where
with:
The theorem is proved, if in the form
As
where
In order to fulfill the conditions
where for any
here,
and:
where
From the relations
for
We will determine the mappings
The relations
We will now consider Taylor’s formula for the case of non-linear mappings between linear normed spaces, formula which for a function
If
Because the mapping
From
and this relation indicates that the problem is solved if the elements
Let us define for every
From this equality we deduce that:
Considering the equality of the Fréchet differentials of the order
On the account of Proposition 8 and of the fact that:
if in the equality
If we introduce:
it is obvious that
On account of the Proposition 12 we deduce that:
From
From
Replacing now in
and from the relations
where
Because:
evidently:
This last equality indicates that the theorem is proved.
Bibliography
- 1
I. K. Argyros, Polynomial Operator Equation in Abstract Spaces and Applications, CRC Press Boca Raton Boston London New York Washington D.C., (1998).
- 2
A. Diaconu, Interpolation dans les espaces abstraits. Méthodes itératives pour la resolution des équation opérationnelles obtenues par l’interpolation inverse (I), Babeş-Bolyai University, Faculty of Mathematics, Research Seminars, Preprint No. 4, 1981, Seminar of Functional Analysis and Numerical Methods, pp. 1–52.
- 3
A. Diaconu, Interpolation dans les espaces abstraits. Méthodes itératives pour la resolution des équation opérationnelles obtenues par l’interpolation inverse (II), Babeş-Bolyai University, Faculty of Mathematics, Research Seminars, Preprint No. 1, 1984, Seminar of Functional Analysis and Numerical Methods, pp. 41–97.
- 4
A. Diaconu, Interpolation dans les espaces abstraits. Méthodes itératives pour la resolution des équation opérationnelles obtenues par l’interpolation inverse (III), Babeş-Bolyai University, Faculty of Mathematics, Research Seminars, Preprint No. 1, 1985, Seminar of Functional Analysis and Numerical Methods, pp. 21–71.
- 5
A. Diaconu, Sur quelques propriétés des dérivées de type Fréchet d’ordre supérieur, Babeş-Bolyai University, Faculty of Mathematics, Research Seminaries, Seminar of Functional Analysis and Numerical Methods, Preprint No. 1, (1983), pp. 13–26.
- 6
A. Diaconu, Remarks on Interpolation in Certain Linear Spaces (I), Studii ın metode de analiză numerică şi optimizare, Chişinău: USM-UCCM., 2, 2(1), (2000), pp. 3–14.
- 7
A. Diaconu, Remarks on Interpolation in Certain Linear Spaces (II), Studii ın metode de analiză numerică şi optimizare, Chişinău: USM-UCCM., 2, 2 (4), (2000), pp. 143–161.
- 8
V. L. Makarov and V. V. Hlobistov, Osnovı teorii polinomialnogo operatornogo interpolirovania, Institut Mathematiki H.A.H. Ukrain, Kiev, (1998) ( in Russian ).
- 9
I. Păvăloiu, Interpolation dans des espaces linéaires normés et application, Mathematica, Cluj, 12(35), 1, (1970), pp. 149–158.
- 10
I. Păvăloiu, Consideraţii asupra metodelor iterative obţinute prin interpolare inversă, Studii şi cercetări matematice, 23, 10, (1971), pp. 1545–1549 (in Romanian).
- 11
I. Păvăloiu, Introducere ın teoria aproximării soluţiilor ecuaţiilor, Editura Dacia, Cluj-Napoca, (1976), ( in Romanian ).
- 12