Optimal Recovery of Holomorphic Functions from Inaccurate Information about Radial Integration ()
1. Introduction
Let W be a subset of a linear space X, let Z be a normed linear space, and T the linear operator that we are trying to recover on from given information. This information is provided by a linear operator where Y is a normed linear space. For any we know some that is near. That is, we know such that
(1)
for some. The value is our inaccurate information. Now we try to approximate the value of from using an algorithm or method,. Define a method to be any mapping, and regard as the approximation to from the information. Our goal is to minimize the difference of
and in, i.e. minimize
However, the size of varies since
can be chosen to be any satisfying (1). Furthermore varies depending on the chosen. So the error of any single method is defined as the worst case error
Now the optimal error is that of the method with the smallest error. Thus the error of optimal recovery is defined as
(2)
For the problems addressed in this paper, let be linear spaces with semi-inner norms and linear operators,. We want to recover for
(where if we let), if we know the values
satisfying for.
Define the extremal problem
(3)
This problem is dual to (2).
2. Construction of Optimal Method and Error
The following results of G. G. Magaril-Il’yaev and K. Yu. Osipenko [1] are applied to several problems of optimal recovery.
Theorem 1: Assume that there exist, such that the solution of the extremal problem
(4)
is the same as in (3). Assume also that for each
there exists
which is a solution to
(5)
Then for all,
and the method
(6)
is optimal.
Theorem 1 gives a constructive approach to finding an optimal method from the information. It follows from results obtained in [1-7] (see also [8] where this theorem was proven for one particular case.)
In order to apply Theorem 1 the values of extremal problems (4) and the dual problem (3) must agree. The following result, also due to G. G. Magaril-Il’yaev and K. Yu Osipenko [1], provides conditions under which the solution of problems (3) and (4) will agree.
Typically, when one encounters extremal problems, one approach is to construct the Lagrange function. For an extremal problem of the form of (4), the corresponding Lagrange function is
Furthermore, is called an extremal element if
for and thus admissible in (4)
and
Theorem 2: Let and be such that
for and 1)
2)
Then is an extremal element and
If we wish to combine Theorems 1 and 2 to determine an optimal error and method then we must show the posed problem is able to satisfy equating extremal problems (3) and (4). Through Theorem 2 we have such a means available.
3. Main Results
Consider the class of functions defined on the unit disc given by
(7)
for, satisfying
(8)
and
(9)
Therefore, any is holomorphic in the unit disc by (6). We define the semi-norm in as
and
(10)
Let, , be a linear operator given by
That is, is the radial integral of. To see that, by (7) we have for all but finitely many,
for some. Thus if then
.
We assume to know given with a level of accuracy. That is, for a given, we know a such that
(11)
The problem of optimal recovery is to find an optimal recovery method of the function in the class from the information satisfying (9). The error of a given method is measured in the norm defined by
Any method is admitted as a recovery method. Let be sequences of non-negative real numbers such that
Define to be the convex hull. Define for by
Lemma 1: The piecewise linear function with points of break , with for given by is such that.
Proof. Assume that It means that
. Since and as there is a such that and. Then the interval between and belongs to. Consequently, and is not a point of break of.
Assume that. Since the interval between and belongs to. Geometrically, the line to will lie above the line. It means that contradicting that is a point of break of.
Note that as then for any fixed the slopes between points and also tends to 0 as
3.1. Inaccuracy in Norm
Consider the points in given by
and define the convex hull of the origin and this collection of points as:
(12)
Let
(13)
thus is a piecewise linear function. Let, be the points of break of with. By (7) the assumption for Lemma 1 is satisfied by and.
Theorem 3: Suppose that with . Let
(14)
Then the error of optimal recovery is
(15)
and
(16)
is an optimal method of recovery. If then
and is an optimal method.
Proof. Consider the dual extremal problem
(17)
which can be written as
where. Define the corresponding Lagrange function as
Let the line segment between successive points
and be given by.
That is. Thus are given by (12). Take any, then by definition of the function we have
Thus for all
and hence for any.
We proceed to the construction of a function admissable in (15) that also satisfies
Assume.
As if and only if and then if and only if or. Let be the indices that satisfy
and
.
We let for, and choose so that they satisfy the conditions
(18)
From these conditions let
(19)
and
(20)
Now if with or and
the function is admissible in (15) and
, that is minimizes
and condition 1) of Theorem 2 is satisfied. Furthermore, by construction, satisfies condition 2) of Theorem 2.
If, that is and, define as in (17). Then as
So let and we have
Thus the function is admissable in (15) and satisfies 1) and 2) of Theorem 2. It should be noted that in this case are simply and.
Now we proceed to the extremal problem
(21)
This problem may be rewritten as
which has solution
So for, by Theorems 1 and 2, (14) is an optimal method and the error of optimal recovery is given by (13). If then
and is an optimal method.
It should be noted that for fixed, that is for a fixed, the terms
will have the property, and as. So smooths approximate values of the coefficients of by the filter.
3.2. Inaccuracy in Norm
Our next problem of optimal recovery remains to recover from inaccurate information pertaining to the radial integral of f. However, the inaccurate information we are given are the values
such that
where is the coefficient of the radial integral,
Denote
We again consider the space of functions given by (5) and and defined by (10) and (11) respectively but now add the condition
(22)
The problem of optimal recovery on the class given by (8) is to determine the optimal error
(23)
(24)
and an optimal method obtaining this error.
Define as the largest index such that
(25)
which by (7) exists, and
(26)
Theorem 4: Suppose with. If
let and.
Then the optimal error is
(27)
and
(28)
is an optimal method. If then
and is an optimal method.
If and then with
and the error of optimal recovery is (22) and (23) is an optimal method. For
, and is an optimal method.
Proof. For the cases with we simply apply the same structure of proof as in Theorem 3. For the case there remains some work.
Our construction will depend on whether or not, that is whether or not with
.
First we notice. Assume not. Then if we also know since for all we assumed. Since we know. Then by definition of we know for,
and substituting we have
which contradicts the definition of. Therefore
and if then
.
In either case, or, the dual problem is of the form
(29)
The corresponding Lagrange function is then
where is the characteristic function of
.
Case 1):
If let correspond to the index satisfying
To determine let be the line through the point that is parallel to the line from the origin to. That is, let
(30)
So for any point of break we have and for any index, we obtain
If then
Thus for the chosen and and any we have.
To construct admissable in (24), let for and define by the system
and since this becomes
So let and.
Then for the function
is admissable in (24) with
Therefore and by construction we have and
so that
and conditions (a) and (b) of Theorem 2 are satisfied.
Case 2):
If then, and, as this is the only point in the set
with a -coordinate of. Furthermore, as is a point of break of we know for all. Since then by the definition of we know. As
(31)
then we obtain equality,.
Define by (25) so and. If we let be then
In addition is admissable in extremal problem (24)
as and.
To justify simply note that as
satisfies (26) and for all then
. So we have
. Since
then minimizes.
For both cases, we now consider extremal problem
(32)
This problem can be written as
which will have solution
So by Theorems 1 and 2 we have obtained the optimal error and an optimal method for all scenarios. In each case i and ii, and are given by (25). In each case, the error of optimal recovery is
which for case 2) simplifies to. Also for each case, a method of optimal recovery is given by where in case 2) this simplifies to since in case 2),.
One may be able to reduce the amount information needed without affecting the error of optimal recovery. Therefore, by reducing the number of terms in the optimal method we reduce the compututaions needed. The following ideas are in [9]. We consider the subset, as the set of all points whose slope to the origin is greater than the slope of for, that is the slope of the line segment between points and. Define the sets
(33)
for where if define
. Now consider the same problem as stated in Theorem 4 using only information. For
, we have and so
. In this situation, with, it was shown that the error of optimal recovery only involves the two points
then the reduction in information from to will not change the error. That is
and if, an optimal method is
(34)
where.
3.3. Varying Levels of Accuracy Termwise
In Theorems 3 and 4 the inaccuracy of the information given is a total inaccuracy. That is, the inaccuracy is an upper bound on the sum total of the inaccuracies in each term, be it a finite or infinite sum. For Theorems 3 and 4 however, there is no way to tell how the inaccuracy is distributed. In particular, with regards to Theorem 4, the situations in which the given information satisfies
or for some particular satisfying
are treated the same. For the next problem of optimal recovery we address this ambiguity. The problem of optimal recovery is to determine an optimal method and the optimal error of recovering, from the information satisfying
for some prescribed and.
To define use conditions (6) and (20) as previously but impose an additional restriction. We add the condition
Define where are the levels of accuracy. If define
(35)
So and furthermore. The case will be treated seperately.
Theorem 5: If let
(36)
then the error of optimal recovery is given by
(37)
and
(38)
is an optimal method.
If then and is an optimal method.
Proof. The dual problem in this situation is
(39)
(40)
with the corresponding Lagrange function
The method of proof will be to first determine
with and admissable in (31) and satisfying 1) and 2) of Theorem 2.
If, define and as follows:
To verify assume in which case
and hence
To show for the chosen and any,
, we consider the cases or.
For we know by assumption
and hence
For
Thus for any,. For the constructed, it can be shown that as desired. and thus minimizes the Lagrange function.
To show is admissable in (31) we can clearly see that for,. It remains to show for. Assume not, then
which occurs if and only if
which contradicts the definition of unless . If then and hence we no longer need the condition in order for to satisfy (31).
Furthermore
and so is admissable in (31).
By the construction of we also have the results
and for while
for. Thus satisfies 2) of Theorem 2 as
We now proceed to the extremal problem
Notice the upper bound on the sum is as for any. This extremal problem will have solution
Therefore the error of optimal recovery is given by
and
is an optimal method.
Now we proceed to the case. Choose and for. Then as for all
Thus for all. Let
and and notice and clearly
so is admissable in (31). Furthermore
and so. Also,
Therefore and
is an optimal method.
The optimal method may not use all of the information provided as may be less than. Thus increasing may not change and hence not change the error or the method. If, then
and we can reduce the amount of information needed for a given optimal error.
If we may be able to reduce the error of optimal recovery if we have more information available. Fix. The greater number of terms we have of then the better we may be able to approximate, that is the smaller the optimal error of recovery. Let
(41)
and for
for any. If we know the first terms with some errors, then further increasing the terms will not yield a decrease in the error of optimal recovery.
3.4. Applications: The Hardy-Sobolev and Bergman-Sobolev Classes
We now apply the general results to the Hardy-Sobolev and Bergman-Sobolev spaces of functions on the unit disc. Let denote the set of functions holomorphic on the unit disc. Define the Hardy space of functions
as the set of all, with where
The Hardy-Sobolev space of functions, , are those such that and
is the class consisting of those
with. The Bergman space of functions
is the space of all such that
That is, is the space of all holomorphic functions in. The Bergman-Sobolev space of functions, , consists of with
and as the class of all
with.
So each space can be considered as the space with
For each space of functions we have the collection of points. If
then for
Therefore for
In this case we consider the collection of points
It is easy to see that if then the piecewise linear function will have points of break
(42)
For the space, the points to consider are
Again let and thus the points of break of will be precisely
For the special case of, the function has only a single point of break at the origin as
so that for. Furthermore, does not satisfy (7) as
Thus, in the applications of the general results, this case will be treated separately.
For notational purposes, let, be the points of break of for the space.
Corollary 1. Let or. If with or then the error of optimal recovery is given by (13) and (14) is an optimal method. If and then and is optimal.
Proof. For the spaces or, if and only if and. Thus if and only if or. Thus apply Theorem 3 to obtain the result for all spaces except. The dual problem in the case leads to a simple Lagrange function. The dual problem is specifically
Therefore the Lagrange function is simply given by
Now if we let and then
for any. So now proceed as in Theorem 3. As any will minimize, choose as in (18). The extremal problem (19) is solved similarly, and as then for.
It should be noted that the optimal method described is stable with respect to the inaccurate information data.
We now apply Theorem 4 to the Hardy-Sobolev spaces and Bergman-Sobolev spaces in which is explicitly defined to be the smallest nonnegative integer satisfying
For the case, for all. Thus does not depend on. So
and hence for any we are in the case.
Corollary 2. Let or. Suppose with. If or then let
be given by (12) and the optimal error is given by
(13) and (23) is an optimal method. If and then and is an optimal method.
Otherwise suppose. If or then the optimal error is given by (13) and (23) is an optimal method with and
. If and then
and is an optimal method.
Proof. As previously stated, if the only break point of is and furthermore as then given by (21) does not exist so we treat this special case. In this case, the dual extremal problem is
and the corresponding Lagrange function is simply
If and then for any
. Now proceed as in the proof of Theorem 4 to obtain the result.
We now apply Theorem 5 to the spaces or for. In this situation will be a non-decreasing sequence for all. Also, for any we have and we are always in the case. For then for both the Hardy and Bergman spaces and so the condition will be satisfied if we know satisfying
Corollary 3. Let or with or and and given by (27). Let, be given by (28). Then the error of optimal recovery is given by (29) and (38) is an optimal method. If and then and is an optimal method.
Proof. For Theorem 5 we simply used conditions (6) and (20), both of which are satisfied by and for all.
As a direct consequence of Theorem 5, we consider the situation in which we have a uniform bound on the inaccuracy of each of the first terms of. That is we take for every. If we define similarly as
and the apriori information is given by the values such that
Again we will only need the values for an optimal method.
As previously noted, since the optimal method and error of optimal recovery only use up to the term then any information beyond may be disregarded if as additional information will not decrease the error of optimal recovery.