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.