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
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
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
This problem is dual to (2).
, 
such that the solution of the extremal problem
there exists
which is a solution to
,
from the information. It follows from results obtained in [1-7] (see also [
. For an extremal problem of the form of (4), the corresponding Lagrange function is
is called an extremal element if
for 
and thus admissible in (4)
and 
be such that
for 
and 1) 
is an extremal element and
given by
, 
satisfying
is holomorphic in the unit disc by (6). We define the semi-norm in 
as
, 
, be a linear operator given by
is the radial integral of
. To see that
, by (7) we have for all but finitely many
,
for some
. Thus if 
then
.
given with a level of accuracy. That is, for a given
, we know a 
such that
in the class 
from the information 
satisfying (9). The error of a given method is measured in the 
norm defined by
is admitted as a recovery method. Let 
be sequences of non-negative real numbers such that
to be the convex hull. Define 
for 
by
with points of break 
, with 
for 
given by 
is such 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
.
. 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
.
then for any fixed 
the slopes between points 
and 
also tends to 0 as
given by
and define the convex hull of the origin and this collection of points as
:
is a piecewise linear function. Let
, 
be the points of break of 
with
. By (7) the assumption for Lemma 1 is satisfied by 
and
.
with 
. Let
then
and 
is an optimal method.
. Define the corresponding Lagrange function as
and 
be given by
.
. Thus 
are given by (12). Take any
, then by definition of the function 
we have
for any
.
admissable in (15) that also satisfies
Assume
.
if and only if 
and 
then 
if and only if 
or
. Let 
be the indices that satisfy
.
for
, and choose 
so that they satisfy the conditions
with 
or 
and
the function 
is admissible in (15) and
, that is 
minimizes 
satisfies condition 2) of Theorem 2.
, that is 
and
, define 
as in (17). Then as 
and we have
is admissable in (15) and satisfies 1) and 2) of Theorem 2. It should be noted that in this case 
are simply 
and
.
, 
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. 
, that is for a fixed
, the terms
and 
as
. So 
smooths approximate values of the coefficients of 
by the filter
.
from inaccurate information pertaining to the radial integral of f. However, the inaccurate information we are given are the values
such that
is the 
coefficient of the radial integral
,
given by (5) and 
and 
defined by (10) and (11) respectively but now add the condition
given by (8) is to determine the optimal error
obtaining this error.
as the largest index such that
with
. If
let 
and
.
then
and 
is an optimal method.
and 
then with
and 
the error of optimal recovery is (22) and (23) is an optimal method. For
, 
and 
is an optimal method.
with 
we simply apply the same structure of proof as in Theorem 3. For the case 
there remains some work.
, that is whether or not 
with
.
. Assume not. Then if 
we also know 
since for all 
we assumed
. Since 
we know
. Then by definition of 
we know for
,
we have
. Therefore
and if 
then
.
or
, the dual problem is of the form
is the characteristic function of
.
let 
correspond to the index satisfying
let 
be the line through the point 
that is parallel to the line from the origin to
. That is, let
and for any index
, we obtain
then
and 
and any 
we have
.
admissable in (24), let 
for 
and define 
by the system
this becomes
and
.
the function
is admissable in (24) with
and by construction we have 
and 
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
.
by (25) so 
and
. If we let 
be 
then
is admissable in extremal problem (24)
and
.
simply note that as
satisfies (26) and 
for all 
then
. So we have
. Since
then 
minimizes
.
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),
.
, 
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
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
.
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
satisfying 
, from the information 
satisfying
and
.
use conditions (6) and (20) as previously but impose an additional restriction. We add the condition
where 
are the levels of accuracy. If 
define
and furthermore
. The case 
will be treated seperately.
let
then 
and 
is an optimal method.
with 
and 
admissable in (31) and satisfying 1) and 2) of Theorem 2.
, define 
and 
as follows:
assume 
in which case
and hence
and any
,
, we consider the cases 
or
.
we know by assumption
and hence
,
. For the constructed
, it can be shown that 
as desired. and thus 
minimizes the Lagrange function.
is admissable in (31) we can clearly see that for
,
. It remains to show 
for
. Assume not, then
unless 
. If 
then 
and hence we no longer need the condition 
in order for 
to satisfy (31).
is admissable in (31).
we also have the results
and 
for 
while 
. Thus 
satisfies 2) of Theorem 2 as
as 
for any
. This extremal problem will have solution
. Choose 
and 
for
. Then as 
for all 
for all
. Let 
and notice 
and clearly
so 
is admissable in (31). Furthermore
. Also,
and 
may be less than
. Thus increasing 
may not change 
and hence not change the error or the method. If
, then
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
. 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.
denote the set of functions holomorphic on the unit disc. Define the Hardy space of functions
as the set of all
, 
with 
where
, are those 
such that 
and
is the class consisting of those 
. The Bergman space of functions
is the space of all 
such that
is the space of all holomorphic functions in
. The Bergman-Sobolev space of functions, 
, consists of 
with
and 
as the class of all
with
.
with
. If
then for 
then the piecewise linear function 
will have points of break
, the points to consider are
and thus the points of break of 
will be precisely
, the function 
has only a single point of break at the origin as
for
. Furthermore, 
does not satisfy (7) as
, 
be the points of break of 
for the space
.
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.
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
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
. 
and Bergman-Sobolev spaces 
in which 
is explicitly defined to be the smallest nonnegative integer satisfying
, 
for all
. Thus 
does not depend on
. So
and hence for any 
we are in the case
.
or
. Suppose 
with
. If 
or 
then let
be given by (12) and the optimal error is given by
and 
then 
and 
is an optimal method.
. 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.
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 
then 
for any
. Now proceed as in the proof of Theorem 4 to obtain the result. 
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
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.
and 
for all
.
terms of
. That is we take 
for every
. If 
we define 
similarly as
such that
for an optimal method.
term then any information beyond may be disregarded if 
as additional information will not decrease the error of optimal recovery.