The Ill-Posedness of Derivative Interpolation and Regularized Derivative Interpolation for Non-Bandlimited Functions ()
1. Introduction
The computation of the derivative is widely applied in science and engineering [1].
In this section, we present the problem of finding the derivative of non-bandlimited signals by the sampling theorem.
Definition 1: Suppose a function
, its Fourier transform
is:
(1)
Definition 2: A function
is said to be
-band-limited if
, for every
. Otherwise, it is non-bandlimited. Here
is the Fourier transform of
[2] [3].
The inversion formula is
(2)
For band-limited signals, we have the Shannon sampling theorem [2] [3].
Shannon Sampling Theorem. If
and is
-band-limited, then it can be exactly reconstructed from its samples
:
(3)
where
and
. Here the convergence is in
and uniformly on
.
In [4], Marks presented an algorithm to find the derivative of band-limited signals by the sampling theorem:
(4)
Here, again, the convergence is in
and uniformly on
.
In [5], a method of numerical differentiation is given by low degree Chebyshev.
In this paper, we will consider the problem of computing
from the samples of
in the presence of noise.
(5)
where
is the exact signal and
is the noise with the bound
,
.
Formula (4) is not reliable due to the ill-poseness. In [6], a regularized derivative interpolation formula is presented for
-bandlimited functions. In this paper, a regularized derivative interpolation formula will be presented for non-bandlimited functions. Its convergence property is proved and applications will be shown by some examples. In the case non-bandlimited functions, the error estimate is different and the step size h of the samples is necessary to be close to zero.
2. The Regularized Derivative Interpolation
In this section, we present the regularized derivative interpolation by the sampling theorem in the pair of spaces
. Here
with the norm of
defined by
and
is the space of bounded sequences with the norm
We define the operator
Here
is the coefficient of
in (4).
Remark 1. The problem of computing
from
is an ill-posed problem.
To solve this ill-posed problem, we introduce the regularized Fourier transform [7] [8] [9] [10] [11]:
Definition 2. For
we define
(6)
where
is the function
multiplied with the weight function
Definition 3. Given
in
, define
The infinite series is uniformly convergent in
for any
.
By the differentiation of
in Definition 3, we obtain the regularized derivative interpolation:
(7)
This derivative is well defined since the infinite series is also uniformly convergent on
.
Lemma 1. If
is non-band-limited and
, then
where
.
It can be seen from the convolution
where
is the Fourier transform of
. For the proof of the convergence of the regularized derivative interpolation we will need the definition of periodic extension of the function
[12].
Definition 4.
denotes the periodic extension of the function
defined on the interval
to the interval
with period
.
The next Lemma is from [12].
Lemma 2. If
, then
for each
.
Remark 2. If
is
-band-limited, Lemma 2 reduces to the Shannon sampling theorem.
Lemma 3. For bounded
on
The proof is in [6].
Lemma 4. Suppose
and
which is defined in lemma 1. For
,
And if a is large enough, by omitting higher order infinitesimal we have
Proof.
Lemma 5. Suppose
and
. For
,
And if a is large enough we have
Proof.
Lemma 6. If
then
Proof. By lemma 4 and 5
So
Lemma 7. Assume
and
. As
, we have
Proof.
By lemma 3
By lemma 6
In order to prove the convergence property of the regularized derivative interpolation we will need the next lemma.
Lemma 8. If
is non-band-limited,
and
, then
Proof.
Then by Lemma 7, we can see the estimate is true.
Lemma 9. If we choose
such that
and
as
, then
The proof is in [6].
We are now in a position to state and prove our main theorem.
Theorem 1. Suppose
where
and
is non-band-limited,
and
. Then if we choose
such that
and
as
, then
Proof. Suppose
. Using Formula (7) and Lemma 8, we obtain
By Lemma 9, we have
Remark 3. According this theorem, if we choose
to be large enough, and
such that
and
as
, we can have good approximation.
3. Derivative Interpolation of Higher Order
In this section, we prove the convergence property of the derivative interpolation formula of high order:
(8)
Lemma 10.
where
and
.
Proof.
Lemma 11.
The proof is similar to the proof of Lemma 10.
Lemma 12. If
, then
and
Proof.
where
.
Lemma 13. If
on
and
Proof.
Lemma 14. If
, then
where
.
Proof.
By Lemma 12,
. Similarly,
.
Lemma 15. If
,
and
, then
where
Proof. By Lemma 13 and 14
Lemma 16. If
is non-band-limited,
,
and
then
where
.
Proof. Since
is uniformly convergent,
Lemma 17. For
, if k is even
and if k is odd
as
.
Proof is in [6].
Lemma 18. For
,
Proof is in [6].
Now we can state and prove a version of Theorem 1 for higher order derivatives.
Theorem 2. Assume
and
. If we choose
such that
and
as
, then we have the estimate
Proof.
where
This implies
Remark 4. We will choose the regularization parameter by the experiment. According to Theorem 2,
depends on
. If we choose
,
,
, the assumptions of Theorem 2 are satisfied. If
is known,
can be determined by discrepancy principle ( [13] ). The GCV and L-curve can be used ( [14] [15] ) if
is not known.
4. Experimental Results
In this section, we give some examples to compare the regularized derivative interpolation by sampling with the Tikhonov regularization method [16] [17].
In practice, only finite terms can be used in (8). So we choose a large integer N, and use next formula in computation:
(9)
where
is the noisy sampling data given in (4) in the section of introduction. Due to the weight function, the series above converges much faster than the series (3) of using Shannon’s sampling theorem. We give the estimate of the truncation error next
So if N is large enough, the truncation error can be very small.
Suppose
.
Then
So
is not a band-limited signal.
In examples 1 and 2 we consider
.
Example 1. We choose the noise
where
,
, and the signals
with
. We choose
. The results of
and
are in Figure 1 and Figure 2.
Example 2. We choose the noise to be white noise that is uniformly distributed in
. We choose
. The results of
and
are in Figure 3 and Figure 4.
Figure 1. The result of Example 1. The solid curve is
. The dashed is the reg derivative
.
Figure 2. The result of Example 1. The solid curve is
. The dashed is the reg derivative
.
Figure 3. The result of Example 2. The solid curve is
. The dashed is the reg derivative
.
Figure 4. The result of Example 2. The solid curve is
. The dashed is the reg derivative
.
5. Conclusion
The computation of derivatives is a highly ill-posed problem. The regularized derivative interpolation by sampling can be applied. The convergence property is proved and tested by some examples. The numerical results are better than Tikhonov regularization method.