The Ill-Posedness of Derivative Interpolation and Regularized Derivative Interpolation for Non-Bandlimited Functions

In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for non-bandlimited signals is presented. The convergence of the regularized derivative interpolation is studied. The numerical results are given and compared with derivative interpolation using the Tikhonov regularization method. The regularized derivative interpolation in this paper is more accurate in computation.


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 ( ) 2 f L ∈ R , its Fourier transform f is: In [4], Marks presented an algorithm to find the derivative of band-limited signals by the sampling theorem: Here, again, the convergence is in 2 L and uniformly on R .
In [5], a method of numerical differentiation is given by low degree Chebyshev.
In this paper, we will consider the problem of computing ( ) k f from the samples of f in the presence of noise.
Formula (4) is not reliable due to the ill-poseness. In [6], a regularized derivative interpolation formula is presented for Ω -bandlimited functions. In this paper, a regularized derivative interpolation formula will be presented for nonbandlimited 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.

The Regularized Derivative Interpolation
In this section, we present the regularized derivative interpolation by the sampling theorem in the pair of spaces ( ) .
The infinite series is uniformly convergent in R for any 0 α > .
By the differentiation of ( ) f t α in Definition 3, we obtain the regularized derivative interpolation: This derivative is well defined since the infinite series is also uniformly convergent on R .
Lemma 1. If f is non-band-limited and It can be seen from the convolution For the proof of the convergence of the regularized derivative interpolation we will need the definition of periodic extension of the function e i t ω [12]. The next Lemma is from [12].
Remark 2. If f is Ω -band-limited, Lemma 2 reduces to the Shannon sampling theorem.
The proof is in [6].
which is defined in lemma 1.
And if a is large enough, by omitting higher order infinitesimal we have Proof. By lemma 4 and 5 In order to prove the convergence property of the regularized derivative interpolation we will need the next lemma.
The proof is in [6].
We are now in a position to state and prove our main theorem.

Derivative Interpolation of Higher Order
In this section, we prove the convergence property of the derivative interpolation formula of high order: Proof.

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: where ( ) f nh 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

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.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.