Error Analysis for Shannon Sampling Series Approximation with Measured Sampled Values

: Errors appear when the Shannon sampling series is applied to reconstruct a signal in practice. In this paper, we study a general model that uses linear functional to cover several errors in one formula, and consider sampling series with measured sampled values for not band-limited signals but satisfying some decay condition. We obtain the uniform truncated error bound of Shannon series approximation for multivariate Besov class. The results show this kind of Shannon series can approximate a smooth signal well.


INTRODUCTION
Since Shannon introduced the sampling series in the context of communication in the land-mark study (Shannon, 1948), the Shannon sampling series has become more and more important.By now, sampling theory has grown to an independent research area.The theorem states that a band-limited signal can be exactly recovered from its sample values.However errors appear when the Shannon is applied to approximate a signal in practice.Typical errors are jitter errors, amplitude errors, truncated error and aliasing error.
Motived by Butzer and Lei (1998), we consider the sampled values which are the results of a linear functional and its integer translates acting on a undergoing signal (Burchard and Lei, 1995).Following (Casey and Walnut, 1994) we call such sampled values measured sampled values because they are closer to the true measurements taken from a signal.This model covers jitter errors, amplitude errors and the errors arising from sampled values obtained by averaging a signal (Boche, 2010).
In this study we investigate the approximation sampling series with measured sampled values for not band-limited functions from Besov class and obtain the uniform bound of the truncation errors.Note that the strong "band-limited" assumption is the first time replaced by a weaker one, a smooth condition.
Let's begin with some concepts.For 1 p ≤ ≤ ∞ , let L p (R d )be the space of all p-th power Lebesgue integrable functions on R d equipped with the usual norm: 1/ ( ) , 1 , : sup ( ) , In the following, ‫ݐ‬ ܴ߳ ௗ denotes vector 1 ( , , ) where, f is the Fourier transform of f in the sense of distribution.In the special case p = 2 it reduces to the Paley-Wiener theorem (Nikolskii, 1975).
The famous Whittaker-Shannon-Nyquist sampling theorem, or simply Shannon theorem, states every signalfunction 2 f B πΩ ∈ can be completely reconstructed from its sampled values taken at instances / 2 k Ω .In this case the representation of f is given by: where, sin ( ) sin / ( ) c t t t π π = and sin (0) 1 c = .The Shannon theorem plays an important role in signal analysis as it provides the foundation for digital signal processing.It has been generalized to the case of nonband-limited signals, irregular sampling, multidimensional, and stochastic signal and so on (Zayed, 1993).
Shannon's expansion requires us to know the exact values of a signal f at infinitely many points and to sum an infinite series.But in practical situations, only finitely many samples are available, so it is natural to use: to approximate ( ) f t and study the bounds on the truncation error, which is defined by: , , ( )( ) : ( ) ( )( ) There were some papers concerned the estimate of truncation error (Zayed, 1993).
We proceed to state our own generalization of Shannon Theorem.We consider multi-dimensional signal.To obtain our main results, we need the error modulus: We organize the study as follows.In section two we derive the truncation error for band-limited functions from the class ( ) B πΩ R .In section three we obtain truncation error for non-band-limited functions from Besov class , ( ( )) In section four we provide some applications.
The main result of this section is the following theorem.
Lemma 1: Let ( ) where, the series on the right-hand side converges uniformly on d R .
Lemma 2: For d t ∈ R , q<1 and Ω >0, we have: For a given class p F πΩ , we show that there exists a applying the decay condition of f itis easy to prove that * ( ) According to Lemma 1 and Lemma 3, we have: We first derive the following estimate of the first term of right hand-side by the definition of ( , ) (2 1) ( , ) Let f satisfy the decay condition (3), then for we have: where, in the last inequality we use the inequality from.
The assumption )( ) ( , ) (ln ) ln Now we take 0 N N > , similarly we have: Thus the proof of Theorem 1 is complete.

TRUNCTION ERROR FOR NOT BAND LIMITED FUNCTIONS FROM BESOV CLASSES
Since the functions encountered in applications of the sampling theory are not always exactly bandlimited.Finding a bound for the error when a non-bandlimited function is approximated by the sampling expansion is a very practical issue.In this paper we study this problem in a considerable generality, that is, we assume that the signal functions belong to Besov class which is very commonly used in many applications.
Let's recall the definition of Besov space.Suppose that l N ∈ , r + ∈ R .For any ( ) [ ] 1 l r = + .We say , ( ) and the following semi-norm is finite: The linear space , ( ) is the Banach space with the norm: : We be the unit ball of the space , ( ) In this section we will develop uniform estimates for , ( )( ) .Our result is the following theorem.
where, the constant s A is taken such that ( ) The family of linear operator s T πΩ plays an important role in the representation and approximation of functions from Besov space.
, satisfying the inequality (3) and , then for any , where, As was mentioned above that under the assumption . This fact, together with (3) ∈ .Therefore we have ( ) By Lemma 3 we have: where, and the estimate of 2 3 I I + as we did in the proof of Theorem 1.
In summary, we have obtained: Thus combining Lemma 4, 6 and (7) we yield the desired result.

Proof of Theorem 2:
We will use the triangle inequality: And, p and 0 q are exponent conjugates, i.e., 0 0 1/ 1/ 1 p q + = .Note that the embedding condition and the decay condition of f implies 0 ( ) d C R .Therefore using the same arguments as in the proof of Theorem 1 we obtain the same error estimates for: ( 1)/ , 0 Therefore the conclusion of theorem 2 follows directly from Nikolskii (1975) and Theorem 3.

Some applications:
In this section we apply Theorem 1 to some practical examples, which will be summarized in the following corollarys.The first one is that the measured sampled values are given by local average.Observe that the equation ( 1) requires values of a signal f that are measured on a discrete set.However, due to the physical limitation, say the inertia, a measuring apparatus may not be able to obtain the exact values of f at epoch tk = ݇/Ω.In practice only local averages of the signal near the sampling point can be measured.In 1992, Gröchenig K. used a sequence of weight functions to average the original signal locally near the sampling point, and later, Aldroubi A., Butzer.P.L and Lei J., Sun W and Zhou X. obtained a lot of interesting results on the local average sampling.Specifically, let λ = {λ k }be a sequence of continuous linear functionals.
The sampled values are given by the local averages of a function may be formulated by the following integral representation: ( ) : , ( ) ( ) where, k u for each k ∈Z is a weight function characterizing the inertia of measuring apparatus.
Particularly, in the ideal case, the function k u is given by Dirac δ -function, ( ) is the exact value of k t Gröchenig first studied the reconstruction of signal from local averages in 1992.After that some authors studied the approximation error when local averages are used as sampled values.Now we assume that the functionals k λ are given by (10) in terms of the weight functions k u and k u satisfy the following properties: ). .Then: We also note that the function ( ) and we complete the proof.
Next we consider the combination of all four errors.The four usual types of errors existing in sampling series are the amplitude error, the time-jitter error, the truncation errors and the aliasing errors.First we give some explanation for the amplitude error and the time-jitter error.
We assume the amplitude error results from quantization, which means the functional value ( ) f t of a function f at moment t is replaced by the nearest discrete value or machine number ( ) f t The quantization size is often known before hand or can be chosen arbitrarily.We may assume that the local error at any moment t is bounded by a constant 0 ε > , i.

CONCLUSION
In this study we show that a function has a smoothness of order r>0 then the approximation order we can get is / ln r d p d − + Ω Ω .We also apply the results to some practical examples.Finally we would like to mention that we expect the sampling series to work better, i.e., without the factor ln d Ω and it would be of great interest to generalize Theorem 3 to the case of the scalar 1 ( .... ), To prove Theorem 2 we will choose an intermediate function which is a good approximation for both f and S f Ω .Now we begin with a description of how to choose this function.For any positive real number v we define the function: 2 proof of Theorem 3 relies on the inequality: