Computing Recomposition of Maps with a New Sampling Asymptotic Formula

Almudena Antuña, Juan L. G. Guirao, Miguel A. López Departamento de Análisis Económico y Finanzas, Universidad de Castilla-La Mancha, Cuenca (Castilla-La Mancha), Spain Departamento de Matemática Aplicada y Estadstica, Universidad Politécnica de Cartagena, Cartagena (Región de Murcia), Spain Departamento de Matemáticas, Universidad de Castilla-La Mancha, Cuenca (Castilla-La Mancha), Spain E-mail: almudena.antuna@uclm.es, juan.garcia@upct.es, mangel.lopez@uclm.es Received April 6, 2011, revised May 5, 2011, accepted May 21, 2011


Introduction and Statement of the Main Results
A central result of the signal theory in engineering is the well-known Shannon-Whittaker-Kotel'nikov's theorem (see for instance [9] or [11]) working for band-limited maps of (i.e., for Paley-Wiener signals), and based on the normalized cardinal sinus map , Another philosopher's stone of the signal processing theory is the Middleton's sampling theorem for band step functions (see [8]).This result was one of the first modifications of the classic Sampling theorem (see [10]) which only works for band-limited maps.After this starting point many different extensions and generalizations of this theorem appeared in the literature trying to obtain approximations of non band-limited signals (see for instance [2] or [4]).Good surveys on these extensions are [3] or [11].
In this paper we follow the spirit of the previous results in the sense of trying to obtain approximations of non band-limited signals by using band-limited ones by increasing the band size.But our approach is completely different to the previous ones in the sense that we keep constant the sampling frequency generalizing in the limit the results of Marvasti et al. [7] and Agud et al. [1] .
In this setting, we state the following asymptotic property of Shannon's sampling theorem type where the convergence is considered in the Cauchy's principal value for the series and pointwise for the limit.
Property 1 Let be a map and We say that f holds the property for  The statement of the main results is: *This work has been partially supported by MCI (Ministerio de Ciencia e Innovación) and FEDER (Fondo Europeo Desarrollo Regional), grant number MTM2008--03679/MTM, Fundación Séneca de la Región de Murcia, grant number 08667/PI/08 and JCCM (Junta de Comunidades de Castilla-La Mancha), grant number PEII09-0220-0222.


, that the Gaussian map holds expression (1) for the three first coefficients of the power series representation of .Note that since the Gaussian map is analytical, for proving formula (1) is enough to show the equality between the coefficients of the power series representation of the Gaussian map and the coefficients of the series stated in the second member of (1) after proving the analitycity of the second member of (1).The statement of our result is the following: be a Gaussian map.Then the three first coefficients of the power series representation of are equal to the three first ones of the second member of expression (1).


The paper is divided into three sections.In Section 2 we present the ideas and results that have inspired us to formulate property and Conjecture 1. Section 3 is devoted to prove Theorem 1 and in Section 4 is proved Theorem 2.

On the Property and Conjecture 1 
We state as a property an approximation in the limit, through potentials of band--limited maps of the original signal, based on [1] and [7].


In [1] is proven that given a sequence     From this is directly deduced that if we consider an odd number and a band-limited signal with bandwidth such that the sequence of coefficients  holds the properties stated in [1], then the signal admits a recomposition of Shannon type in the form where clearly the sampling frequency can be choosen bigger than the Nyquist one.
Our aim is to provide a method for approximating non band-limited signal by band-limited ones and keeping the frequency of the sampling constant.And our idea is to take limits in (2) obtaining an equality of the form expressed as a property . In Section 3 we prove that property is held by any constant map for every .Thus, the universe of non-trivial signals which hold the conjecture is nonempty (note that ).Our feeling is that there are a big number of representative signals in engineering processes which satisfy property .
  We state as Conjecture 1 to prove that any signal of Gaussian type holds the statement.Note that the Gaussian map, which is mathematically important in itself, plays an important role in the signal theory because the Gaussian map is the unique function which reachs the minimum of the product of the temporal and frecuential width.This minimum is given by the Uncertainty Principle, see [6].We believe in the working of Conjecture 1 and we support it through Theorem 2 where we show the equality between the three first coefficients of the power series representation of the Gaussian map and property .For proving completely the conjecture, by the analyticity of the Gaussian map, is enough to prove that expression defines an analytical map and to show that the equality works for the rest of coefficients.

Proof of Theorem 1
The following lemma will play a key role in the proof of Theorem 1.
Proof.First of all we shall show that the result works for every t   .Indeed, if , the result is straight because of Therefore, from now on we assume that \ t    .Taking simetric terms in the series we obtain On the other hand, for a given is known Finally, replacing (4) in expression (3) the proof is over for every real number .
t The prove of the result for complex numbers is a consequence of the use of the Analytic Prologation Principle.For applying it, is enough to prove that the series k is an analytic function.Indeed, by (3) the series can be written in the form Obviously, the first term of the previous sum is an analytic map.For proving the analyticity of the second term of the sum we shall prove that the series which guarantees the uniformly convergency of the series in and the proof is over.L Remark 4 We underline that the fact of the series k defines an analytic function is a direct consequence of the application of the Uniform Convergence Principle for cardinal Series, see [5, pag. 70] or [11, pag. 22] for a more up-to-date reference.We present a direct approach in the proof of Lemma 3 for completness of the arguments.
. By Lemma 3 we have Thus, is shown that f holds property ending the proof.

Proof of Theorem 2
In the sequel we denote by J a set of consecutive natural numbers in the form  which eventually can be . By   # J we denote the cardinal of the set J and we assume the arithmetic of the infinity (i.e., , ), therefore by J m be an increasing bounded sequence of real numbers holding the following conditions: 1) , a b , eventually f can be equal to zero.Then for every sequence and for every > 0 and where = 2 Proof.For proving (5) we assume, without loss of generality, that and  \ 0 n n n J  is a decreasing sequence.We shall use the following notation Taking Proceeding in a similar way < , < and < .3 3 Now, it is easily deduced that which is just as we want to show.(5) The proof of ( 6) follows in an analogous way.
Note that for every and every . For a given > 0


there exist On the other hand, using the power series representation of the exponential function and the Newton's binomial,  and if we have the following inequality , using the last inequality and the proof is over.(12) The following proposition will play a key role in the proof of Theorem 2.
and using the previous inequality and ( 7), if  γ then Proof.We consider the functions  is a decreasing sequence on .It is easily deduced that using the Intermediate Value Theorem for suitable On the one hand, given > 0  clearly there exists On the other hand, using Lemma 5 for , and , there exists Since arctan is a continuous map on , and replacing and (16) in we obtain (15) Proof of Theorem 2. The aim of the proof is to show that the limit of the three first nonzero coefficients of the power series representations of      , g t n can be written in the form .
where is introduced in Proposition 7 and now by such result we obtain where We will take the limit in each part separately.Since   To support our feeling on the truth of the Conjecture 1 we prove, without loss of generality for = 1 the roots of unity of order and 3