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polynomials [GRAY 89] Expansion coef cients are easily calculated by recurrence Because of this property, it is possible to calculate hundreds of thousands of terms (290,000 in [GRAY 89] to approximate a lter having two roots on the unit circle) Moreover, linear recurrences still exist for the coef cients of the impulse response of general fractional lters They remain linear, but the coef cients are af ne functions of time In simple cases, the quadratic upper bound for the rest of the truncated series are easy to determine Nevertheless, the number of terms increases with the memory range To construct a very long memory process, a moving average process of a gigantic order is necessary However, the simplicity of this procedure largely accounts for its success In a traditional way, the analyticity of F and a criterion of in nite norm are both used Other approximations are studied to solve stochastic problems that involve the properties of F on the imaginary axis in continuous time or on the unit circle in discrete time Rational approximations are rarer although more promising The principle lies in the search for an ARMA lter which minimizes a certain criterion, in general of type L2 In [WHI 86], Whit eld brings together several ideas of approximations by a rational fraction Certain criteria include a ponderation function in the criterion which essentially ensures the approximation in a frequency band The procedures are built on linear or non-linear least square algorithms, recursive or not The integral is replaced by a sum, or approximated by a trapezoid method Various authors carry out approximations with an interesting intuitive sense Let us quote Oustaloup [OUS 91], who chooses 2n + 1 zeros zj and 2n + 1 real poles pj , pj z so that ratios zj and j+1 do not depend on j pj In [BON 92, CUR 86] approximations of Hankel matrices H were studied, whose rst line consists of the desired impulse response Calculations are carried out satisfactorily for the processes with intermediate memory We then have access to an upper bound H for the approximation error Baratchart et al [BARA 91] perfected a powerful algorithm, which we describe brie y A linear system is considered, with constant coef cients, strictly causal, single entry and single exit Let (f1 , f2 , , fm , ) be its impulse response and f de ned + in the complex plane by f (z) = m=1 fm z m We assume that f belongs to Hardy space H2 , ie, it is square integrable on the unit circle We then seek a rational fraction of maximum order n (to be determined) in H2 that minimizes the criterion f r 2 2 The problem at hand is that of minimizing a functional n (q), where q is a polynomial.

Another approach similar to SAODV based on the use of digital signatures has been proposed in [51] The focus here is on securing on demand routing protocols in general The authors in [51] assume that there is a trusted certi cate server, T, which creates and distributes a certi cate for each node before the node joins the network All the nodes in the network have access to the true public key of T which they can use to verify the authenticity of the certi cates The certi cate of node A has the following format certA [IPA, PuKA, t, e] PrKT, where IPA is the IP address of Node A, PuKA is the public key of node A, t is the time the certi cate was created, and e is the time the certi cate expires [ ] PrKT denotes that the certi cate was signed with T s private key When a node, say node A, wishes to nd a path to node X, it broadcasts a message called the route discovery packet (RDP), which is similar to the AODV route request message The RDP message has the following format: RDP, IPx, certA , NA , t PrKA

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where F is a set of the Cantor type A proof of this result, as well as an opening point of discussion on the set of the GMF, is found in [AYA 99] Lastly, a process can be associated with a multifractal sequence in the following manner DEFINITION 611 We will call a GMBM associated with a multifractional sequence noted by (h) = (hn )n N any process permitting the harmonizable representation: Y(h) (t) = e it 1

| | 2 +h0 (t)

W (d ) e it 1 (654) W (d )

| | 2 +hn (t)

.

The comparison of this de nition with that of MBM is instructive: if it is supposed that functions (hn ) of the multifractional sequence are all equal to a function h verifying Hypothesis 64, then it is noted that the GMBM is a MBM with near normalization: 1 Y(h) (t) = Bh (t) v h(t) It is also noted that the law of a GMBM depends on the whole of the multifractional sequence and not only on the generalized multifractional function limit In fact, writing the GMBM in series form, let us guess that the nth function of the multifractional sequence governs the behavior of the GMBM at scale 1/2n 1 To clarify this idea, let us suppose that the white noise which intervenes in formula (654) is W (d ) and let us consider the development in series of the GMBM which is deduced from it:

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