fast implementation of the continuous wavelet transform with integer scales we describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m in this approach the input signal and the analyzing wavelet are both represented by polynomial splines the algorithm uses a combination of moving sum and zero padded filters and its complexity per scale is o n where n is the signal length the computation is exact and the implementation is noniterative across scales we also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection first and second derivative operators or gabor like time frequency signal analysis

fuzzy groups some group theoretic analogs abstract we introduce the concepts of 1 order of a fuzzy subgroup of a finite group 2 a fuzzy abelian group and 3 a fuzzy solvable group we obtain some results involving these notions which are analogs of results from group theory also we prove analogs of two basic theorems of group theory namely cayley s theorem and lagrange s theorem

multivariate scattered data fitting during the week of 2 8 march 1995 a meeting entitled international conference on scattered data fitting was held in cancun mexico the meeting was attented by 42 researchers from europe and the americas and 34 lectures were presented this special issue of journal of computational and applied mathematics includes 14 papers based on presentations at the conference

second order polynomial estimators from uncertain observations using covariance information this paper presents recursive least mean squared error second order polynomial filtering and fixed point smoothing algorithms to estimate a signal from uncertain observations when only the information on the moments up to fourth order of the signal and observation noise is available the estimators require the autocovariance and crosscovariance functions of the signal and their second order powers in a semidegenerate kernel form and the probability that the signal exists in the observed values

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

invertibility conditions of a class of moment matrices and applications we consider a multivariable polynomial with random variables as arguments all the monoids of this polynomial are arranged to define a random vector for which we study the correlation matrix we establish necessary and sufficient invertibility conditions of this matrix and investigate some applications

deconvolution and estimation of transfer function phase and coefficients for nongaussian linear processes nongaussian linear processes are considered it is shown that the phase of the transfer function can be estimated under broad conditions this is not true of gaussian linear processes and in this sense gaussian linear processes are atypical the asymptotic behavior of a phase estimate is determined the phase estimates make use of bispectral estimates these ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum phase a number of computational illustrations are given

two direction poly scale refinability the concept of two direction poly scale refinable functions with dilation factor a and shift invariant spaces is introduced the existence of solutions of two direction poly scale refinable equations is investigated in particular a necessary and sufficient condition for the orthonormality of the solutions is established some properties of two direction poly scale refinable functions are discussed finally two examples of two direction poly scale refinable functions are given

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

nonlinear infinite extent interpolation of stationary processes the problem of minimum mean square interpolation of discrete time stationary stochastic processes using a system consisting of a finite number p of nonlinear instantaneous transformations followed by a p input one output infinite extent linear filter is studied the expressions of the optimal interpolator and of the approximation error are derived and generalize the corresponding relations known in linear interpolation theory some extensions including the estimation problem with several missing observations are also investigated

blind identifiability of a quadratic stochastic system quadratic systems appear in various fields of signal processing in particular in detection and estimation in this paper we establish some conditions to determine the order and identify the coefficients of any discrete and finite extent quadratic system driven by a sequence of independent and identically distributed random variables the input sequence is assumed unobservable and the conditions are based on some properties of the output cumulant functions up to the third order necessary and sufficient conditions ensuring the determination of the order are given a sufficient condition of uniqueness to recover the parameters is presented lastly some examples are given

wavelet based schemes for linear advection dispersion equation abstract in this paper two wavelet based adaptive solvers are developed for linear advection dispersion equation the localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock like behaviour the first framework is based on wavelet galerkin and the second is based on multiscale decomposition of finite element method coiflet wavelet filter is incorporated in both the methods the main advantage of both the adaptive methods is the elimination of spurious oscillations at very high peclet number

tensor product basis approximations for volterra filters the paper studies approximations for a class of nonlinear filters known as volterra filters although the volterra filter provides a relatively simple and general representation for nonlinear filtering it is often highly overparameterized due to the large number of parameters the utility of the volterra filter is limited the overparameterization problem is addressed in the paper using a tensor product basis approximation tpba in many cases a volterra filter may be well approximated using the tpba with far fewer parameters hence the tpba offers considerable advantages over the original volterra filter in terms of both implementation and estimation complexity furthermore the tpba provides useful insight into the filter response the paper studies the crucial issue of choosing the approximation basis several methods for designing an appropriate approximation basis and error bounds on the resulting mean square output approximation error are derived certain methods are known to be nearly optimal

correspondence theorem for fuzzy ideals abstract the ring r j of cosets of a fuzzy ideal j of a ring r is defined and shown to be isomorphic to a factor ring of r in a natural way if f r r is an epimorphism of rings a one to one order preserving correspondence is established between the fuzzy ideals of r and those of r which are constant on the kernel of f this generalizes the correspondence theorem of non fuzzy ideals

on the definition of a fuzzy subgroup abstract using the idea of quasi coincidence of a fuzzy point with a fuzzy set some new concepts of a fuzzy subgroup are introduced and their acceptibility is investigated some fundamental properties of one such viable fuzzy subgroup are obtained

optimal linear quadratic systems for detection and estimation the problem of linear quadratic systems for detection has long been solved by assuming the deflection criterion and gaussian noise it is shown here that the gaussian assumption can be removed and a complete solution is presented for an arbitrary probability distribution with finite fourth order moments the optimal solution can always be obtained by solving a linear system of equations some properties of the optimal systems are developed for particular examples of nongaussian noise it is shown that there is a strong relationship between linear quadratic optimal detection and optimal estimation which extends results known for the purely linear case

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

polyspectrum modeling using linear or quadratic filters the polyspectrum modeling problem using linear or quadratic filters is investigated in the linear case it is shown that if the output pth order cumulant function of a filter driven by a white noise is of finite extent then the filter necessarily has a finite extent impulse response it is proved that every factorable polyspectrum with a non gaussian white noise can also be modeled with a quadratic filter driven by a gaussian white noise it is shown that if the quadratic filter has a finite extent impulse response then the output pth order cumulant function is of finite extent and if the output pth order cumulant function of a quadratic filter is of finite extent then the impulse response may or may not be of finite extent it is shown that there exist finite and infinite extent pth order cumulant functions that are not factorable but can be modeled with quadratic filters

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

generalized fuzzy quotient subgroups abstract in this paper we consider the fuzzy subgroup of a given group defined by generalized triangular norms and conorms introduced by yager 1986 the definitions of a anti fuzzy subgroup and a anti pseudo fuzzy subgroup are reviewed then a generalized fuzzy cayley theorem as well as a partial generalization of fuzzy lagrange theorem are given also a unified approach for studying fuzzy quotient subgroups is discussed some correspondence theorems for several kinds of fuzzy subgroups are proved and examples of generalized triangular norms conorms are given

semi orthogonal wavelet packet bases for parallel least squares approximation a generalization to the wavelet packet case of the semi orthogonal wavelets given by c k chui is presented and a simple numerical algorithm for their practical evaluation is developed to provide a motivation it is shown how using these new bases the least squares approximation problem in a multiresolution space can easily be solved by means of an efficient parallel algorithm

fuzzy subgroups some characterizations the notion of a fuzzy set was first introduced in 1965 by zadeh 12 since then the theory of fuzzy sets has gone through remarkably rapid strides with well over 4000 papers by now see for example the bibliography 3 several textbooks for example 2 and 6 and an international journal solely devoted to it s the theory has found wideranging applications in such diverse fields as automata control theory decision theory and social behavior pattern studies to name just a few for an interesting expository account see rosenfeld lo and also the book review 7 in 9 rosenfeld introduced the notion of a fuzzy group and showed that many concepts of group theory can be extended in an elementary manner to develop the theory of fuzzy groups in particular he characterized all the fuzzy subgroups of cyclic groups of prime order 9 proposition 5 101 fuzzy groups were further investigated by das in 1 where he characterized all the fuzzy subgroups of finite cyclic groups l theorem 4 21 in this connection he introduced the notion of level subgroups of a fuzzy subgroup which is based on the notion of a level subset introduced earlier by zadeh for a finite group das showed that these level subgroups of a fuzzy subgroup form a chain 1 corollary 3 11 and also constructed a fuzzy subgroup whose chain of level subgroups is maximal in the following special cases when the group is i a direct product of cyclic groups of prime orders ii abelian l theorem 3 41 we analyze the level subgroups of a fuzzy subgroup in more detail and investigate whether the family of level subgroups of a fuzzy subgroup deter

not so fuzzy fuzzy ideals abstract a characterization of rings which are both artinian and noetherian is obtained they are proved to be the rings whose fuzzy ideals assume only finitely many distinct values prime fuzzy ideals have been defined and conditions have been obtained under which they have only two distinct values

fuzzy congruences and fuzzy normal subgroups abstract a complete description of the congruences on a group in terms of its normal subgroups can be seen in papers by a rosenfeld and w j howie in this paper we shall introduce the concept of fuzzy congruences on a groupoid and a group and characterize fuzzy congruences on a group in terms of fuzzy normal subgroups

fast algorithms for periodic spline wavelets on sparse grids we consider boolean sums of univariate interpolation operators which define multivariate jth order blending interpolation operators on sparse grids sample spaces are defined as range of the blending operators sample and wavelet spaces have significantly lower dimension and good approximation order for certain function spaces fast decomposition and reconstruction algorithms for bivariate spline wavelets based on algorithms for univariate functions are described operation counts for the algorithms are given and it is shown that the complexity depends linearly on the dimension of sample spaces

polynomial estimation of the amplitude of a signal the problem of estimating the amplitude of a signal is addressed using higher order statistics the probability distribution of the noise is assumed to be unknown so that the maximum likelihood estimator cannot be calculated the estimator is taken as a polynomial of the observation the coefficients of which are determined so that the estimate is unbiased with minimum variance this method generalizes the linear approach and the estimate variance is reduced the ease of linear quadratic estimation is detailed and numerical examples are presented

fuzzy subgroup some characterizations ii abstract we consider an analysis of fuzzy subgroups in terms of the corresponding family of level subgroups given a finite chain of subgroups of a group g we prove that there exists a fuzzy subgroup of g whose level subgroups are exactly the subgroups of this chain as a corollary we obtain an interpretation of the number of chains of subgroups of a group g in which a subgroup h is a member when the group g is a supersolvable group some further interpretation of this number is obtained