extension of the berlekamp massey algorithm to n dimensions abstract we present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n dimensional array over any field where the minimality is defined with respect to the partial order over the n dimensional lattice the algorithm is an extension of our two dimensional version of the berlekamp massey algorithm to more than two dimensions the n dimensional theory is based on more general concepts which can be reduced into those of the two dimensional theory in the previous paper in a typical case the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a groebner basis of the ideal defined by the array and consequently the structure of an n dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it

on the spectra of a class of self affine measures on r2 abstract for i 1 2 let 0 i 1 for an expanding real matrix m d i a g 1 1 2 1 and a three element digit set d z 2 with cardinality d let m d be the self affine measure defined by m d 1 d d d m d m d let f 3 2 1 3 l 1 l 2 t l 1 l 2 n 0 l 1 l 2 2 and z m d x r 2 d d e 2 i d x 0 if a f 3 2 z m d a mod z 2 and x 0 1 2 d d e 2 i d x d 0 0 t then l 2 m d has an exponential orthonormal basis if and only if i 1 3 z 0 for i 1 2

spectral property of a class of moran measures on r abstract let b 2 be a positive integer let d be a finite subset of z and n k k 1 n be a sequence of strictly increasing numbers a moran measure b d n k is a borel probability measure generated by the moran iterated function system moran ifs f k d x b n k 1 n k x d d d k n n 0 0 in this paper we study one of the basic problems in fourier analysis associated with b d n k more precisely we give some conditions under which the measure b d n k is a spectral measure i e there exists a discrete subset r such that e e 2 i x is an orthonormal basis for l 2 b d n k

iterated function systems ruelle operators and invariant projective measures we introduce a fourier based harmonic analysis for a class of discrete dynamical systems which arise from iterated function systems our starting point is the following pair of special features of these systems 1 we assume that a measurable space x comes with a finite to one endomorphism r x x which is onto but not one to one 2 in the case of affine iterated function systems ifss in r d this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets b l in r d of the same cardinality which generate complex hadamard matrices our harmonic analysis for these iterated function systems ifs x is based on a markov process on certain paths the probabilities are determined by a weight function w on x from w we define a transition operator r w acting on functions on x and a corresponding class h of continuous r w harmonic functions the properties of the functions in h are analyzed and they determine the spectral theory of l 2 for affine ifss we establish orthogonal bases in l 2 these bases are generated by paths with infinite repetition of finite words we use this in the last section to analyze tiles in r d

on the minimal partial realization of 2d discrete linear shift invariant systems the partial realization problem for 2d discrete linear shift invariant systems is discussed the authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2d linear system characterized by a given 2d impulse response array the method used is based on the 2d berlekamp massey algorithm which has a close connection with the 2d hankel matrix derived from the 2d array while it compares in efficiency with other methods for identification and approximation of 2d linear systems it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree

on the spectra of sierpinski type self affine measures abstract for a class of sierpinski type self affine measures on r 2 a characterization for all maximal orthogonal sets of exponentials is given by this characterization a sufficient condition for a maximal orthogonal set to be a basis of l 2 is given

analysis of orthogonality and of orbits in affine iterated function systems we introduce a duality for affine iterated function systems aifs which is naturally motivated by group duality in the context of traditional harmonic analysis our affine systems yield fractals defined by iteration of contractive affine mappings we build a duality for such systems by scaling in two directions fractals in the small by contractive iterations and fractals in the large by recursion involving iteration of an expansive matrix by a fractal in the small we mean a compact attractor x supporting hutchinson s canonical measure and we ask when is a spectral measure i e when the hilbert space l 2 mu has an orthonormal basis onb of exponentials e lambda lambda in lambda we further introduce a fourier duality using a matched pair of such affine systems using next certain extreme cycles and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary fourier series and which serve as spectra for x our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for x our results in turn are illustrated with concrete sierpinski like fractals in dimensions 2 and 3

the cardinality of orthogonal exponentials of planar self affine measures with three element digit sets abstract in this paper we consider the planar self affine measures m d generated by an expanding matrix m m 2 z and an integer digit set d 0 0 1 2 1 2 with 1 2 2 1 0 we show that if det m 3 z then the mutually orthogonal exponential functions in l 2 m d is finite and the exact maximal cardinality is given

local nash inequality and inhomogeneity of heat kernels the local nash inequality is introduced as a natural extension of the classical nash inequality yielding a space homogeneous upper heat kernel estimate the local nash inequality contains local information of the heat kernel and is a necessary condition for the space inhomogeneous heat kernel estimate involving the volume of balls like the one obtained by li and yau for a complete riemannian manifold with non negative ricci curvature under the volume doubling property the local nash inequality combined with the exit time estimate is shown to be equivalent to a sub gaussian off diagonal upper estimate of the heat kernel allowing space inhomogeneity

quasiperiodic spectra and orthogonality for iterated function system measures we extend classical basis constructions from fourier analysis to attractors for affine iterated function systems ifss this is of interest since these attractors have fractal features e g measures with fractal scaling dimension moreover the spectrum is then typically quasi periodic but non periodic i e the spectrum is a small perturbation of a lattice due to earlier research on ifss there are known results on certain classes of spectral duality pairs also called spectral pairs or spectral measures it is known that some duality pairs are associated with complex hadamard matrices however not all ifss x admit spectral duality when x is given we identify geometric conditions on x for the existence of a fourier spectrum serving as the second part in a spectral pair we show how these spectral pairs compose and we characterize the decompositions in terms of atoms the decompositions refer to tensor product factorizations for associated complex hadamard matrices

an extension of łaba wang s theorem abstract for a positive integer b 2 and two finite subsets d c of z with the same cardinality we say that the pair b 1 d c is a compatible pair if the matrix e 2 i d c b d d c c is orthogonal let n j j 1 n be any sequence of positive integers in this paper we will show under the condition of compatible pair b 1 d c that the homogeneous moran measure b d n j b n 1 d b n 1 n 2 d b n 1 n 2 n k d has an orthonormal basis consisting of exponentials this extends a result of aba and wang 2002 22 and establishes the proof of conjecture 1 1 of fu and wen 2015 18

on the spectra of a cantor measure abstract we analyze all orthonormal bases of exponentials on the cantor set defined by jorgensen and pedersen in j anal math 75 1998 185 228 a complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one to one correspondence with the spectral labelings of the infinite binary tree with the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum an orthonormal basis this result not only provides us an easy and efficient way to construct various of new spectra for the cantor measure but also extends many previous results in the literature in fact most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition we also obtain two new conditions for a labeling tree to generate a spectrum when other digits digits not necessarily in 0 1 2 3 are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling these new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis

self affine tiles in ℝn abstract a self affine tile in r n is a set t of positive measure with a t d d t d where a is an expanding n n real matrix with det a m an integer and d d d 2 d m r n is a set of m digits it is known that self affine tiles always give tilings of r n by translation this paper extends known characterizations of digit sets d yielding self affine tiles it proves several results about the structure of tilings of r n possible using such tiles and gives examples showing the possible relations between self replicating tilings and general tilings which clarify results of kenyon on self replicating tilings

roof duality for polynomial 0 1 optimization the purpose of this note is to generalize the roof duality theory of hammer hansen and simeone to the case of polynomial 0 1 optimization 0 1pp by reformulating 0 1pp and expanding some of their definitions we show that most of the results for quadratic 0 1 problem 0 1qp can be extended to the general polynomial case

spectral theory for borel sets in rn of finite measure abstract let be a pair of subsets in r n such that has finite positive lebesgue measure the pair is called a spectral pair if the functions e i x for form an orthogonal total family in l 2 i e when restricted to our present work continues earlier papers on simultaneous diagonalization of vector fields for the special case when is also assumed open however the setting here is more general and the results are stated in the measure theoretic category in considering fundamental domains for lattices for example we allow distinct lattice point translates of the domain to overlap on sets of measure zero rather than the traditional condition of empty overlap two classes of examples of spectral pairs are known fundamental domains and simple factors the latter may be identified in terms of a certain finite group action we show that every spectral pair can be factored in the sense of cartesian products as a product of two pairs of lower dimension where the first factor is a fundamental domain and the second a simple factor the dimensions add a simple factor yields in a natural way a finite covering of an associated compact torus but not every such covering corresponds to a spectral pair we give a complete spectral characterization of the finite coverings and identify those which correspond to simple factors the latter identification is phrased in terms of generators and relations for certain c algebras which are analogous to but different from algebras studied earlier by cuntz and arveson

rational approximation of 2 d linear discrete systems in this paper we present an efficient procedure to obtain a rational model for a 2 d linear shift invariant discrete system using first and second order data from it this procedure is a modification of the nonlinear least squares approximation and it generalizes the pade approximants and the spectral estimation modeling procedures the parameters of the approximating filter are obtained by solving a system of linear equations by means of an efficient recursive algorithm which is developed using the relation of the approximation problem with the theory of orthogonal polynomials on the unit bidisk we discuss some of the algebraic properties of the solution and apply them to define cases for which the bibo stability of the approximating filters is ensured the proposed procedure finds applications in the design and stabilization of 2 d recursive digital filters and in the autoregressive moving average arma modeling of stationary random fields

the cardinality of certain μm d orthogonal exponentials abstract the self affine measures m d corresponding to the case i m p i 3 d 0 e 1 e 2 e 3 in the space r 3 and the case ii m p i 2 d 0 e 1 e 2 e 1 e 2 in the plane r 2 are non spectral where p 1 is odd i n is the n n identity matrix and e 1 e n are the standard basis of unit column vectors in r n one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that in both cases i and ii there are at most 4 mutually orthogonal exponentials in l 2 m d each and the number 4 is the best

fourier frequencies in affine iterated function systems abstract we examine two questions regarding fourier frequencies for a class of iterated function systems ifs these are iteration limits arising from a fixed finite families of affine and contractive mappings in r d and the ifs refers to such a finite system of transformations or functions the iteration limits are pairs x where x is a compact subset of r d the support of and the measure is a probability measure determined uniquely by the initial ifs mappings and a certain strong invariance axiom the two questions we study are 1 existence of an orthogonal fourier basis in the hilbert space l 2 x and 2 explicit constructions of fourier bases from the given data defining the ifs

self similarity of volume measures for laplacians on p c f self similar fractals our main goal in this paper is to obtain a precise analogue of weyl s asymptotic formula for the eigenvalue distribution of laplacians on a certain class of finitely ramified or p c f self similar fractals building in particular on the work of 7 9 22 24 our main result consists in precisely identifying for the class of decimable fractals the volume measures constructed by the second author in 24 for general p c f fractals and showing that they are self similar

spectral theory of finite volume domains in rn abstract let be an arbitrary open subset of r n of finite positive measure and assume the existence of a subset r n such that the exponential functions e exp i 1x1 nxn 1 n form an orthonormal basis for l 2 with normalized measure assume 0 and define subgroups k and a of r n by k 0 r n 2 z a a r n ua m u a m where ut is the unitary representation of r n on l 2 given by ute eit e t r n and where m is the multiplication algebra of l on l2 assume that a is discrete then there is a discrete subgroup d a of dimension n a fundamental domain d for d and finite sets of representers r r r each containing 0 r for a k in k0 and r for a k in a such that is disjoint union of translates of d a r a d neglecting null sets and r d0 if r is a set of representers for d a in d then r k is a translation set for i e r n direct sum neglecting null sets the case a r n corresponds to d d0 and k this last case corresponds in turn to a function theoretic assumption of forelli

on vanishing sums of roots of unity an unsolved problem in number theory asked the following for a given natural number m what are the possible integers n for which there exist mth roots of unity 1 n c such that 1 n 0 we show in this paper that the set of all possible n s is exactly the collection of n combinations of the prime divisors of m where n denotes the set of all non negative integers the proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity couched in the setting of integral group rings of finite cyclic groups our techniques also recovered with ease some of the classical results on vanishing sums of roots of unity such as those of redei de bruijn and schoenberg

lattice parameter autoregressive modeling of two dimensional fields part i the quarter plane case quarter plane autoregressive lattice parameter modeling of two dimensional fields is developed as a natural extension of one dimensional lattice parameter theory starting with a given data field four prediction error fields are generated their linear combination is used to define and calculate 2 d lattice parameter reflection factors for successive lattice parameter model stages in addition to developing the basic theory the presentation includes the relationships between the quarter plane two dimensional transfer function and the lattice parameters the lattice parameter synthesis model for generating a two dimensional field from a random field input and lattice parameter stability conditions the theory has been confirmed by computer simulations the extension of this lattice parameter theory for the asymmetric half plane case is to be presented in a subsequent paper

monic representations of the cuntz algebra and markov measures abstract we study representations of the cuntz algebras o n while for fixed n the set of equivalence classes of representations of o n is known not to have a borel cross section there are various subclasses of representations which can be classified we study monic representations of o n that have a cyclic vector for the canonical abelian subalgebra we show that o n has a certain universal representation which contains all positive monic representations a large class of examples of monic representations is based on markov measures we classify them and as a consequence we obtain that different parameters yield mutually singular markov measure extending the classical result of kakutani the monic representations based on the kakutani measures are exactly the ones that have a one dimensional cyclic s i invariant space

when does a bernoulli convolution admit a spectrum abstract in this paper we solve a long standing problem on bernoulli convolutions in particular we show that the bernoulli convolution with contraction rate 0 1 admits a spectrum if and only if is the reciprocal of an even integer

on the μm d orthogonal exponentials abstract let m d be a self affine measure corresponding to a given affine iterated function system d x m 1 x d d d in the present paper we will study the problem of how to determine the l 2 m d space has finite or infinite orthogonal exponentials such research is motivated by a conjecture on the non spectrality of m d we first obtain a non spectral criterion which extends the result of dutkay and jorgensen in opposition to the condition of this criterion we then obtain some conditions which imply the infinite orthogonal systems in l 2 m d these are necessary for further investigation on the spectrality of m d as an application we completely settle the corresponding problem for the generalized planar sierpinski family

sierpinski type spectral self similar measures abstract we consider the sierpinski type self similar measure on r 2 with contraction ratio 0 1 we show that is a spectral measure if and only if 1 3 p for some integer p 0 a similar characterization for bernoulli convolution is due to dai 2 over which 1 2 p