beurling dimension and self similar measures abstract in this paper the authors study the beurling dimension of bessel sets and frame spectra of some self similar measures on r d and obtain their exact upper bound of the dimensions which is the same given by dutkay et al 2011 8 the upper bound is attained in usual cases and some examples are given to explain our theory

spectrality of one dimensional self similar measures with consecutive digits abstract assume 0 1 and m is a prime let be the self similar measure defined by a 1 m j 0 m 1 1 a j a b we prove that l 2 contains an orthonormal basis of exponential functions if and only if 1 m k for some k n

a class of spectral moran measures abstract let d k k 1 be a sequence of digit sets in n and let b k k 1 be a sequence of integer numbers bigger than 1 we call the family f k d k x b k 1 x d d d k k 1 a moran iterated function system ifs which is a natural generalization of an ifs we prove under certain conditions in terms of b k d k that the associated moran measure is a spectral measure i e there exists a countable set n such that e 2 i x is an orthonormal basis for l 2

spectrality of a class of infinite convolutions for a finite set d z and an integer b 2 we say that b d is compatible with c z if e 2 idc b d d c c is a hadamard matrix let e 1 e a e a denote the uniformly discrete probability measure on e we prove that the class of infinite convolution moran measure b dk b 1d1 b 2d2 is a spectral measure provided that there is a common c z compatible to all the b dk and c c 0 1 b 1 we also give some examples to illustrate the hypotheses and results in particular the last condition on c is essential

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

spectral property of cantor measures with consecutive digits abstract we consider equally weighted cantor measures q b arising from iterated function systems of the form b 1 x i i 0 1 q 1 where q b we classify the q b so that they have infinitely many mutually orthogonal exponentials in l 2 q b in particular if q divides b the measures have a complete orthogonal exponential system and hence spectral measures improving the construction by dutkay et al 2009 3 we characterize all the maximal orthogonal sets when q divides b via a maximal mapping on the q adic tree in which all elements in are represented uniquely in finite b adic expansions and we can separate the maximal orthogonal sets into two types regular and irregular sets for a regular maximal orthogonal set we show that its completeness in l 2 q b is crucially determined by the certain growth rate of non zero digits in the tail of the b adic expansions of the elements furthermore we exhibit complete orthogonal exponentials with zero beurling dimensions these examples show that the technical condition in theorem 3 5 of dutkay et al 2011 4 cannot be removed for an irregular maximal orthogonal set we show that under some conditions its completeness is equivalent to that of the corresponding regularized mapping

spectral structure and spectral eigenvalue problems of a class of self similar spectral measures abstract let be a probability measure with compact support in r the measure is called a spectral measure if there exists a countable set r called a spectrum of such that the family of exponential functions e 2 i x forms an orthonormal basis for l 2 in this paper we study the structure of spectra and the real number t such that both and t are spectra for a class of self similar spectral measures which have symmetric spectra some examples are given to explain our theory

weighted fourier frames on self affine measures abstract continuing the ideas from our previous paper 6 we construct parseval frames of weighted exponential functions for self affine measures

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

spectrality and non spectrality of the riesz product measures with three elements in digit sets abstract let be a borel probability measure with compact support in r the is called a spectral riesz spectral frame spectral measure if there exists a set r such that the family of exponential functions e e 2 i x forms an orthonormal basis riesz basis frame for l 2 in this paper we study the spectrality and the non spectrality of a class of singular measures which is one of the fundamental works for the analysis on l 2 for a clear expression we use the simplest setting in this paper although the most results of ours can be extended to more general cases even to higher dimensions

the existence of fourier basis for some moran measures abstract let b n n 1 be a sequence of integers bigger than 1 and let d n n 1 be a sequence of digit sets in z where d n 0 r n 2 r n q n 1 r n the family of functions f n d x b n 1 x d d d n n 1 is called a moran iterated function system ifs in this paper we prove that the associated moran measure generated by an infinite convolution of atomic measures with equal distribution b n d n b 1 1 d 1 b 1 b 2 1 d 2 b 1 b 2 b n 1 d n is a spectral measure if r n q n b n

spectral property of self affine measures on mathbb r n abstract we study spectral properties of the self affine measure m d generated by an expanding integer matrix m m n z and a consecutive collinear digit set d 0 1 q 1 v where v z n 0 and q 2 is an integer some sufficient conditions for m d to be a spectral measure or to have infinitely many orthogonal exponentials are given moreover for some special cases we can obtain a necessary and sufficient condition on the spectrality of m d our study generalizes the one dimensional results proved by dai et al 4 5

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

a class of spectral self affine measures with four element digit sets abstract the self affine measure m d associated with an expanding matrix m m n z and a finite digit set d z n is uniquely determined by the self affine identity with equal weight in this paper we construct a class of self affine measures m d with four element digit sets in the higher dimensions n 3 such that the hilbert space l 2 m d possesses an orthogonal exponential basis that is m d is spectral such a spectral measure cannot be obtained from the condition of compatible pair this extends the corresponding result in the plane

tiling properties of spectra of measures we investigate tiling properties of spectra of measures i e sets lambda in mathbb r such that e 2 pi i lambda x lambda in lambda forms an orthogonal basis in l 2 mu where mu is some finite borel measure on mathbb r such measures include lebesgue measure on bounded borel subsets finite atomic measures and some fractal hausdorff measures we show that various classes of such spectra of measures have translational tiling properties this lead to some surprizing tiling properties for spectra of fractal measures the existence of complementing sets and spectra for finite sets with the coven meyerowitz property the existence of complementing hadamard pairs in the case of hadamard pairs of size 2 3 4 or 5 in the context of the fuglede conjecture we prove that any spectral set is a tile if the period of the spectrum is 2 3 4 or 5

on spectral n bernoulli measures abstract for 0 1 and n 1 an integer let be the self similar measure defined by i 0 n 1 1 n 1 i we prove that l 2 has an exponential orthonormal basis if and only if 1 q for some q 0 and n divides q the special case is the cantor measure with 1 2 k and n 2 16 which was proved recently to be the only spectral measure among the bernoulli convolutions with 0 1 4

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

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

translational absolute continuity and fourier frames on a sum of singular measures abstract a finite borel measure in r d is called a frame spectral measure if it admits an exponential frame or fourier frame for l 2 it has been conjectured that a frame spectral measure must be translationally absolutely continuous which is a criterion describing the local uniformity of a measure on its support in this paper we show that if any measures and without atoms whose supports form a packing pair then t is translationally singular and it does not admit any fourier frame in particular we show that the sum of one fourth and one sixteenth cantor measure 4 16 does not admit any fourier frame we also interpolate the mixed type frame spectral measures studied by lev and the measure we studied in doing so we demonstrate a discontinuity behavior for any anticlockwise rotation mapping r with 2 the two dimensional measure 4 0 0 16 r 1 supported on the union of x axis and y cot x always admit a fourier frame furthermore we can find e 2 i x such that it forms a fourier frame for with frame bounds independent of nonetheless 2 does not admit any fourier frame

yet another characterization of solutions of the algebraic riccati equation abstract this paper deals with a characterization of the solution set of algebraic riccati equation are over reals for both controllable and uncontrollable systems we characterize all solutions using simple linear algebraic arguments it turns out that solutions of are of maximal rank have lower rank solutions encoded within it we demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution we characterize situations where there are no full rank solutions to the are we also characterize situations when the number of solutions to the are is finite when they are infinite and when they are bounded we also explore the poset structure on the solution set of are which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix we provide several examples that bring out the essence of these results

spectral property of certain fractal measures abstract let 0 a j b j 0 1 2 mod 3 be a sequence of digit sets in z and let n j 3 r j be a sequence of integers bigger than 1 we call f j d x n j 1 x d d 0 a j b j j 0 a moran iterated function system which is a generalization of an ifs we prove that the associated moran measure is spectral

characterization of solutions of non symmetric algebraic riccati equations abstract we give a new characterization of the solution set of non symmetric algebraic riccati equations involving real matrices our characterization involves the use of invariant subspaces of the coefficient matrices we also give a poset structure on the solutions of are and explore some properties of this poset partially ordered set

weighted fourier frames on fractal measures abstract we generalize an idea of picioroaga and weber 10 to construct parseval frames of weighted exponential functions for self affine measures

on the orthogonal exponential functions of a class of planar self affine measures abstract for an expanding real matrix m 1 c 0 1 m 2 r and a digit set d 0 0 t 1 0 t 0 1 t let m d be the self affine measure generated by m and d in this paper we show that l 2 m d admits an infinite orthogonal set of exponential functions if and only if q p 1 r and c 1 for some positive integers p q r with p 3 z gcd p q 1 and q moreover if l 2 m d does not admit any infinite orthogonal set of exponential functions we estimate the number of orthogonal exponential functions in l 2 m d and give the exact maximal cardinality

on spectral cantor measures a probability measure in rd is called a spectral measure if it has an orthonormal basis consisting of exponentials in this paper we study spectral cantor measures we establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral cantor measure these results extend the studies by jorgensen and pedersen j anal math 75 1998 185 228 and strichartz j d analyse math 81 2000 209 238

uniformity of measures with fourier frames abstract we examine fourier frames and more generally frame measures for different probability measures we prove that if a measure has an associated frame measure then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support to be more precise by considering certain absolute continuity properties of the measure and its translation we recover the characterization on absolutely continuous measures g d x with fourier frames obtained in 24 moreover we prove that the frame bounds are pushed away by the essential infimum and supremum of the function g this also shows that absolutely continuous spectral measures supported on a set if they exist must be the standard lebesgue measure on up to a multiplicative constant we then investigate affine iterated function systems ifss we show that if an ifs with no overlap admits a frame measure then the probability weights are all equal moreover we also show that the aba wang conjecture 20 is true if the self similar measure is absolutely continuous finally we will present a new approach to the conjecture of liu and wang 29 about the structure of non uniform gabor orthonormal bases of the form g g j

spectra of a class of self affine measures abstract in the present paper we will study the spectral property of a class of self affine measures under the condition of compatible pair we first answer a question of dutkay and jorgensen concerning the relationship between spectral self affine measure and compatible pair we then consider the spectra of bernoulli convolutions and obtain a sharp result which extends the corresponding result of jorgensen kornelson and shuman finally we provide a structural property for the integer spectrum of a spectral self affine measure

μm d orthogonality and compatible pair abstract let m d be a self affine measure associated with an expanding integer matrix m m n z and a finite subset d z n in the present paper we study the m d orthogonality and compatible pair conditions as well as relations between them the research here is based on the structure of vanishing sums of roots of unity and is closely related to the problem of spectral self affine measure

non spectral problem for a class of planar self affine measures abstract the self affine measure m d corresponding to an expanding matrix m m n r and a finite subset d r n is supported on the attractor or invariant set of the iterated function system d x m 1 x d d d the spectral and non spectral problems on m d including the spectrum tiling problem implied in them have received much attention in recent years 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 if a b c z a 1 c 1 and a c z 3 z m a b 0 c and d 0 0 1 0 0 1 then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this extends several known conclusions the proof of such result depends on the characterization of the zero set of the fourier transform m d and provides a way of dealing with the non spectral problem

non spectral self affine measure problem on the plane domain the self affine measure m d corresponding to an expanding integer matrix m abcd andd 00 10 01 is supported on the attractor or invariant set of the iterated function system d x m 1 x d d d in the present paper we show that if a d 2 4 ad bc and ad bc is not a multiple of 3 then there exist at most 3 mutually orthogonal exponential functions in l2 m d and the number 3 is the best this extends several known results on the non spectral self affine measure problem the proof of such result depends on the characterization of the zero set of the fourier transform m d and provides a way of dealing with the non spectral problem

orthogonal exponentials translations and bohr completions abstract we are concerned with an harmonic analysis in hilbert spaces l 2 where is a probability measure on r n the unifying question is the presence of families of orthogonal complex exponentials e x exp 2 i x in l 2 this question in turn is connected to the existence of a natural embedding of l 2 into an l 2 space of bohr almost periodic functions on r n in particular we explore when l 2 contains an orthogonal basis of e functions for in a suitable discrete subset in r n i e when the measure is spectral we give a new characterization of finite spectral sets in terms of the existence of a group of local translation we also consider measures that arise as fixed points in the sense of hutchinson of iterated function systems ifss and we specialize to the case when the function system in the ifs consists of affine and contractive mappings in r n we show in this case that if is then assumed spectral then its partitions induced by the ifs at hand have zero overlap measured in this solves part of the aba wang conjecture as an application of the new non overlap result we solve the spectral pair problem for bernoulli convolutions advancing in this way a theorem of ka sing lau in addition we present a new perspective on spectral measures and orthogonal fourier exponentials via the bohr compactification

on spectral cantor moran measures and a variant of bourgain s sum of sine problem abstract in this paper we show that if we have a sequence of hadamard triples n n b n l n with b n 0 1 n n 1 for n 1 2 except an extreme case then the associated cantor moran measure n n b n 1 n 1 b 1 1 n 1 n 2 b 2 1 n 1 n 2 n 3 b 3 n n with support inside 0 1 always admits an exponential orthonormal basis e e 2 i x for l 2 where is obtained from suitably modifying l n here n is the convolution of the first n dirac measures and n denotes the tail term we show that the completeness of e in general depends on the equi positivity of the sequence of the pull backed tail of the cantor moran measure n n n 1 n n 1 such equi positivity can be analyzed by the integral periodic zero set of the weak limit of n this result offers a new conceptual understanding of the completeness of exponential functions and it improves significantly many partial results studied by recent research whose focus has been specifically on b n 4 using the bourgain s example that a sum of sine can be asymptotically small we show that in the extreme case there exists some cantor moran measure such that the equi positive condition fails and the fourier transform of the associated n uniformly converges on some unbounded set

tiling the line with translates of one tile a region t is a closed subset of the real line of positive finite lebesgue measure which has a boundary of measure zero call a region t a tile if bbb r can be tiled by measure disjoint translates of t for a bounded tile all tilings of bbb r with its translates are periodic and there are finitely many translation equivalence classes of such tilings the main result of the paper is that for any tiling of bbb r by a bounded tile any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling from it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups

on spectral eigenvalue problem of a class of generalized cantor measures abstract let q 4 r with integer r 1 let a a 1 a 2 b b 1 b 2 0 1 2 3 with a b i e a n b n for all n 1 we consider the following generalized cantor measure q a b 1 q a 1 b 1 1 q 2 a 2 b 2 where e 1 e a e a denotes the uniformly discrete probability on e in this paper we investigate the spectral eigenvalues of the second type of the measure q a b the spectrality of q a b is settled by he et al 2017 17 more precisely we characterize the possible real numbers t satisfying that there exists a countable set r such that and t are spectra of q a b simultaneously i e the sets e e 2 i x and e t e 2 i t x are both orthonormal basis for l 2 q a b

tree structure of spectra of spectral self affine measures abstract for an integral self affine spectral measure if the zeros of its fourier transform are all integral vectors it is proven that any its spectrum has a tree structure for any subset with such tree structure a sufficient condition and a necessary condition for the subset to be a spectrum are given respectively applications are given to some known results as special cases

renormalization group theory and variational calculations for propagating fronts we study the propagation of uniformly translating fronts into a linearly unstable state both analytically and numerically we introduce a perturbative renormalization group rg approach to compute the change in the propagation speed when the fronts are perturbed by structural modification of their governing equations this approach is successful when the fronts are structurally stable and allows us to select uniquely the numerical experimentally observable propagation speed for convenience and completeness the structural stability argument is also briefly described we point out that the solvability condition widely used in studying dynamics of nonequilibrium systems is equivalent to the assumption of physical renormalizability we also implement a variational principle due to hadeler and rothe which provides a very good upper bound and in some cases even exact results on the propagation speeds and which identifies the transition from linear to nonlinear marginal stability as parameters in the governing equation are varied

spectra of cantor measures the cantor measures mu q b with 2 q b q in mathbb z are few of known singular continuous measures that admits a spectrum the spectra of cantor measures mu q b are not unique and their structure is not well studied it has been observed that maximal orthogonal sets of the cantor measure mu q b have tree structure and they can be built by selecting maximal tree appropriately a challenging problem is when a maximal orthogonal set becomes a spectrum in this paper we introduce a measurement on the tree structure associated with a maximal orthogonal set lambda of the cantor measure mu q b and we use boundedness and linear increment of that measurement to justify whether lambda is a spectrum or not as applications of our justification we provide a characterization for the expanding set k lambda of a spectrum lambda to be a spectrum again furthermore we construct a spectrum lambda such that for some integer k the shrinking set lambda k is a maximal orthogonal set but not a spectrum

orthogonal exponentials on the generalized three dimensional sierpinski gasket abstract the self affine measure m d corresponding to the expanding integer matrix m p 0 m 0 p 0 0 0 p and d 0 0 0 1 0 0 0 1 0 0 0 1 is supported on the generalized three dimensional sierpinski gasket t m d where p is odd in the present paper we show that there exist at most 7 mutually orthogonal exponential functions in l 2 m d this generalizes the result of dutkay and jorgensen d e dutkay p e t jorgensen analysis of orthogonality and of orbits in affine iterated function systems math z 256 2007 801 823 on the non spectral self affine measure problem by using the same method we also obtain that for self affine measure m d corresponding to the expanding integer matrix m p 0 0 p and d 0 0 1 0 0 1 1 1 where p is odd there exist at most 5 mutually orthogonal exponential functions in l 2 m d

total linear least squares and the algebraic riccati equation abstract it is shown that the solution to a total linear least squares problem satisfies a quadratic matrix equation which turns into an algebraic riccati equation when the matrix of unknowns is square if there is an additional symmetry constraint on the solution the optimal solution is given by the anti stabilizing solution of this riccati equation

tiling the integers with translates of one finite set abstract a set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set we consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers for sets of prime power size it was solved by d newman 1977 j number theory 9 107 111 we solve it for sets of size having at most two prime factors the conditions are always sufficient but it is unknown whether they are necessary for all finite sets

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

scaling of spectra of self similar measures with consecutive digits abstract we consider the equally weighted cantor measures p q generated by the iterated function system ifs f i x x p i q i 0 q 1 where 2 q z and q p r it is known that if q divides p then p q is a spectral measure with a spectrum p q 0 1 q 1 p 0 1 q 1 p 2 0 1 q 1 finite sum dai he and lai 2013 6 in this paper the authors study some positive integers b such that b p q is also a spectrum of p q

spectral measures with arbitrary hausdorff dimensions abstract in this paper we consider spectral properties of riesz product measures supported on homogeneous cantor sets and we show the existence of spectral measures with arbitrary hausdorff dimensions including non atomic zero dimensional spectral measures and one dimensional singular spectral measures

spectral sets and factorizations of finite abelian groups aspectral setis a subset ofrnwith lebesgue measure 0 such that there exists a set of exponential functions which form an orthogonal basis ofl2 the spectral set conjecture of b fuglede states that a set 0 is a spectral set if and only if tilesrnby translation we study sets which tilernusing a rational periodic tile set s zn a where a 1 n1 z 1 nn zis finite we characterize geometrically bounded measurable sets that tilernwith such a tile set certain tile sets s have the property that every bounded measurable set which tilesrnwith s is a spectral set with a fixed spectrum s we call sa universal spectrum for such s we give a necessary and sufficient condition for a rational periodic set to be a universal spectrum for s which is expressed in terms of factorizationsa b gwhereg zn1 znn anda a modzn in dimensionn 1 we show that s has a universal spectrum whenevern1is the order of a good group in the sense of hajos and for various other sets s

spectrality of infinite bernoulli convolutions let bk ak k 1 be a sequence of positive integers with upper bound and let e be the uniformly discrete probability measure on the finite set e for 0 1 the infinite convolution ak bk a1 b1 2 a2 b2 is called an infinite bernoulli convolution in this paper we investigate whenever there exists such that e 2 i x is an orthonormal basis for l2 ak bk

fourier duality for fractal measures with affine scales for a family of fractal measures wend an explicit fourier d uality the measures in the pair have compact support in r d and they both have the same matrix scaling but the two use dierent translation vectors one by a subset b in r d and the other by a related subset l among other things we show that there is then a pair of innite disc rete sets l and b in r d such that the l fourier exponentials are orthogonal in l 2 b and the b fourier exponentials are orthogonal in l 2 l these sets of orthogonal frequencies are typically lac unary and they will be obtained by scaling in the large the nature of our duality is explored below both in higher dimensions and for examples on the real line our duality pairs do not always yield orthonormal fourier bases in the respective l 2 hilbert spaces but depending on the geometry of certainnite orbit s we show that they do in some cases we further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these ane fractal measures

spectrality of the planar sierpinski family abstract let be a borel probability measure with compact support in r 2 is called a spectral measure if there exists a countable set r 2 such that e e 2 i x is an orthonormal basis for l 2 in this note we prove that the integral sierpinski measure a d is a spectral measure if and only if a d is admissible this completely settles the spectrality of integral sierpinski measures in r 2

on the fourier orthonormal bases of cantor moran measures abstract let d n p n n 1 be a sequence of integers so that 0 d n p n for n 1 the infinite convolution of probability measures with finite support and equal distribution p n d n p 1 1 0 d 1 p 1 p 2 1 0 d 2 is a borel probability measure cantor moran measure in this paper we study the existence of fourier basis for l 2 p n d n i e find a discrete set such that e e 2 i x is an orthonormal basis for l 2 p n d n we give some sufficient conditions for this aim and some examples to explain the theory

spectral property of the bernoulli convolutions for 0 1 let be the bernoulli convolution associated with jorgensen and pedersen p jorgensen s pedersen dense analytic subspaces in fractal l2 spaces j anal math 75 1998 185 228 proved that if 1 q where q is an even integer then l2 has an exponential orthonormal basis we show that for any 0 1 l2 contains an infinite orthonormal set of exponential functions if and only if is the nth root of a fraction p q where p is an odd integer and q is an even integer

on fourier frame of absolutely continuous measures let be a compactly supported absolutely continuous probability measure on rn we show that l2 k d admits a fourier frame if and only if its radon nikodym derivative is bounded above and below almost everywhere on the support k as a consequence we prove that if is an equal weight absolutely continuous self similar measure on r1 and l2 k d admits a fourier frame then the density of must be a characteristic function of self similar tile in particular this shows for almost everywhere 1 2 1 the l2 space of the bernoulli convolutions cannot admit a fourier frame

spectral structure of digit sets of self similar tiles on mathbb r 1 we study the structure of the digit sets d for the integral selfsimilar tiles t b d we call such a d a tile digit set with respect to b so far the only available classes of such tile digit sets are the complete residue sets and the product forms our investigation here is based on the spectrum of the mask polynomial pd i e the zeros of pd on the unit circle by using the fourier criteria of self similar tiles of kenyon and protasov as well as the algebraic techniques of cyclotomic polynomials we characterize the tile digit sets through some product of cyclotomic polynomials kernel polynomials which is a generalization of the product form to higher order

orthogonal exponential functions of self similar measures with consecutive digits in r abstract suppose that 0 1 and m 2 is an integer let m be the self similar measure defined by m 1 m j 0 m 1 m 1 j in this paper we prove that l 2 m contains an infinite orthonormal set of exponential functions if and only if q p 1 r for some p q r n with gcd p q 1 and gcd p m 1

spectrality of certain moran measures with three element digit sets abstract let d n 0 a n b n 0 1 2 m o d 3 p n 3 z n 1 satisfy sup n 1 max a n b n p n it is well known that there exists a unique borel probability measure p n d n generated by the following infinite convolution product p n d n p 1 1 d 1 p 1 p 2 1 d 2 in the weak convergence in this paper we give some conditions to ensure that there exists a discrete set such that the exponential function system e 2 i x forms an orthonormal basis for l 2 p n d n

spectral properties of self similar measures with product form digit sets abstract in this paper we consider the spectral properties of self similar measures r d generated by the integer r n q and the product form digit set d 0 1 n 1 n p 0 1 n 1 where the integers q p 1 and n 2 we show that r d is a spectral measure if and only if q p

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

scaling of spectra of a class of random convolution on r abstract given a borel probability measure on r and a real number p we call p a spectral eigenvalue of the measure if there exists a discrete set such that the sets e e 2 i x and e p e 2 i p x are both orthonormal basis for hilbert space l 2 in the present paper we determine the spectral eigenvalues of a class of random convolution on r

spectrality of moran measures with four element digit sets abstract let e 1 e a e a denote the uniformly discrete probability measure on a finite set e we prove that the infinite convolution moran measure b d k b 1 d 1 b 2 d 2 admits an orthonormal basis of exponential provided that d k k 1 is a uniformly bounded sequence of 4 digit spectral sets b 2 l 1 q with q 1 an odd integer and l sufficiently large depends on d k we also give some examples to illustrate the result

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

defect correction method for the solution of algebraic riccati equations the solution of discrete and continuous algebraic riccati equations is considered it is shown that if an approximate solution is obtained then the defect for this solution again solves an algebraic riccati equation of the same form and that the system properties of detectability and stabilizability are inherited by this defect equation on the basis of these results a general defect correction method is proposed and numerical examples are given for the use of this method in combination with a bunse gerstner and v mehrmann s 1986 sr method

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