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

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

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

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

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

spectral gaps for sets and measures if x is a closed subset of the real line denote by gx the supremum of the size of the gap in the fourier spectrum of a measure taken over all non trivial finite complex measures supported on x in this paper we attempt to find gx in terms of x

on equations for union free regular languages in this paper we consider the variety uf generated by all algebras of binary relations equipped with the operations of composition reflexive transitive closure and the empty set and the identity relation as constants this variety coincides with the variety generated by the union free reducts of kleene algebras of languages and its free objects are formed by union free regular languages that is regular languages represented by regular expressions having no occurrence of we show that the variety uf is not finitely based the situation does not change if we consider the variety uf generated by the above algebras of binary relations equipped with the conversion operation

bayesian bootstrap for proportional hazards models we propose two bayesian bootstrap extensions the binomial and poisson forms for proportional hazards models the binomial form bayesian bootstrap is the limit of the posterior distribution with a beta process prior as the amount of the prior information vanishes and thus can be considered as a default nonparametric bayesian analysis it is also the same as lo s bayesian bootstrap for censored data when covariates are absent the poisson form bayesian bootstrap is equivalent to the bayesian analysis with cox s profile likelihood when the baseline distribution is discrete thus when the data set has many ties simulation study suggests that the binomial form bayesian bootstrap performs better than standard frequentist procedures in the frequentist sense an advantage of the proposed bayesian bootstrap procedures over the standard bayesian analysis is conceptual and computational simplicity finally it is shown that both bayesian bootstrap posteriors are asymptotically equivalent to the sampling distribution of the maximum likelihood estimator

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

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

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

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

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

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

a problem on completeness of exponentials let be a nite positive measure on the real line for a 0 denote by ea the family of exponential functions ea fe ist j s2 0 a g

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

spectral properties of a class of moran measures abstract let b b n n n and q q n n n be a sequence of positive integers respectively satisfying b n q n 1 let b q b 1 1 0 1 q 1 1 b 1 b 2 1 0 1 q 2 1 b q is called a moran measure which is a generalization of self affine measures in particular we call b q the n bernoulli convolution if b n b and q n n with 1 b r for all n n it is known 1 that b q is a spectral measure if q n divides b n for all n n in this paper we give a simpler proof to this result by developing the method given by dai 3 and simultaneously we construct a spectrum for the measure b q when q n divides b n for all n n moreover we give an equivalent condition for the n bernoulli convolution with 1 b r and a sufficient condition for the moran type measures b q with 1 b n z to admit an infinitely many orthogonal exponentials we also provide an equivalent characterization for the maximal orthogonal exponential sets of b q by defining a mixed tree mapping of the b q which is an important tool for us to characterize the maximality of orthogonal exponential set

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