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 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

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

spin borromean surgeries in 1986 matveev defined the notion of borromean surgery for closed oriented 3 manifolds and showed that the equivalence relation generated by this move is characterized by the pair first betti number linking form up to isomorphism we explain how this extends for 3 manifolds with spin structure if we replace the linking form by the quadratic form defined by the spin structure we then show that the equivalence relation among closed spin 3 manifolds generated by spin borromean surgeries is characterized by the triple first betti number linking form up to isomorphism rochlin invariant modulo 8

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

on abelian quantum invariants of links in 3 manifolds from a finite abelian group g a quadratic form ong and an element c c 1 ldots c n in g n we define a topological invariant tau of a pair m l where m 3 is a closed oriented 3 manifold and l an oriented framedn component link inm the main result consists in an explicit formula for this invariant based on a reciprocity formula for gauss sums which features a special linking pairing this pairing depends on both the quadratic form q and the linking pairing of m a necessary and sufficient condition for the invariant to vanish is described in terms of a characteristic class for this pairing we also discuss torsion spin c structures and related structures which appear in this context

extrapolation of time domain responses from three dimensional conducting objects utilizing the matrix pencil technique in this paper we use the matrix pencil approach to extrapolate time domain responses from three dimensional 3 d conducting objects that arise in the numerical solution of electromagnetic field problems by modeling the time functions as a sum of complex exponentials we can eliminate some of the instabilities that arise in late times for the electric field integral equation in the time domain however this method can also be utilized for extending the responses obtained using a finite difference time domain fdtd formulation

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

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

on one dimensional self similar tilings and pq tiles let b 2 be an integer base d 0 d 1 d b 1 z a digit set and t t b d the set of radix expansions it is well known that if t has nonvoid interior then t can tile r with some translation set j t is called a tile and d a tile digit set there are two fundamental questions studied in the literature i describe the structure of j ii for a given b characterize d so that t is a tile we show that for a given pair b d there is a unique self replicating translation set j c z and it has period b m for some m n this completes some earlier work of kenyon our main result for ii is to characterize the tile digit sets for b pq when p q are distinct primes the only other known characterization is for b p l due to lagarias and wang the proof for the pq case depends on the techniques of kenyon and de bruijn on the cyclotomic polynomials and also on an extension of the product form digit set of odlyzko

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

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

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

a comparison of performance of three orthogonal polynomials in extraction of wide band response using early time and low frequency data the objective of this paper is to generate a wideband and temporal response of three dimensional composite structures by using a hybrid method that involves generation of early time and low frequency information the data in these two separate time and frequency domains are mutually complementary and contain all the necessary information for a sufficient record length utilizing a set of orthogonal polynomials the time domain signal be it the electric or the magnetic currents or the near far scattered electromagnetic field could be expressed in an efficient way as well as the corresponding frequency domain responses the available data is simultaneously extrapolated in both domains computational load for electromagnetic analysis in either domain time or frequency can be thus significantly reduced three orthogonal polynomial representations including hermite polynomial laguerre function and bessel function are used in this approach however the performance of this new method is sensitive to two important parameters the scaling factor l sub 1 and the expansion order n it is therefore important to find the optimal parameters to achieve the best performance a comparison is presented to illustrate that for the classes of problems dealt with the choice of the laguerre polynomials has the best performance as illustrated by a typical scattering example from a dielectric hemisphere

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

abelian bf theory and turaev viro invariant the u 1 bf quantum field theory is revisited in the light of deligne beilinson cohomology we show how the u 1 chern simons partition function is related to the bf one and how the latter on its turn coincides with an abelian turaev viro invariant significant differences compared to the non abelian case are highlighted

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

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

on the beurling dimension of exponential frames abstract we study fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems we prove that under a mild technical condition the beurling dimension of a fourier frame coincides with the hausdorff dimension of the fractal

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

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

model subspaces techniques to study fourier expansions in l2 spaces associated to singular measures abstract let be a probability measure on t that is singular with respect to the haar measure in this paper we study fourier expansions in l 2 t using techniques from the theory of model subspaces of the hardy space since the sequence of monomials z n n n is effective in l 2 t it has a parseval frame associated via the kaczmarz algorithm our first main goal is to identify the aforementioned frame with boundary values of the frame p z n for the model subspace h h 2 h 2 where p is the orthogonal projection from the hardy space h 2 onto h the study of fourier expansions in l 2 t also leads to consider positive kernels in the hardy space our second main goal is to study the set of measures which reproduce a kernel contained in a model subspace we completely characterize this set when the kernel is the reproducing kernel of a model subspace and we study the consequences of this characterization

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

spectrality of a class of cantor moran measures abstract let p p n n 1 be a sequence of positive integers with p n 1 and d d n n 1 be a sequence of subsets of n with card d n 2 or 3 the infinite convolution of probability measures p d p 1 1 d 1 p 1 p 2 1 d 2 is a borel probability measure which is singular with respect to the lebesgue measure on r in this paper we prove the spectrality of such measure i e to find a set r such that the set e 2 i x is an orthonormal basis of l 2 p d

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

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

multi tiling and riesz bases abstract let s be a bounded riemann measurable set in r d and be a lattice by a theorem of fuglede if s tiles r d with translation set then s has an orthogonal basis of exponentials we show that under the more general condition that s multi tiles r d with translation set s has a riesz basis of exponentials the proof is based on meyer s quasicrystals

linking forms reciprocity for gauss sums and invariants of 3 manifolds we study invariants of 3 manifolds derived from finite abelian groups equipped with quadratic forms these invariants arise in turaev s theory of modular categories and generalize those of h murakami t ohltsuki and m okada the crucial algebraic tool is a new reciprocity formula for gauss sums generalizing classical formulas of cauchy kronecker krazer and siegel we use this reciprocity formula to give an explicit formula for the invariants and to generalize them to higher dimnensions

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

direct extrapolation of a causal signal using low frequency and early time data in this paper we provide three direct procedures to extrapolate the early time and the low frequency response of a causal signal simultaneously in the time and frequency domain compared with the extrapolation by orthonormal basis functions direct extrapolation is straightforward and we do not need to evaluate the basis functions and search for the optimal scaling factor and the optimal number of basis functions we show that the extrapolation introduced by adve and sarkar is equivalent to a neumann series solution of an integral equation of the second kind it is further shown that this iterative neumann expansion is an error reducing method we propose to solve this integral equation efficiently by employing a conjugate gradient iterative scheme the convergence of this scheme is also demonstrated we provide the matrix equations and show the equivalence to the integral equations and demonstrate that the method of singular value decomposition svd of solving the matrix equation provides accurate and stable results finally a number of illustrative numerical examples are presented and the performances of the three direct methods are compared

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

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

non spectrality of planar self affine measures with three elements digit set abstract the self affine measure m d associated with an affine iterated function system d x m 1 x d d d is uniquely determined the problems of determining the spectrality or non spectrality of a measure m d have been 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 for an expanding integer matrix m m 2 z and the three elements digit set d given by m a b d c and d 0 0 1 0 0 1 if a c b d 3 z then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this confirms the three elements digit set conjecture on the non spectrality of self affine measures in the plane

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

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

simultaneous time and frequency domain extrapolation in this paper given the early time response and the low frequency response of a causal system we simultaneously extrapolate them in the time and frequency domains the approach is iterative and is based on a simple discrete fourier transform simultaneous extrapolation in time and frequency domains is further enhanced by using the matrix pencil technique in the time domain and the cauchy method in the frequency domain the results are further enhanced through the hilbert transform hence enforcing the physical constraints of the system and thereby guaranteeing a causal extrapolation in time it is therefore possible to generate information over a larger domain from limited data it is important to note that through this extrapolation no new information is created the early time and low frequency data are complementary and contain all the desired information the key is to extract this information in an efficient and accurate manner the electric current on a scatterer is used as an example for the method

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

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