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

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

estimation of noisy telegraph processes nonlinear filtering versus nonlinear smoothing corresp in the estimation problem of a two state stationary markov process with gaussian white noise added the optimal smoother is a two filter smoother in a special case the performance of the optimal nonlinear filter and smoother is evaluated analytically some asymptotic results are also derived

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

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

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

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

the cauchy problem for the bgk equation with an external force abstract in this paper we study the cauchy problem for the bgk equation with an external force firstly we establish an l existence result for this equation and obtain some weighted l estimates then by means of the regularizing effects to the initial datum we construct the approximate solutions and obtain some uniform estimates of the approximate solutions finally by using compactness method and passing to the limit we prove the existence theorems of the l 1 and l p solutions and establish the propagation properties of the l p moments

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

bessel sequences of exponentials on fractal measures jorgensen and pedersen have proven that a certain fractal measure has no infinite set of complex exponentials which form an orthonormal set in l2 we prove that any fractal measure obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a riesz basic sequence or more generally a bessel sequence in l2 such that the frequencies have positive beurling dimension

continuous and discrete fourier frames for fractal measures motivated by the existence problem of fourier frames on fractal measures we introduce bessel and frame measures for a given finite measure on r d as extensions of the notions of bessel and frame spectra that correspond to bases of exponential functions not every finite compactly supported borel measure admits frame measures we present a general way of constructing bessel frame measures for a given measure the idea is that if a convolution of two measures admits a bessel measure then one can use the fourier transform of one of the measures in the convolution as a weight for the bessel measure to obtain a bessel measure for the other measure in the convolution the same is true for frame measures but with certain restrictions we investigate some general properties of frame measures and their beurling dimensions in particular we show that the beurling dimension is invariant under convolution with a probability measure and under a certain type of discretization moreover if a measure admits a frame measure then it admits an atomic one and hence a weighted fourier frame we also construct some examples of frame measures for self similar measures

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

cauchy problem for the ellipsoidal bgk model of the boltzmann equation ellipsoidal bgk model es bgk is a generalized version of the original bgk model designed to reproduce the physically correct prandtl number in the navier stokes limit in this paper we study the cauchy problem for the es bgk model under the condition of finite initial mass energy and entropy equivalence type estimates for the temperature tensor are crucially used

new finite dimensional filters and smoothers for noisily observed markov chains new finite dimensional filters and smoothers that are related to the wonham filter of a noisily observed markov chain are obtained in particular finite dimensional recursive filters and smoothers are given for the number of jumps from state i to state j for the occupation time of state i and for a stochastic integral related to the drift in the observations these filters allow easy application of the em algorithm for the estimation of the parameters of the markov chain and observation process

on the cauchy problem of the vlasov poisson bgk system global existence of weak solutions global in time existence of weak solutions to the cauchy problem of the three dimensional vlasov poisson bgk system is shown for initial data belonging to the space lp 3 3 with p 9 and having finite second order velocity moments this result solves partially the well posed problem for the vlasov poisson bgk system proposed by b perthame higher moments for kinetic equations the vlasov poisson and fokker planck cases math meth appl sci 13 441 452 1990

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

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

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

global weak solutions to the cometary flow equation with a self generated electric field abstract we study the cauchy problem of a cometary flow equation with a self generated electric field this kinetic model originates from the theory of astrophysical plasmas and can be viewed as a perturbation by a wave particle collision operator of the classical vlasov poisson system by asymptotic methods in kinetic theory global existence of nonnegative weak solutions to the cauchy problem in three space variables is established for bounded initial data having finite second order velocity moments

cauchy problem for the vlasov poisson boltzmann system the dynamics of dilute electrons can be modelled by the fundamental vlasov poisson boltzmann system which describes mutual interactions of the electrons through collisions in the self consistent electric field in this paper it is shown that any smooth perturbation of a given global maxwellian leads to a unique global in time classical solution when either the mean free path is small or the background charge density is large moreover the solution converges to the global maxwellian when time tends to infinity the analysis combines the techniques used in the study of conservation laws with the decomposition of the boltzmann equation introduced in 17 19 by obtaining new entropy estimates for this physical model

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

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

the vlasov maxwell boltzmann system near maxwellians perhaps the most fundamental model for dynamics of dilute charged particles is described by the vlasov maxwell boltzmann system in which particles interact with themselves through collisions and with their self consistent electromagnetic field despite its importance no global in time solutions weak or strong have been constructed so far it is shown in this article that any initially smooth periodic small perturbation of a given global maxwellian which preserves the same mass total momentum and reduced total energy 22 leads to a unique global in time classical solution for such a master system the construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation the linear collision operator l not its time integration is positive definite for any solutionf t x v with small amplitude to the vlasov maxwell boltzmann system 8 and 12 as a by product such an estimate also yields an exponential decay for the simpler vlasov poisson boltzmann system 24

on the 3 d vlasov poisson system with point charges global solutions with unbounded supports and propagation of velocity spatial moments abstract a new energy functional which is shown to decay to zero as time tends to infinite is introduced for the three dimensional vlasov poisson plasma in the presence of heavy point charges with repulsive interaction moreover a new kind of pointwise energy of a plasma particle relative to each point charge is also constructed and the variation of its root is shown to be controlled by the electrostatic fields based on those results we prove global existence and uniqueness of a classical solution possibly having infinite kinetic energy but with compact velocity spatial support specified in theorem 1 1 furthermore for the single point charge vlasov poisson plasma global existence and polynomial propagation of velocity spatial moments of weak solutions are also established

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

exact filters for the estimation of the number of transitions of finite state continuous time markov processes the problem of estimating the number of transitions of finite state continuous time markov processes observed by a noisy sensor is considered a finite dimensional exact filter is derived and using the em algorithm an extension of the baum welch algorithm for the discrete time case an application is made to the problem of estimating the unknown transition matrix of a finite state continuous time markov process

near optimal signal detection for finite state markov signals with application to magnetic resonance force microscopy detection of a finite state markov signal in additive white gaussian noise awgn can be done in an intuitive manner by applying an appropriate filter and using an energy detector one might not expect this heuristic approach to signal detection to be optimal however in this paper we show that for a certain type of finite state markov signal namely the discrete time dt random telegraph this filtered energy detector is approximately optimal under the following conditions of symmetric transition probabilities low signal to noise ratio snr long observation time and small probability of transition between two consecutive time instances when these last three conditions hold but the transition probabilities are not symmetric we show that a variant of the filtered energy detector is approximately optimal it is also shown under low snr conditions that the likelihood ratio test lrt for a finite state dt markov signal in awgn reduces to the matched filter statistic with the minimum mean squared error mmse predictor signal values used in place of the known signal values using this result we propose a general methodology for obtaining an approximation to the lrt of a finite state dt markov signal in awgn specifically instead of the conditional mean also mmse estimators affine estimators with lowest mean squared error are used this work is relevant to magnetic resonance force microscopy an emerging technology that uses ultrasensitive force sensing to detect magnetic resonance sensitivity down to the single spin level was demonstrated in a recent experiment

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

a characterization of boundary representations of positive matrices in the hardy space via the abel product abstract spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure we consider in this paper the reverse for a positive matrix in the hardy space of the unit disc we consider which measures if any yield a boundary representation of the positive matrix we prove a characterization of those representing measures via a matrix identity by introducing a new operator product called the abel product

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

scalar spectral measures associated with an operator fractal we study a spectral theoretic model on a hilbert space l2 where is a fixed cantor measure in addition to we also consider an independent scaling operator u acting in l2 to make our model concrete we focus on explicit formulas we take to be the bernoulli infinite convolution measure corresponding to scale number 14 we then define the unitary operator u in l2 from a scale by 5 operation the spectral theoretic and geometric properties we have previously established for u are as follows i u acts as an ergodic operator ii the action of u is not spatial and finally iii u is fractal in the sense that it is unitarily equivalent to a countable infinite direct sum of twisted copies of itself in this paper we prove new results about the projection valued measures and scalar spectral measures associated to u and its constituent parts our techniques make use of the representations of the cuntz algebra o2 on l2

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

on the bgk equation with some force fields in lp space abstract the cauchy problem for the bgk equation with two kinds of force fields is studied in this paper the first one considers a given force field f t x v which is assumed to be divergence free with respect to the velocity variable and has an l p integrability it is shown that there exists a nonnegative distributional solution which extends a previous result obtained by the authors the second one deals with a lorentz field e t x v b t x with e t x and b t x belonging to l p space which gives a similar existence result to the cauchy problem

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

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

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

global existence of classical solutions to the vlasov poisson boltzmann system the time evolution of the distribution function for the charged particles in a dilute gas is governed by the vlasov poisson boltzmann system when the force is self induced and its potential function satisfies the poisson equation in this paper we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global maxwellian moreover the convergence rate in time to the global maxwellian is also obtained through the energy method the proof is based on the theory of compressible navier stokes equations with forcing and the decomposition of the solutions to the boltzmann equation with respect to the local maxwellian introduced in 23 and elaborated in 31

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

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

lp solutions to the cauchy problem of the bgk equation the bgk model of the boltzmann equation plays an important role in the kinetic theory of rarefied gases some existence and uniqueness results of solutions to its cauchy problem were established for large initial data under various circumstances see for example perthame b global existence to the bgk model of boltzmann equation j differ equations 82 191 205 1989 in this paper by establishing some weighted lp estimates of the hydrodynamical quantities of a gas we prove the existence theorem of the lp solutions to the cauchy problem and establish the propagation properties of the lp moments for this kind of solutions

cauchy problem for the ellipsoidal bgk model for polyatomic particles abstract we establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal bgk model which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work

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