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

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

stochastic linear quadratic adaptive control for continuous time first order systems in this paper we discuss the linear quadratic lq adaptive control problem for the following continuous time first order scalar stochastic system dxt axt dt but dt c dwt with cost function min lim sup jt u u u t where jt u 1t 0t q1x2s q2u2s ds q1 0 0 based on the self convergence property of continuous time weighted least squares cwls algorithm 9 and the similar parameter modification method 13 the stability of the closed loop system is achieved without invoking any excitations here self convergence means that the convergence is automatic in the sense that no excitation conditions on the signals or measurements is needed it is a terminology from 10 the strong consistency of cwls and the optimality of the adaptive control are established by incorporating with diminishing excitations

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

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

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

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

non spectral problem for the planar self affine measures abstract in this paper we consider the non spectral problem for the planar self affine measures m d generated by an expanding integer matrix m m 2 z and a finite digit set d z 2 let p 2 be a positive integer e p 2 1 p i j t 0 i j p 1 and z d 2 x 0 1 2 d d e 2 i d x 0 we show that if z d 2 e p 2 0 and gcd det m p 1 then there exist at most p 2 mutually orthogonal exponential functions in l 2 m d in particular if p is a prime then the number p 2 is the best

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

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

finite bath fluctuation theorem we demonstrate that a finite bath fluctuation theorem of the crooks type holds for systems that have been thermalized via weakly coupling them to a bath with energy independent finite specific heat we show that this theorem reduces to the known canonical and microcanonical fluctuation theorems in the two respective limiting cases of infinite and vanishing specific heat of the bath the result is elucidated by applying it to a two dimensional hard disk colliding elastically with few other hard disks in a rectangular box with perfectly reflecting walls

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

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

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

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

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

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

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