self similar set identification in the time scale domain we study the wavelet transform of deterministic self similar signals and derive their properties as well as a new algorithm for identification of the self similarity parameter we also include such applications as characterization and analysis of real chaotic signals in the presence of additive noise

wavelet based fractal analysis of dna sequences abstract the fractal scaling properties of dna sequences are analyzed using the wavelet transform mapping nucleotide sequences onto a dna walk produces fractal landscapes that can be studied quantitatively by applying the so called wavelet transform modulus maxima method this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized oscillating boxes from the scaling behavior of partition functions that are defined from the wavelet transform modulus maxima this method allows us to determine the singularity spectrum of the considered signal and thereby to achieve a complete multifractal analysis moreover by considering analyzing wavelets that make the wavelet transform microscope blind to patches of different nucleotide composition that are observed in genomic sequences we demonstrate and quantify the existence of long range correlations in the noncoding regions although the fluctuations in the patchy landscape of the dna walks reconstructed from both noncoding and protein coding regions are found homogeneous with gaussian statistics our wavelet based analysis allows us to discriminate unambiguously between the fluctuations of the former which behave like fractional brownian motions from those of the latter which cannot be distinguished from uncorrelated random brownian walks we discuss the robustness of these results with respect to various legitimate codings of the dna sequences finally we comment about the possible understanding of the origin of the observed long range correlations in noncoding dna sequences in terms of the nonequilibrium dynamical processes that produce the isochore structre of the genome

on helson matrices moment problems non negativity boundedness and finite rank we study helson matrices also known as multiplicative hankel matrices that is infinite matrices of the form m nm n m 1m nm n m 1 where is a sequence of complex numbers helson matrices are considered as linear operators on l2 n l2 n by interpreting helson matrices as hankel matrices in countably many variables we use the theory of multivariate moment problems to show that m is non negative if and only if is the moment sequence of a measure on r assuming that does not grow too fast we then characterize the non negative bounded helson matrices m as those where the corresponding moment measures are carleson measures for the hardy space of countably many variables finally we give a complete description of the helson matrices of finite rank in parallel with the classical kronecker theorem on hankel matrices

random weighted shifts abstract in this paper we initiate the study of a fundamental yet untapped random model of non selfadjoint bounded linear operators acting on a separable complex hilbert space we replace the weights w n 1 in the classical unilateral shift t defined as t e n w n e n 1 where e n n 1 form an orthonormal basis of a complex hilbert space by a sequence of i i d random variables x n n 1 that is w n x n this paper answers basic questions concerning such a model we propose that this model can be studied in comparison with the classical hardy bergman dirichlet spaces in function theoretic operator theory we calculate the spectra and determine their fine structures section 3 we classify the samples up to four equivalence relationships section 4 we introduce a family of random hardy spaces and determine the growth rate of the coefficients of analytic functions in these spaces section 5 we compare them with three types of classical operators section 6 this is achieved in the form of generalized von neumann inequalities the invariant subspaces are shown to admit arbitrarily large indices and their semi invariant subspaces model arbitrary contractions almost surely we discuss a beurling type theorem section 7 we determine various non selfadjoint algebras generated by t section 8 their dynamical properties are clarified section 9 their iterated aluthge transforms are shown to converge section 10 in summary they provide a new random model from the viewpoint of probability theory and they provide a new class of analytic functional hilbert spaces from the viewpoint of operator theory the technical novelty in this paper is that the methodology used draws from three largely separate sources probability theory functional hilbert spaces and the approximation theory of bounded operators

banach spaces of general dirichlet series abstract we study when the spaces of general dirichlet series bounded on a half plane are banach spaces and show that some of those classes are isometrically isomorphic between themselves in a precise way let n be a strictly increasing sequence of positive real numbers such that lim n n we denote by h n the complex normed space of all dirichlet series d s n b n n s which are convergent and bounded on the half plane re s 0 endowed with the norm d sup re s 0 d s if there exists q 0 such that inf n n 1 q n q 0 then h n is a banach space further if there exists a strictly increasing sequence r n of positive numbers such that the sequence log r n is q linearly independent n r for n p and n is the increasing rearrangement of the sequence n then h n is isometrically isomorphic to h b c 0 with this condition we explain more explicitly the optimal cases of the difference among the abscissas c b u and a

zeros of functions in hilbert spaces of dirichlet series the dirichlet hardy space fancyscript h 2 consists of those dirichlet series sum n a n n s for which sum n a n 2 1 2 is a necessary and sufficient condition for the existence of a nontrivial function f in fancyscript h 2 vanishing on a given bounded sequence the proof implies in fact a stronger result every function in the hardy space h 2 of the half plane text re s 1 2 can be interpolated by a function in fancyscript h 2 on such a blaschke sequence analogous results are proved for the hilbert space fancyscript d alpha of dirichlet series sum n a n n s for which sum n a n 2 d n alpha 1 2 partial results are then obtained for the zeros of functions in fancyscript h p l p analogues of fancyscript h 2 for 2 p infty based on certain contractive embeddings of fancyscript d alpha in fancyscript h p

fractal wavelet dimensions and localization in this paper we want to give a new definition of fractal dimensions as small scale behavior of theq energy of wavelet transforms this is a generalization of previous multi fractal approaches with this particular definition we will show that the 2 dimension correlation dimension of the spectral measure determines the long time behavior of the time evolution generated by a bounded self adjoint operator acting in some hilbert space it will be proved that for we have mathop lim inf limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 and that mathop lim sup limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 wherek 2 are the upper and lower correlation dimensions of the spectral measure associated with and a quantitative version of the rage theorem shall also be given

on several q special matrices including the q bernoulli and q euler matrices abstract in the spirit of our earlier articles 12 14 11 and our work 13 we define two dual q bernoulli polynomials with corresponding vector and matrix forms following aceto trigiante 1 the q l matrix the indefinite q integral of the q pascal matrix is the link between the q cauchy and the q bernoulli matrix the q analogue of the bernoulli complementary argument theorem can be expressed in matrix form through the diagonal a n matrix for the q euler polynomials corresponding results are obtained the umbral calculus for generating functions of q appell polynomials is shown to be equivalent to a transform method which maps polynomials to matrices a true q analogue of arponen 6 this is manifested by the vein 21 matrix which occurs as the transform of the q difference operator the aceto trigiante shifted q bernoulli matrix has a simple connection to the q bernoulli arponen matrix through the q pascal matrix we reintroduce certain q stirling numbers z q from 12 which will be needed for the polynomial matrix definitions

multifractality of entangled random walks and non uniform hyperbolic spaces multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a non symmetric lattice of obstacles are investigated using the equivalence of the model to random walks on a locally non symmetric tree statistical properties of topological invariants such as drift and return probabilities have been studied by means of a renormalization group rg technique the comparison of the analytical rg results with numerical simulations as well as with the rigorous results of gerl and woess demonstrates clearly the validity of our approach it is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space which has an exponentially large number of states has been broken

norms of composition operators on the h2 space of dirichlet series abstract we consider composition operators c on the hardy space of dirichlet series h 2 generated by dirichlet series symbols we prove two different subordination principles for such operators one concerns affine symbols only and is based on an arithmetical condition on the coefficients of the other concerns general symbols and is based on a geometrical condition on the boundary values of both principles are strict in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties in particular we generalize a result of j h shapiro on the norm of composition operators on the classical hardy space of the unit disc based on our techniques we also improve the recently established upper and lower norm bounds in the special case that s c r 2 s a number of other examples are given

approximation numbers of composition operators on h p spaces of dirichlet series by a theorem of gordon and hedenmalm generates a bounded composition operator on the hilbert space h2 of dirichlet series nbnn s with square summable coefficients bn if and only if s c0s s where c0 is a nonnegative integer and a dirichlet series with the following mapping properties maps the right half plane into the half plane res 1 2 if c0 0 and is either identically zero or maps the right half plane into itself if c0 is positive it is shown that the nth approximation numbers of bounded composition operators on h2 are bounded below by a constant times rn for some 0 0 when c0 is positive both results are best possible the case when c0 0 is bounded and smooth up to the boundary of the right half plane and sup re 1 2 is discussed in depth it includes examples of non compact operators as well as operators belonging to all schatten classes sp for s c1 j 1dcqjqj s with qj independent integers it is shown that the nth approximation number behaves as n d 1 2 possibly up to a factor log n d 1 2 estimates rely mainly on a general hilbert space method involving finite linear combinations of reproducing kernels a key role is played by a recently developed interpolation method for h2 using estimates of solutions of the equation finally by a transference principle from h2 of the unit disc explicit examples of compact composition operators with approximation numbers decaying at essentially any sub exponential rate can be displayed

the multiplicative hilbert matrix abstract it is observed that the infinite matrix with entries m n log m n 1 for m n 2 appears as the matrix of the integral operator h f s 1 2 f w w s 1 d w with respect to the basis n s n 2 here s is the riemann zeta function and h is defined on the hilbert space h 0 2 of dirichlet series vanishing at and with square summable coefficients this infinite matrix defines a multiplicative hankel operator according to helson s terminology or alternatively it can be viewed as a bona fide small hankel operator on the infinite dimensional torus t by analogy with the standard integral representation of the classical hilbert matrix this matrix is referred to as the multiplicative hilbert matrix it is shown that its norm equals and that it has a purely continuous spectrum which is the interval 0 these results are in agreement with known facts about the classical hilbert matrix it is shown that the matrix m 1 p n p 1 p log m n 1 has norm sin p when acting on l p for 1 p however the multiplicative hilbert matrix fails to define a bounded operator on h 0 p for p 2 where h 0 p are h p spaces of dirichlet series it remains an interesting problem to decide whether the analytic symbol n 2 log n 1 n s 1 2 of the multiplicative hilbert matrix arises as the riesz projection of a bounded function on the infinite dimensional torus t

volterra operators on hardy spaces of dirichlet series for a dirichlet series symbol g s n 1bnn s the associated volterra operator tg acting on a dirichlet series f s n 1ann s is defined by the integral f sf w g w dw we show that tg is a bounded operator on the hardy space hp of dirichlet series with 0 0 whenever tg is bounded in particular such g belong to hp for every p we relate the boundedness of tg to several other bmo type spaces bmoa in half planes the dual of h1 and the space of symbols of bounded hankel forms moreover we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m homogeneous symbols as well as for general symbols finally we consider the action of tg on reproducing kernels for appropriate sequences of subspaces of h2 our proofs employ function and operator theoretic techniques in one and several variables a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols g

conserved quantities and adaptation to the edge of chaos certain dynamical systems such as the shift map and the logistic map have an edge of chaos in their parameter spaces on one side of this edge the dynamics are chaotic for many parameter values on the other side of the edge they are periodic we find that discrete time dynamical systems with wavelet filtered feedback from the dynamical variable to the parameters are attracted to a narrow parameter range near the edge of chaos the periodic boundary regime we show that the migration from the chaotic regime to the periodic boundary regime can be attributed to a conserved quantity and find that such adaptation to the edge of chaos is accompanied by a depopulation of the chaotic regime we use this conserved quantity to determine the location of the periodic boundary regime and show that its size is proportional to the size of the feedback further we compute the dynamics of the probability density for the parameter for a specific example

invariances laplacian like wavelet bases and the whitening of fractal processes in this contribution we study the notion of affine invariance specifically invariance to the shifting scaling and rotation of the coordinate system as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional brownian motion fbm fields in particular using a rigorous and powerful distribution theoretic formulation we extend previous results of blu and unser 2006 to the multivariate case showing that polyharmonic splines and fbm processes can be seen as the deterministic vs stochastic solutions to an identical fractional partial differential equation that involves a fractional laplacian operator we then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional laplacian which also turns out to be the whitening operator for fbm fields this fact allows us to study the probabilistic properties of the wavelet transform coefficients of fbm like processes leading for instance to ways of estimating the hurst exponent of a multiparameter process from its wavelet transform coefficients we provide theoretical and experimental verification of these results to complement the toolbox available for multiresolution processing of stochastic fractals we also introduce an extended family of multidimensional multiresolution spaces for a large class of separable and nonseparable lattices of arbitrary dimensionality

wavelet analysis of transients in fractal superlattices fractal superlattices are designed by alternating dielectric layers according to an iterative fractal process the discrete self similarity of such stratified structures can be remotely detected from interrogation by an incident pulse in this paper the impulse response of one dimensional cantor superlattices is computed and the wavelet transform is applied to the reflected signal in order to explore its temporal distribution of singularities for a sufficiently narrow pulse the skeleton of the wavelet transform modulus maxima exhibits a hierarchical structure that makes apparent the iterative process governing the construction rule of the interrogated fractal superlattice and its geometry in the reflected signal such hierarchy reveals the existence of singularities that are distributed on the governing cantor set finally the similarity dimension is extracted from reflection data and a strategy for estimating the stage of growth is developed

composition operators on weighted bergman spaces of dirichlet series abstract we study boundedness and compactness of composition operators on weighted bergman spaces of dirichlet series particularly we obtain in some specific cases upper and lower bounds of the essential norm of these operators and a criterion of compactness on classical weighted bergman spaces moreover a sufficient condition of compactness is obtained using the notion of carleson s measure

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

more on the analysis of local regularity through wavelets in this paper we want to extend the results of pointwise analysis through wavelet transforms to the class of functions where the local fluctuation is bounded by any submultiplicative function this generalizez the results obtained before in the well known case of holder regularity

functional calculus using wavelet transforms it is shown how the wavelet transform may be used to compute for a function s the symbol s a for any not necessarily self adjoint operator a whose spectrum is contained in the upper half plane for self adjoint operators it is shown that this functional calculus coincides with the usual one in particular it is shown how the exponential eita can be written in terms of the resolvent rz a z 1 of a as follows eita 1 c 0 da an 2 dbg at eitbrb ian a with c 2i 0 d i n 1g e and n n and the integral is understood as the cesaro limit this shows explicitly how the behavior for large t is determined by the behavior of rz at iz 1 t

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

revisiting multifractality of high resolution temporal rainfall using a wavelet based formalism 1 we reexamine the scaling structure of temporal rainfall using wavelet based methodologies which as we demonstrate offer important advantages compared to the more traditional multifractal approaches such as box counting and structure function techniques in particular we explore two methods based on the continuous wavelet transform cwt and the wavelet transform modulus maxima wtmm the partition function method and the newer and more efficient magnitude cumulant analysis method we also report the results of a two point magnitude correlation analysis which is able to infer the presence or absence of multiplicativity as the underlying mechanism of scaling the diagnostic power of these methodologies for small samples signals with short ranges of scaling and signals for which high frequency fluctuations are superimposed on a low frequency component all common attributes of geophysical signals is carefully documented application of these methodologies to several midwestern convective storms sampled every 5 s over several hours provides new insights they reveal the presence of a very intermittent multifractal structure a wide spectrum of singularities in rainfall fluctuations between the scales of 5 min and the storm pulse duration of the order of 1 2 hours for the analyzed storms the two point magnitude statistical analysis suggests that this structure is consistent with a multiplicative cascading mechanism which however is local in nature that is it applies only within each storm pulse but not over the whole storm duration

lower bounds for the maximum of the riemann zeta function along vertical lines let alpha in 1 2 1 be fixed we prove that begin aligned max 0 le t le t zeta alpha it ge exp left frac c alpha log t 1 alpha log log t alpha right end aligned for all sufficiently large t where we can choose c alpha 0 18 2 alpha 1 1 alpha the same result has already been obtained by montgomery with a smaller value for c alpha however our proof which uses a modified version of soundararajan s resonance method together with ideas of hilberdink is completely different from montgomery s this new proof also allows us to obtain lower bounds for the measure of those t in 0 t for which zeta alpha it is of the order mentioned above

addendum to a hilbert space of dirichlet series and systems of dilated functions in l 2 0 1 addendum to a hilbert space of dirichlet series and systems of dilated functions in l 2 0 1

the cesaro space of dirichlet series and its multiplier algebra abstract we consider the space h c e s p of all dirichlet series whose coefficients belong to the cesaro sequence space c e s p consisting of all complex sequences whose absolute cesaro means are in l p for 1 p it is a banach space of analytic functions for which we study the maximal domain of analyticity and the boundedness of point evaluations we identify the algebra of analytic multipliers on h c e s p as the wiener algebra of dirichlet series shifted to the vertical half plane c 1 q s c s 1 q where 1 p 1 q 1

a matrix approach to sheffer polynomials abstract this paper deals with a unified matrix representation for the sheffer polynomials the core of the proposed approach is the so called creation matrix a special subdiagonal matrix having as nonzero entries positive integer numbers whose exponential coincides with the well known pascal matrix in fact sheffer polynomials may be expressed in terms of two matrices both connected to it as we will show one of them is strictly related to appell polynomials while the other is linked to a binomial type sequence consequently different types of sheffer polynomials correspond to different choices of these two matrices

an optimal condition for the lil for trigonometric series by a classical theorem salem zygmund 6 erdos gal 3 if nk is a sequence of positive integers satisfying nk i nk q 1 k 1 2 then cos nkx obeys the law of the iterated logarithm i e 1 limsup nloglogn 1 2 e cosnkx 1 a e n oo k 1 can be essentially weakened here but the problem of finding the precise gap condition for the lil 1 has remained open in this paper we find using combinatorial methods an optimal gap condition for the upper half of the lil i e the inequality 1 in 1

multifractal characterization of stochastic resonance we use a multifractal formalism to study the effect of stochastic resonance in a noisy bistable system driven by various input signals to characterize the response of a stochastic bistable system we introduce a new measure based on the calculation of a singularity spectrum for a return time sequence we use wavelet transform modulus maxima method for the singularity spectrum computations it is shown that the degree of multifractality defined as a width of singularity spectrum can be successfully used as a measure of complexity both in the case of periodic and aperiodic stochastic or chaotic input signals we show that in the case of periodic driving force singularity spectrum can change its structure qualitatively becoming monofractal in the regime of stochastic synchronization this fact allows us to consider the degree of multifractality as a new measure of stochastic synchronization also moreover our calculations have shown that the effect of stochastic resonance can be catched by this measure even from a very short return time sequence we use also the proposed approach to characterize the noise enhanced dynamics of a coupled stochastic neurons model

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

contractive inequalities for bergman spaces and multiplicative hankel forms we consider sharp inequalities for bergman spaces of the unit disc establishing analogues of the inequality in carleman s proof of the isoperimetric inequality and of weissler s inequality for dilations by contractivity and a standard tensorization procedure the unit disc inequalities yield corresponding inequalities for the bergman spaces of dirichlet series we use these results to study weighted multiplicative hankel forms associated with the bergman spaces of dirichlet series reproducing most of the known results on multiplicative hankel forms associated with the hardy spaces of dirichlet series in addition we find a direct relationship between the two type of forms which does not exist in lower dimensions finally we produce some counter examples concerning carleson measures on the infinite polydisc

dynamical integral transform on fractal sets and the computation of entropy we introduce an integral transform of wavelet type which we call dynamical integral transform and we show that it can be used to compute the second renyi entropy for a large class of invariant measures the method is then generalized to the whole spectrum of the renyi entropies and establishes a correspondence between thermodynamic formalism and the dynamical integral transform of expanding strange sets numerical examples are presented

local properties of hilbert spaces of dirichlet series we show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the hilbert space of dirichlet series defined using these as weights this extends results recently obtained describing the local behavior of dirichlet series with square summable coefficients in terms of local integrability boundary behavior carleson measures and interpolating sequences as these spaces can be identified with functions spaces on the infinite dimensional polydisk this gives new results on the dirichlet and bergman spaces on the infinite dimensional polydisk as well as the scale of besov sobolev spaces containing the drury arveson space on the infinite dimensional unit ball we use both techniques from the theory of sampling in paley wiener spaces and classical results from analytic number theory c 2011 elsevier inc all rights reserved less

multipliers of dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables let mathscr h infty be the set of all ordinary dirichlet series d sum n a n n s representing bounded holomorphic functions on the right half plane a completely multiplicative sequence b n of complex numbers is said to be an ell 1 multiplier for mathscr h infty whenever sum n a n b n infty for every d in mathscr h infty we study the problem of describing such sequences b n in terms of the asymptotic decay of the subsequence b p j where p j denotes the jth prime number given a completely multiplicative sequence b b n we prove among other results b is an ell 1 multiplier for mathscr h infty provided b p j 1 for all j and overline lim n frac 1 log n sum j 1 n b p j 2 1 and conversely if b is an ell 1 multiplier for mathscr h infty then b p j 1 for all j and overline lim n frac 1 log n sum j 1 n b p j 2 le 1 here b stands for the decreasing rearrangement of b following an ingenious idea of harald bohr it turns out that this problem is intimately related with the question of characterizing those sequences z in the infinite dimensional polydisk mathbb d infty the open unit ball of ell infty for which every bounded and holomorphic function f on mathbb d infty has an absolutely convergent monomial series expansion sum alpha frac partial alpha f 0 alpha z alpha moreover we study analogous problems in hardy spaces of dirichlet series and hardy spaces of functions on the infinite dimensional polytorus mathbb t infty

riesz means in hardy spaces on dirichlet groups given a frequency lambda lambda n we study when almost all vertical limits of a mathcal h 1 dirichlet series sum a n e lambda ns are riesz summable almost everywhere on the imaginary axis equivalently this means to investigate almost everywhere convergence of fourier series of h 1 functions on so called lambda dirichlet groups and as our main technical tool we need to invent a weak type 1 infty hardy littlewood maximal operator for such groups applications are given to h 1 functions on the infinite dimensional torus mathbb t infty ordinary dirichlet series sum a n n s as well as bounded and holomorphic functions on the open right half plane which are uniformly almost periodic on every vertical line

wavelet transform and scale space filtering of fractal images in this correspondence we consider wavelet transform and scale space filtering as special cases of general scale shift mapping ssm then the ssm is used as a tool for characterizing the geometrical complexity of uniform fractals this tool can reveal the construction rule of fractals and estimate fractal dimension finally we use the results obtained from scale shift mapping to estimate iterated function system ifs codes of a class of fractal images some examples have been illustrated to demonstrate its good performance

on the determination of box dimensions by means of wavelet transforms sufficient conditions are established for the determination of box dimensions of graphs from the decrease of the corresponding wavelet transforms as an application wavelet weierstrass functions are constructed

barycentric fixed mass method for multifractal analysis we present a method to estimate the multifractal spectrum of point distributions the method incorporates two motivated criteria barycentric pivot point selection and nonoverlapping coverage in order to reduce edge effects improve precision and reduce computation time implementation of the method on synthetic benchmarks demonstrates the superior performance of the proposed method compared with existing alternatives routinely used in the literature finally we use the method to estimate the multifractal properties of the widely studied growth process of diffusion limited aggregation dla and compare our results with recent and earlier studies our tests support the conclusion of a genuine but weak multifractality of the central core of dla clusters with d q decreasing from 1 75 0 01 for q 10 to 1 65 0 01 for q 10

on the equivalence between a minimal codomain cardinality riesz basis construction a system of hadamard sylvester operators and a class of sparse binary optimization problems piecewise low order polynomial riesz basis families are constructed such that they share the same coefficient functionals of smoother orthonormal bases in a localized indexing subset it is shown that a minimal cardinality basis codomain can be realized by inducing sparsity via l1 regularization in the distributional derivatives of the basis functions and that the optimal construction can be found numerically by constrained binary optimization over a suitably large dictionary furthermore it is shown that a subset of these solutions are equivalent to a specific constrained analytical solution derived via sylvester type hadamard operators

self affine fractal functions and wavelet series abstract we consider functions represented by series g g c g g 1 x of wavelet type where g is a group generated by affine functions l 1 l n and is piecewise affine by means of those functions we characterize the class of self affine fractal functions previously studied by barnsley et al we compute their global and local holder exponents and investigate points of non differentiability wavelet representations for various continuous nowhere differentiable and singular functions are presented another application is the construction of functions with prescribed local holder exponents at each point

a matrix approach to polynomials we present a matrix formalism to study univariate polynomials the structure of this formalism is beautiful enough to be worth seeing on its own yet we give another motivation to this by presenting three new theorems and applying the formalism to give new proofs of some known results

pointwise analysis of riemann s nondifferentiable function we will show how to analyse the local regularity of functions with the help of the wavelet transform these results will be applied to the function of riemann where we show the existence of a dense set of points where this function is differentiable on another dense set we show the existence of local singularities of cusp type on a third set we show differentiability to the right left on the remaining set the functions will be shown to be not differentiable

an umbral approach to find q analogues of matrix formulas a general introduction is given to the logarithmic q analogue formulation of mathematical expressions with a special focus on its use for matrix calculations the fundamental definitions relevant t

multifractal detrended cross correlation analysis of coding and non coding dna sequences through chaos game representation abstract we propose a new approach combining the chaos game representation and the two dimensional multifractal detrended cross correlation analysis methods to examine multifractal behavior in power law cross correlation between any pair of nucleotide sequences of unequal lengths in this work we analyzed the characteristic behavior of coding and non coding dna sequences of eight prokaryotes the results show the presence of strong multifractal nature between coding and non coding sequences of all data sets we found that this integrative approach helps us to consider complete dna sequences for characterization and further it may be useful for classification clustering identification of class affiliation of nucleotide sequences etc with high precision

variants of a theorem of helson on general dirichlet series abstract a result of helson on dirichlet series a n e n s states that whenever a n is 2 summable and n satisfies a certain condition introduced by bohr then for almost all homomorphism r t the dirichlet series a n n e n s converges on the open right half plane r e 0 for ordinary dirichlet series a n n s hedenmalm and saksman related this result with the famous carleson hunt theorem on pointwise convergence of fourier series and bayart extended it within his theory of hardy spaces h p of such series the aim here is to prove variants of helson s theorem within our recent theory of hardy spaces h p 1 p of dirichlet series to be more precise in the reflexive case 1 p we extend helson s result to dirichlet series in h p without any further condition on the frequency and in the non reflexive case p 1 to the wider class of frequencies satisfying the so called landau condition more general than bohr s condition in both cases we add relevant maximal inequalities finally we try to indicate how our study of helson s theorem can be employed to understand that for a fixed but arbitrary frequency certain key stones of the structure theory of dirichlet series in fact turn out to be equivalent

basis properties of the p q sine functions we improve the currently known thresholds for basisness of the family of periodically dilated p q sine functions our findings rely on a beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity we also determine refined bounds on the riesz constant associated with this family these results seal mathematical gaps in the existing literature on the subject

multiresolution analysis of fluctuations in non stationary time series through discrete wavelets abstract we illustrate the efficacy of a discrete wavelet based approach to characterize fluctuations in non stationary time series the present approach complements the multifractal detrended fluctuation analysis mf dfa method and is quite accurate for small size data sets as compared to polynomial fits in the mf dfa a single daubechies wavelet is used here for detrending purposes the natural built in variable window size in wavelet transforms makes this procedure well suited for non stationary data we illustrate the working of this method through the analysis of binomial multifractal model for this model our results compare well with those calculated analytically and obtained numerically through mf dfa to show the efficacy of this approach for finite data sets we also do the above comparison for gaussian white noise time series of different sizes in addition we analyze time series of three experimental data sets of tokamak plasma and also spin density fluctuations in 2d ising model

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

thermodynamics of fractal signals based on wavelet analysis application to fully developed turbulence data and dna sequences we use the continuous wavelet transform to generalize the multifractal formalism to fractal functions we report the results of recent applications of the so called wavelet transform modulus maxima wtmm method to fully developed turbulence data and dna sequences we conclude by briey describing some works currently under progress which are likely to be the guidelines for future research c 1998 elsevier science b v all rights reserved

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

fixed scale transformation approach to critical fluctuations and fractal growth abstract a geometrical approach to the study of critical phenomena calls for a fractal description of critical fluctuations the fixed scale transformation fst approach introduces such a description incorporating both the scale and translational invariance properties of the critical state besides providing a theoretical understanding of the genesis of fractal structures it enables one to compute the fractal dimension of the critical clusters to a high degree of accuracy in terms of the critical statistics of the underlying model laplacian models of irreversible fractal growth to which the fst approach was first applied are presented within the general framework of critical phenomena this geometrical approach links the growth rules which are fully nonlocal in both space and time measured in number of particles added to the morphology of the grown cluster and its scaling behaviour possible directions towards a more comprehensive theory of fractal growth are discussed

wavelet analysis of a gaussian kolmogorov signal several attempts have been made recently to use wavelets transforms for extracting histograms of scaling exponents from experimental turbulence data similar techniques are here applied to a gaussian signal having a kolmogorov 5 3 energy spectrum this is an instance of the class of fractional brownian motions having scaling exponent 1 3 almost everywhere for the gaussian signal a spurious non trivial histogram is obtained by applying a mexican hat wavelet transform analysis on the other hand we show that the use of complex wavelets and even more so the application of an optimal wavelet transform method strongly reduce the spurious fluctuations observed in the processing of the gaussian signal

complex symmetric composition operators on a hilbert space of dirichlet series abstract on the hilbert space of dirichlet series with square summable coefficients there are no non normal complex symmetric composition operators induced by non constant symbols this is in sharp contrast with the phenomenon on the classical hardy space over the unit disk

discrete wavelet approach to multifractality abstract a new and simple method is presented to study local scaling properties of measures defined on regular and fractal supports the method based on a discrete wavelet analysis wa complements the well known multifractal analysis ma extensively used in many physical problems the present wavelet approach is particularly suitable for problems where the multifractal analysis does not provide conclusive results as e g in the case of measures corresponding to anderson localized wave functions in one dimension examples of different types of measures are also discussed which illustrate the usefulness of the wa to classify non multifractal measures according to additional characteristic exponents which can not be obtained within the ma

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

factorizations induced by complete nevanlinna pick factors abstract we prove a factorization theorem for reproducing kernel hilbert spaces whose kernel has a normalized complete nevanlinna pick factor this result relates the functions in the original space to pointwise multipliers determined by the nevanlinna pick kernel and has a number of interesting applications for example for a large class of spaces including dirichlet and drury arveson spaces we construct for every function f in the space a pluriharmonic majorant of f 2 with the property that whenever the majorant is bounded the corresponding function f is a pointwise multiplier

multiresolution wavelet analysis of heartbeat intervals discriminates healthy patients from those with cardiac pathology we applied multiresolution wavelet analysis to the sequence of times between human heartbeats r ensuremath r intervals and have found a scale window between 16 and 32 heartbeat intervals over which the widths of the r ensuremath r wavelet coefficients fall into disjoint sets for normal and heart failure patients this has enabled us to correctly classify every patient in a standard data set as belonging either to the heart failure or normal group with 100 accuracy thereby providing a clinically significant measure of the presence of heart failure from the r ensuremath r intervals alone comparison is made with previous approaches which have provided only statistically significant measures

scaling features of texts images and time series in the given paper we consider the scaling features of long letter sequences like human writings discretized images and discretized 3nancial data using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate di5erent scalings for large and small 6uctuations we discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency dax future data c 2001 elsevier science b v all rights reserved

factorial stirling matrix and related combinatorial sequences abstract in this paper some relationships between the stirling matrix the vandermonde matrix the benoulli numbers and the eulerian numbers are studied from a matrix representation of k s n k which will be called the factorial stirling matrix where s n k are the stirling numbers of the second kind as a consequence a number of interesting and useful identities are obtained

a general view of multipliers and composition operators this paper defines a multiplier pair which is a very general setting in which the notions of multipliers and composition operators can be studied we prove some results in this general setting and we present many new interesting examples in which these notions can be interpreted

receiver operating characteristic analysis reveals superiority of scale dependent wavelet and spectral measures for assessing cardiac dysfunction receiver operating characteristic analysis was used to assess the suitability of various heart rate variability hrv measures for correctly classifying electrocardiogram records of varying lengths as normal or revealing the presence of heart failure scale dependent hrv measures were found to be substantially superior to scale independent measures scaling exponents for discriminating the two classes of data over a broad range of record lengths hours to minutes a jittered integrate and fire model built around a fractal gaussian noise kernel provides a realistic though not perfect simulation of heartbeat sequences s0031 9007 98 08087 9

a hardy space analysis of the baez duarte criterion for the rh abstract in this article methods from sub hardy hilbert spaces such as the de branges rovnyak spaces and local dirichlet spaces are used to investigate baez duarte s hilbert space reformulation of the riemann hypothesis rh

difference in nature of correlation between nasdaq and bse indices abstract we apply a recently developed wavelet based approach to characterize the correlation and scaling properties of non stationary financial time series this approach is local in nature and it makes use of wavelets from the daubechies family for detrending purpose the built in variable windows in wavelet transform makes this procedure well suited for the non stationary data we analyze daily price of nasdaq composite index for a period of 20 years and bse sensex index over a period of 15 years it is found that the long range correlation as well as fractal behavior for both the stock index values differ from each other significantly strong non statistical long range correlation is observed in bse index whose removal revealed a gaussian random noise character for the corresponding fluctuation the nasdaq index on the other hand showed a multifractal behavior with long range statistical correlation

modified zeta functions as kernels of integral operators abstract the modified zeta functions n k n s where k n converge absolutely for re s 1 these generalise the riemann zeta function which is known to have a meromorphic continuation to all of c with a single pole at s 1 our main result is a characterisation of the modified zeta functions that have pole like behaviour at this point this behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces l 2 i for symmetric and bounded intervals i r we also consider the special case when the set k n is assumed to have arithmetic structure in particular we look at local l p integrability properties of the modified zeta functions on the abscissa re s 1 for p 1

statistics of event by event fluctuations abstract following hwa and wu r c hwa y wu phys rev c 60 1999 0544904 we characterize the fluctuation behavior of the hadron density produced during quark hadron phase transition as modeled by a 2d ising model using a recently developed discrete wavelet based approach the scaling behavior is studied at temperatures below at and above t c at t c we find the hurst exponent h 1 as observed in a recent experimental finding l qin m ta chung phys rev d 72 2005 014011 however as compared to the r s analysis which yields only the hurst exponent our local approach finds a correlation behavior and multifractal properties at temperatures below at and above t c we find evidence for a transition from brownian to fractional browian motion near t c the correlation behavior compares well with the results obtained from a continuous wavelet based average wavelet co efficient method as well as with fourier power spectral analysis

structural five fold symmetry in the fractal morphology of diffusion limited aggregates abstract the statistical self similarity of the geometry of diffusion limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform this study reveals the existence of a predominant structural five fold symmetry in the internal frozen region as well as in the active outer region of the interface this observation is corroborated by a statistical analysis of the screening effects that govern diffusion limited aggregation dla growth in linear and sector shaped cells the existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering we report on the discovery of fibonacci sequences in the inner extinct region of large mass off lattice dla clusters with a branching ratio which converges asymptotically to the golden mean we suggest an interpretation of the dla morphology as a quasifractal counterpart of the well ordered snowflake fractal architecture

spectral fluctuation characterization of random matrix ensembles through wavelets a recently developed wavelet based approach is employed to characterize the scaling behaviour of spectral fluctuations of random matrix ensembles as well as complex atomic systems our study clearly reveals anti persistent behaviour and supports the fourier power spectral analysis it also finds evidence for multi fractal nature in the atomic spectra the multi resolution and localization nature of the discrete wavelets ideally characterizes the fluctuations in these time series some of which are not stationary

critical lil behavior of the trigonometric system it is a classical fact that for rapidly increasing n k the sequence cos n k x behaves like a sequence of i i d random variables actually this almost i i d behavior holds if n k grows faster than e c k below this speed we have strong dependence while there is a large literature dealing with the almost i i d case practically nothing is known on what happens at the critical speed n k e c k critical behavior and what is the probabilistic nature of cos n k x in the strongly dependent domain in our paper we study the critical lil behavior of cos n k x i e we investigate how classical fluctuational theorems like the law of the iterated logarithm and the kolmogorov feller test turn to nonclassical laws in the immediate neighborhood of n k e c k