on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

gaussian heat kernel upper bounds via the phragmen lindelof theorem we prove that in presence of l gaussian estimates so called davies gaffney estimates on diagonal upper bounds imply precise off diagonal gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

calculation of a constant q spectral transform the frequencies that have been chosen to make up the scale of western music are geometrically spaced thus the discrete fourier transform dft although extremely efficient in the fast fourier transform implementation yields components which do not map efficiently to musical frequencies this is because the frequency components calculated with the dft are separated by a constant frequency difference and with a constant resolution a calculation similar to a discrete fourier transform but with a constant ratio of center frequency to resolution has been made this is a constant q transform and is equivalent to a 1 24 oct filter bank thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range this transform against log frequency to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted this is compared to the conventio

atomic and molecular decompositions of anisotropic besov spaces in this work we develop the theory of weighted anisotropic besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms this study extends the isotropic littlewood paley methods of dyadic transforms of frazier and jawerth 19 21 to non isotropic settings

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

a penalty function promoting sparsity within and across groups we introduce a new penalty function that promotes signals composed of a small number of active groups where within each group only a few high magnitude coefficients are nonzero we derive the threshold function associated with the proposed penalty and study its properties we discuss how the proposed penalty threshold function can be useful for signals with isolated nonzeros such as audio with isolated harmonics along the frequency axis or reflection functions in exploration seismology where the nonzeros occur on the boundaries of subsoil layers we demonstrate the use of the proposed penalty threshold functions in a convex denoising and a nonconvex deconvolution formulation we provide convergent algorithms for both formulations and compare the performance with state of the art methods

variable weak hardy spaces and their applications abstract let p r n 0 be a variable exponent function satisfying the globally log holder continuous condition in this article the authors first introduce the variable weak hardy space on r n wh p r n via the radial grand maximal function and then establish its radial or non tangential maximal function characterizations moreover the authors also obtain various equivalent characterizations of wh p r n respectively by means of atoms molecules the lusin area function the littlewood paley g function or g function as an application the authors establish the boundedness of convolutional type and non convolutional order calderon zygmund operators from h p r n to wh p r n including the critical case when p n n or when p n n where p ess inf x r n p x

reproducing pairs and the continuous nonstationary gabor transform on lca groups in this paper we introduce and investigate the concept of reproducing pairs as a generalization of continuous frames reproducing pairs yield a bounded analysis and synthesis process while the frame condition can be omitted at both stages moreover we will investigate certain continuous frames resp reproducing pairs on lca groups which can be described as a continuous version of nonstationary gabor systems and state sufficient conditions for these systems to form a continuous frame resp reproducing pair as a byproduct we identify the structure of the frame operator resp resolution operator we will apply our results to systems generated by a unitary action of a subset of the affine weyl heisenberg group in this setup will also serve as a nontrivial example of a system for which whereas continuous frames exist no dual system with the same structure exists even if we drop the frame property

hardy spaces associated to operators satisfying davies gaffney estimates and bounded holomorphic functional calculus abstract let x be a space of homogeneous type assume that an operator l has a bounded holomorphic functional calculus on l 2 x and the kernel of the heat semigroup e t l t 0 satisfies the davies gaffney estimates without the assumption that l is self adjoint we develop a theory of hardy spaces h l p x 0 p 1 which includes a molecular decomposition an atomic decomposition a square function characterization duality of hardy and lipschitz spaces and a marcinkiewicz type interpolation theorem as applications we show that l has a bounded holomorphic functional calculus on h l p x for all p 0 and certain riesz transforms associated to l are bounded from h l p x to l p x for all 0 p 2

some weighted inequalities on product domains we extend the results of r fefferman 3 on the bidisc to higher product domains via induction as an application we extend the weighted inequality for calderon zygmund operators on the bidisc to higher product domains and we also extend the result of the littlewood paley operator corresponding to the arbitrary disjoint rectangles to the weighted case 0 introduction in 3 r fefferman discovered a general technique which can be used to prove weighted results in r1 x r1 the product domain for product weights ap12 ri x r1 with 2 p oo his technique however explicitly used the special geometric properties of r2 in this paper we generalize his technique to higher dimeasion i e ri x r x r and beyond to be more specific we reduce fefferman s technique to a verification of one main estimate for a linear operatort f qqt tf x 1 dx t cf m f2 dx t q we will explain the notation in a later section having this estimate we can conclude that t is a bounded map on lp w for 2 p oo and a product weight w e ap12 verifying the main estimate for various operators t is not entirely trivial we work on two operators we prove the lp w boundedness for 2 p 00 w e api2 for the product domain generalization of calderon zygmund operators and lastly we prove the following littlewood paley result theorem let r be a set of disjoint rectangles with sides parallel to the coordinate axes let xr be their characteristic functions define si f 74 xr1 4 f we have the operator a f x si f x i bounded on lp w 2 p oo and w e ap12 r1 x r received by the editors february 9 1988 1980 mathematics subject classification 1985 revision primary 42b20 secondary 42b25 1990 american mathematical society 0002 9947 90 1 00 25 per page

besov spaces with variable smoothness and integrability abstract in this article we introduce besov spaces with variable smoothness and integrability indices we prove independence of the choice of basis functions as well as several other basic properties we also give sobolev type embeddings and show that our scale contains variable order holder zygmund spaces as special cases we provide an alternative characterization of the besov space using approximations by analytic functions

hormander type theorems for multi linear and multi parameter fourier multiplier operators with limited smoothness abstract the main purpose of this paper is three fold first of all we are concerned with the limited smoothness conditions in the spirit of hormander on the multi linear and multi parameter coifman meyer type fourier multipliers studied by c muscalu j pipher t tao c thiele 2004 2006 where they established the l r estimates for the multiplier operators under the assumption that the multiplier has smoothness of sufficiently large order under our limited smoothness assumption we will prove the l p 1 l p n l r boundedness with 1 p 1 1 p n 1 r for 1 p 1 p n and 0 r second our proof of l r estimates also offers a different and more direct approach than the one given in muscalu et al 2004 2006 where they use the deep analysis of multi linear and multi parameter paraproducts third we also prove a hormander type multiplier theorem in the weighted lebesgue spaces for such operators when the fourier multiplier is only assumed with limited smoothness

superposition frames for adaptive time frequency analysis and fast reconstruction in this paper we introduce a broad family of adaptive linear time frequency representations termed superposition frames and show that they admit desirable fast overlap add reconstruction properties akin to standard short time fourier techniques this approach stands in contrast to many adaptive time frequency representations in the existing literature which while more flexible than standard fixed resolution approaches typically fail to provide for efficient reconstruction and often lack the regular structure necessary for precise frame theoretic analysis our main technical contributions come through the development of properties which ensure that our superposition construction provides for a numerically stable invertible signal representation our primary algorithmic contributions come via the introduction and discussion of two signal adaptation schemes based on greedy selection and dynamic programming respectively we conclude with two short enhancement examples that serve to highlight potential applications of our approach

a discrete transform and triebel lizorkin spaces on the bidisc we use a discrete transform to study the triebel lizorkin spaces on bidisc fpaq jq and establishes the boundedness of transform sek p jq and tt jq q fp we also define the almost diagonal operator and prove its boundedness with the use of discrete transform and journe lemma we get the atomic decomposition of jq for 0 1 a e r is established too the case for p q is similar 0 introduction in this paper we use a discrete transform to study the triebel lizorkin spaces on the bidisc in 1 we define the sequence space fp and the distribution space fp 7 our fundamental formula is f eq f qq yiq where q runs over all dyadic rectangles and qq yq are translates and dilates of x y associated with q respectively fourier transforms of x y have compact support and suppqq q c l 2 2v11 1 4 in 2 we define almost diagonal operators and prove the boundedness of such operators on fp by duality we also define 3 m rectangle molecules received by the editors july 8 1993 and in revised form december 28 1993 originally communicated to the proceedings of the ams by andrew bruckner 1991 mathematics subject classification primary 42b20 supported by nnsf zpnsf and the doctoral foundation in china 1995 american mathematical society 0002 9947 95 1 00 25 per page

characterization of analytic wavelet transforms and a new phaseless reconstruction algorithm we obtain a characterization of all wavelets leading to analytic wavelet transforms wt the characterization is obtained as a byproduct of the theoretical foundations of a new method for wavelet phase reconstruction from magnitude only coefficients the cornerstone of our analysis is an expression of the partial derivatives of the continuous wt which results in phase magnitude relationships similar to the short time fourier transform setting and valid for the generalized family of cauchy wavelets we show that the existence of such relations is equivalent to analyticity of the wt up to a multiplicative weight and a scaling of the mother wavelet the implementation of the new phaseless reconstruction method is considered in detail and compared to previous methods it is shown that the proposed method provides significant performance gains and a great flexibility regarding accuracy versus complexity in addition we discuss the relation between scalogram reassignment operators and the wavelet transform phase gradient and present an observation on the phase around zeros of the wt

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

weighted norm inequalities off diagonal estimates and elliptic operators part i general operator theory and weights this is the first part of a series of four articles in this work we are interested in weighted norm estimates we put the emphasis on two results of different nature one is based on a good inequality with two parameters and the other uses calderon zygmund decomposition these results apply well to singular non integral operators and their commutators with bounded mean oscillation functions singular means that they are of order 0 non integral that they do not have an integral representation by a kernel with size estimates even rough so that they may not be bounded on all lp spaces for 1 p 8 pointwise estimates are then replaced by appropriate localized lp lq estimates we obtain weighted lp estimates for a range of p that is different from 1 8 and isolate the right class of weights in particular we prove an extrapolation theorem a la rubio de francia for such a class and thus vector valued estimates

affine and quasi affine frames for rational dilations in this paper we extend the investigation of quasi affine systems which were originally introduced by ron and shen j funct anal 148 1997 408 447 for integer expansive dilations to the class of rational expansive dilations we show that an affine system is a frame if and only if the corresponding family of quasi affine systems are frames with uniform frame bounds we also prove a similar equivalence result between pairs of dual affine frames and dual quasi affine frames finally we uncover some fundamental differences between the integer and rational settings by exhibiting an example of a quasi affine frame such that its affine counterpart is not a frame

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

atomic and molecular decompositions of anisotropic triebel lizorkin spaces weighted anisotropic triebel lizorkin spaces are introduced and studied with the use of discrete wavelet transforms this study extends the isotropic methods of dyadic transforms of frazier and jawerth 1985 1989 to non isotropic settings associated with general expansive matrix dilations and a weights in close analogy with the isotropic theory we show that weighted anisotropic triebel lizorkin spaces are characterized by the magnitude of the transforms in appropriate sequence spaces we also introduce non isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces thus extending isotropic results of frazier and jawerth

multiparameter operators and sharp weighted inequalities this article is concerned with the operators from harmonic analysis which are naturally associated to a multiple parameter family of dilations we are especially interested here in dealing with questions from the theory of such operators whose answers cannot be obtained by a reduction to the case of product operators we also introduce a new tool in order to carry out this multiparameter analysis which comes from the classical theory this is the concept of sharp weighted inequalities where for operators such as the hilbert transform or classical square function one asks for hilbert space inequalities which imply the correct estimates of the norm on lp of these operators as p becomes large 0 introduction in this article our main goal is to study classes of oper ators associated with multiparameter groups of dilations on rn in some sense the consideration of these operators is a natural next step or simplest case after those of the classical calderon zygmund theory and the product space theory it turns out that in order to obtain estimates for the multiparameter operators we shall be led to the following type of extremely classical question how does one obtain l2 estimates on r1 for the hilbert transform or other singular integrals which yield sharp estimates on the lp rl operator norm of h as p approaches 1 or oc these sharp weighted inequalities are of some independent interest and their study will be seen to be essentially equivalent to the deep work of chang wilson and wolff 3 on the exponential square integrability of functions whose area integral is bounded they also have other applications for example they provide an immediate extension of the chang wilson wolff estimates to product spaces as given by pipher in 12 at this point let us be more specific about the exact nature of the multipa rameter operators which we shall consider consider in r3 the dilations given by ps t x y z sx ty9 stz for s91 0 then there is a maximal operator first considered by zygmund ml and singular integrals tl introduced by ricci and stein 13 naturally associated to these dilations the maximal operator af3 is defined by

hardy spaces with variable exponents and generalized campanato spaces in the present paper we define hardy spaces with variable exponents on rn by the grand maximal function and then investigate their several properties the present paper will connect harmonic analysis with function spaces with variable exponents we obtain the atomic decomposition and the molecular decomposition with these decomposition proved we investigate the littlewood paley characterization also we specify the dual spaces of hardy spaces with variable exponents they will turn out to be campanato spaces with variable growth conditions the present paper covers local hardy spaces with variable exponents

complex interpolation of besov spaces and triebel lizorkin spaces with variable exponents abstract this paper concerns the complex interpolation of besov spaces and triebel lizorkin spaces with variable exponents

flexible tree structured signal expansions using time varying wavelet packets addresses the problem of finding the best time varying filter bank tree structured representation for a signal the tree is allowed to vary at regular intervals and the spacing of these changes can be arbitrarily short the question of how to choose tree structured representations of signals based on filter banks is considered wavelets and their adaptive version known as wavelet packets represent one approach that is popular wavelet packets are subband trees where the tree is chosen to match the characteristics of the signal variations where the tree varies over time have been proposed as the double tree and the time frequency tree algorithms time variation adds a further level of adaptivity in all of the approaches proposed so far the tree must be either fixed for the whole duration of the signal or fixed for its dyadic subintervals the solution that we propose as it allows more flexible variation is an advance on the wavelet packet algorithm the double tree algorithm and the recently proposed time frequency tree algorithm our solution is based on casting it in a dynamic programming dp setting focusing on compression applications we use a lagrangian cost of distortion spl lambda spl times rate as the objective function and explain our algorithm in detail pointing out its relation to existing approaches to the problem we demonstrate that the new algorithm indeed searches a larger library of representations than previously possible and that overcoming the constraint of dyadic time segmentations gives a significant improvement in practice

weighted hardy spaces in the three parameter case abstract in this paper we use the idea of the discrete littlewood paley theory developed by han and lu to carry out the three parameter weighted hardy spaces theory under a rather weak condition on the product weight w a and obtain the boundedness of singular integral operators on the weighted hardy spaces

an analytic wavelet transform with a flexible time frequency covering we develop a rational dilation wavelet transform for which the dilation factor the q factor and the redundancy can be easily specified the introduced transform contains hilbert transform pairs of atoms therefore it is also suitable for oscillatory signal processing the transform may be modified to obtain a tight chirplet frame for discrete time signals a fast implementation that makes use of an equivalent filter bank makes the transform suitable for long signals examples on natural signals are provided to demonstrate the utility of the transform

new orlicz hardy spaces associated with divergence form elliptic operators abstract let l be the divergence form elliptic operator with complex bounded measurable coefficients the positive concave function on 0 of strictly critical lower type p 0 1 and t t 1 1 t 1 for t 0 in this paper the authors study the orlicz hardy space h l r n and its dual space bmo l r n where l denotes the adjoint operator of l in l 2 r n several characterizations of h l r n including the molecular characterization the lusin area function characterization and the maximal function characterization are established the carleson measure characterization and the john nirenberg inequality for the space bmo l r n are also given as applications the authors show that the riesz transform l 1 2 and the littlewood paley g function g l map h l r n continuously into l the authors further show that the riesz transform l 1 2 maps h l r n into the classical orlicz hardy space h r n for p n n 1 1 and the corresponding fractional integral l for certain 0 maps h l r n continuously into h l r n where is determined by and and satisfies the same property as all these results are new even when t t p for all t 0 and p 0 1

estimates for rough fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators we study the boundedness of rough fourier integral and pseudodifferential operators defined by general rough hormander class amplitudes on banach and quasi banach lplp spaces thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for hormander class bilinear pseudodifferential operators and certain classes of bilinear as well as multilinear fourier integral operators for these classes of amplitudes the boundedness of the aforementioned fourier integral operators turn out to be sharp furthermore we also obtain results for rough multilinear operators

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

riesz transforms associated to schrodinger operators on weighted hardy spaces let w be some ap weight and enjoy reverse holder inequality and let l v be a schrodinger operator on rn where v lloc1 rn is a non negative function on rn in this article we introduce weighted hardy spaces hl w1 rn associated to l in terms of the area function characterization and prove their atomic characters we show that the riesz transform l 1 2 associated to l is bounded on lwp rn for 1 p 2 and bounded from hl w1 rn to the classical weighted hardy space hw1 rn

boundedness criterion of journeʼs class of singular integrals on multiparameter hardy spaces abstract this article concerns nonconvolutional type operators also known as journe s type operators associated with a multiparameter family of dilations given by x 1 x 2 x m 1 x 1 2 x 2 m x m where x 1 r n 1 x 2 r n 2 x m r n m and m 3 we are especially interested in the boundedness of such operators on the multiparameter hardy spaces this work is motivated by pipher s result on the boundedness of these operators from the multiparameter h p spaces to l p spaces for 0 p 1 and journe s counter example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter hardy spaces journe s work shows that there is a sharp difference between the situations for two and three or more parameters we establish in this paper the necessary and sufficient conditions under which the singular integral operators in journe s class are bounded on multiparameter h p spaces 0 p 1 with arbitrary number of parameters

vector valued nonstationary gabor frames abstract as an extension of gabor frames nonstationary gabor nsg frames were recently introduced in adaptive signal analysis they allow for efficient reconstruction with flexible sampling and varying window functions in this paper we generalize the notion of nsg frames from l 2 r to the vector valued hilbert space l 2 r c l and investigate the resulting vector valued nsg frames we derive a walnut s representation of the mixed frame operator and provide some necessary sufficient conditions for a vector valued nsg system to be a frame for l 2 r c l furthermore we show the existence of painless vector valued nsg frames and of vector valued nsg frames with fast decaying window functions

new molecular characterizations of anisotropic musielak orlicz hardy spaces and their applications abstract let r n 0 0 be an anisotropic growth function and a a general expansive matrix on r n let h a r n be the anisotropic musielak orlicz hardy space associated with a defined via the non tangential grand maximal function in this article the authors establish a new molecular characterization of h a r n via its known atomic characterizations which positively answer a question mentioned by li et al 2016 34 as an application the authors obtain a criterion on the boundedness of linear operators on h a r n which further implies the boundedness of integral anisotropic calderon zygmund operators on h a r n in addition the boundedness of these operators from h a r n to the musielak orlicz space l r n is also presented

a new class of function spaces connecting triebel lizorkin spaces and q spaces abstract let s r 0 p 1 and q 1 in this paper we introduce a new class of function spaces f p q s r n which unify and generalize the triebel lizorkin spaces with both p 1 and p and q spaces by establishing the carleson measure characterization of q space we then determine the relationship between triebel lizorkin spaces and q spaces which answers a question posed by dafni and xiao in g dafni j xiao some new tent spaces and duality theorem for fractional carleson measures and q r n j funct anal 208 2004 377 422 moreover via the hausdorff capacity we introduce a new class of tent spaces f t p q s r n 1 and determine their dual spaces f w p q s q r n where s r p q 1 max p q 1 0 q max p q and t denotes the conjugate index of t 1 as an application of this we further introduce certain hardy hausdorff spaces f h p q s r n and prove that the dual space of f h p q s r n is just f p q s q r n when p q 1

weighted l p estimates for rough bi parameter fourier integral operators abstract fourier integral operators play an important role in applications to partial differential equations the main purpose of this paper is to investigate the weighted l p boundedness of the bi parameter fourier integral operators or fios for short of the following form t f x r n 1 r n 2 e i x a x f d d where for x x 1 x 2 r n 1 r n 2 r n 1 0 and r n 2 0 the amplitude function a x l b s m and the phase function is of the form x 1 x 1 2 x 2 when 1 l 2 r n 1 r n 1 0 2 l 2 r n 2 r n 2 0 satisfy certain differential conditions we will establish the weighted l w p boundedness of t under the assumption that the weight w is in the class of the product muckenhoupt a p r n 1 r n 2 weights

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

implementation of discretized gabor frames and their duals the usefulness of gabor frames depends on the easy computability of a suitable dual window this question is addressed under several aspects several versions of schulz s iterative algorithm for the approximation of the canonical dual window are analyzed for their numerical stability for gabor frames with totally positive windows or with exponential b splines a direct algorithm yields a family of exact dual windows with compact support it is shown that these dual windows converge exponentially fast to the canonical dual window

littlewood paley characterization and duality of weighted anisotropic product hardy spaces the authors study anisotropic product hardy spaces hwp a associated with a pair a a1 a2 of expansive dilations and a class of product muckenhoupt weights a a on rn rm this article is a continuation of their earlier work in bownik et al 2010 8 the authors establish the littlewood paley g function characterization and the transform characterization of hwp a 0 p 1 the authors also introduce the weighted anisotropic product campanato space lp w a and establish its transform characterization as an application the authors identify the dual space of hwp a with lp w a the results of this article improve the existing results for weighted product hardy spaces on r r and are new even in the weighted isotropic setting

duality and interpolation of anisotropic triebel lizorkin spaces we study properties of anisotropic triebel lizorkin spaces associated with general expansive dilations and doubling measures on mathbb r n using wavelet transforms this paper is a continuation of bownik in j geom anal 2007 to appear trans am math soc 358 1469 1510 2006 where we generalized the isotropic methods of dyadic varphi transforms of frazier and jawerth j funct anal 93 34 170 1990 to non isotropic settings by working at the level of sequence spaces we identify the duals of anisotropic triebel lizorkin spaces we also obtain several real and complex interpolation results for these spaces

heat kernel based decomposition of spaces of distributions in the framework of dirichlet spaces classical and nonclassical besov and triebel lizorkin spaces with complete range of indices are developed in the general setting of dirichlet space with a doubling measure and local scale invariant poincare inequality this leads to a heat kernel with small time gaussian bounds and holder continuity which play a central role in this article frames with band limited elements of sub exponential space localization are developed and frame and heat kernel characterizations of besov and triebel lizorkin spaces are established this theory in particular allows the development of besov and triebel lizorkin spaces and their frame and heat kernel characterization in the context of lie groups riemannian manifolds and other settings

an atomic decomposition of weighted hardy spaces associated to self adjoint operators abstract let w be an a p weight on r n 1 p let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n associated to l which was introduced in 28 the main contribution of this article is to weaken conditions on the weight w of the weighted hardy spaces h l w 1 r n and improve the result in 28

and variants of multiparameter calderon zygmund theory we consider calderon zygmund operators on product domains under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on h p r r r 0 p 1 if its behaviour on l 2 and on certain scalar valued and vector valued rectangle atoms is known another result concerns an extension of the authors results on l p variants of calderon zygmund theory 1 23 to the product domain setting as an application one obtains estimates for fourier multipliers and pseudo differential operators

new besov type spaces and triebel lizorkin type spaces including q spaces let s tau in mathbb r and q in 0 infty we introduce besov type spaces dot b s tau p q mathbb r n for p in 0 infty and triebel lizorkin type spaces dot f s tau p q mathbb r n rm for p in 0 infty which unify and generalize the besov spaces triebel lizorkin spaces and q spaces we then establish the varphi transform characterization of these new spaces in the sense of frazier and jawerth using the varphi transform characterization of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n we obtain their embedding and lifting properties moreover for appropriate we also establish the smooth atomic and molecular decomposition characterizations of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n for s in mathbb r p in 1 infty q in 1 infty and tau in 0 frac 1 max p q via the hausdorff capacity we introduce certain hardy hausdorff spaces b dot h s tau p q mathbb r n and prove that the dual space of b dot h s tau p q mathbb r n is just dot b s tau p q mathbb r n where t denotes the conjugate index of t in 1 infty

system bandwidth and the existence of generalized shift invariant frames abstract we consider the question whether given a countable family of lattices j j j in a locally compact abelian group g there exist functions g j j j such that the resulting generalized shift invariant system g j j j j is a tight frame of l 2 g this paper develops a new approach to the study of generalized shift invariant system via almost periodic functions based on a novel unconditional convergence property from this theory we derive characterizing relations for tight and dual frame generators we introduce the system bandwidth as a measure of the total bandwidth a generalized shift invariant system can carry and we show that the so called calderon sum is uniformly bounded from below for generalized shift invariant frames without the unconditional convergence property we show counter intuitively that even orthonormal bases can have arbitrary small system bandwidth our results show that the question of existence of frame generators for a general lattice system is rather subtle and depends on analytical and algebraic properties of the lattice system

an efficient algorithm for the calculation of a constant q transform an efficient method of transforming a discrete fourier transform dft into a constant q transform where q is the ratio of center frequency to bandwidth has been devised this method involves the calculation of kernels that are then applied to each subsequent dft only a few multiples are involved in the calculation of each component of the constant q transform so this transformation adds a small amount to the computation in effect this method makes it possible to take full advantage of the computational efficiency of the fast fourier transform fft graphical examples of the application of this calculation to musical signals are given for sounds produced by a clarinet and a violin

frames their relatives and reproducing kernel hilbert spaces this paper considers different facets of the interplay between reproducing kernel hilbert spaces rkhs and stable analysis synthesis processes first we analyze the structure of the reproducing kernel of a rkhs using frames and reproducing pairs second we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space our proof uses the rkhs structure of the range of the analysis operator this in turn implies that all the attempts to extend the notion of riesz basis to general measure spaces are fruitless since every such family can be identified with a discrete riesz basis finally we show how the range of the analysis operators of a reproducing pair can be equipped with a rkhs structure

anisotropic variable hardy lorentz spaces and their real interpolation abstract let p r n 0 be a variable exponent function satisfying the globally log holder continuous condition q 0 and a be a general expansive matrix on r n in this article the authors first introduce the anisotropic variable hardy lorentz space h a p q r n associated with a via the radial grand maximal function and then establish its radial or non tangential maximal function characterizations moreover the authors also obtain characterizations of h a p q r n respectively in terms of the atom and the lusin area function as an application the authors prove that the anisotropic variable hardy lorentz space h a p q r n serves as the intermediate space between the anisotropic variable hardy space h a p r n and the space l r n via the real interpolation this together with a special case of the real interpolation theorem of h kempka and j vybiral on the variable lorentz space further implies the coincidence between h a p q r n and the variable lorentz space l p q r n when ess inf x r n p x 1

weighted norm inequalities off diagonal estimates and elliptic operators part iii harmonic analysis of elliptic operators this is the third part of a series of four articles on weighted norm in equalities o diagonal estimates and elliptic operators for l in some class of elliptic operators we study weighted norm l p inequalities for singular non integral opera tors arising from l those are the operators l for bounded holomorphic functions the riesz transforms rl 1 2 or 1 2 l 1 2 and its inverse l 1 2 1 2 some quadratic functionals gl and gl of littlewood paley stein type and also some vector valued inequalities such as the ones involved for maximal l p regularity for each we obtain sharp or nearly sharp ranges of p using the general theory for bound edness of part i and the o diagonal estimates of part ii we also obtain commutator results with bmo functions

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

function spaces of variable smoothness and integrability abstract in this article we introduce triebel lizorkin spaces with variable smoothness and integrability our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years vector valued maximal inequalities do not work in the generality which we pursue and an alternate approach is thus developed using it we derive molecular and atomic decomposition results and show that our space is well defined i e independent of the choice of basis functions as in the classical case a unified scale of spaces permits clearer results in cases where smoothness and integrability interact such as sobolev embedding and trace theorems as an application of our decomposition we prove optimal trace theorem in the variable indices case

on the analysis of accumulation curves summary identifying and counting the total number of biological species observed when plotted against a measure of the effort used to record them gives rise to a species accumulation curve we investigate estimation of the total number of species and other relevant properties of accumulation by elaborating on the multinomial model nakamura and peraza 1998 journal of agricultural biological and environmental statistics 3 17 36 that includes specification of a beta density for recording probabilities we consider a unified description including complete and incomplete aggregated curves a more general scheme for recording and a bayesian framework to allow for inclusion of the biologist s knowledge with regard to typical recording probabilities and the total number of species the beta distribution is used as a prior but recording probabilities are not restrained to be beta distributed the methods yield either closed analytical expressions for inference or relatively simple numerical procedures predictive distributions of future recordings that would eventually lead to an optimal decision theoretical rule for stopping collection effort may be easily calculated a case study regarding species of bats is considered including some guidelines for elicitation of an informative prior

calderon reproducing formulas and new besov spaces associated with operators abstract let l be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of poisson type acting on l 2 x where x is a possibly non doubling space of polynomial upper bound on volume growth the aim of this paper is to introduce a new class of besov spaces associated with the operator l so that when l is the laplace operator or its square root acting on the euclidean space r n the new besov spaces are equivalent to the classical besov spaces depending on the choice of l the new besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers l since our approach does not require the doubling volume property of the underlying space it is applicable to any subset of euclidean spaces without any smoothness requirement on the boundary we will also develop a number of calderon reproducing formulas which play an important role in the theory of function spaces and are of independent interest as an application we study besov spaces associated with schrodinger operators with non negative potentials satisfying reverse holder estimates

theory implementation and applications of nonstationary gabor frames signal analysis with classical gabor frames leads to a fixed time frequency resolution over the whole time frequency plane to overcome the limitations imposed by this rigidity we propose an extension of gabor theory that leads to the construction of frames with time frequency resolution changing over time or frequency we describe the construction of the resulting nonstationary gabor frames and give the explicit formula for the canonical dual frame for a particular case the painless case we show that wavelet transforms constant q transforms and more general filter banks may be modeled in the framework of nonstationary gabor frames further we present the results in the finite dimensional case which provides a method for implementing the above mentioned transforms with perfect reconstruction finally we elaborate on two applications of nonstationary gabor frames in audio signal processing namely a method for automatic adaptation to transients and an algorithm for an invertible constant q transform

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

besov type spaces with variable smoothness and integrability abstract in this article the authors introduce besov type spaces with variable smoothness and integrability the authors then establish their characterizations respectively in terms of transforms in the sense of frazier and jawerth smooth atoms or peetre maximal functions as well as a sobolev type embedding as an application of their atomic characterization the authors obtain a trace theorem of these variable besov type spaces

gradient estimates for the p x laplacean system we prove calderon and zygmund type estimates for a class of elliptic problems whose model is the non homogeneous p x laplacean system div jduj p x 2 du div jf j p x 2 f

frequency domain design of overcomplete rational dilation wavelet transforms the dyadic wavelet transform is an effective tool for processing piecewise smooth signals however its poor frequency resolution its low q factor limits its effectiveness for processing oscillatory signals like speech eeg and vibration measurements etc this paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied the new wavelet transform can attain higher q factors desirable for processing oscillatory signals or the same low q factor of the dyadic wavelet transform the new wavelet transform is modestly overcomplete and based on rational dilations like the dyadic wavelet transform it is an easily invertible constant q discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for l2 r the wavelet can be made to resemble a gabor function and can hence have good concentration in the time frequency plane the construction of the new wavelet transform depends on the judicious use of both the transform s redundancy and the flexibility allowed by frequency domain filter design

estimates for singular integrals on homogeneous groups we consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups we prove certain estimates for the operators that imply lp boundedness of them by an extrapolation argument under a sharp condition for the kernels also we prove some weighted lp inequalities for the operators

calderon zygmund theory for product domains hp spaces this article deals with the boundedness properties of singular integrals that generalize the double hilbert transform on product domains the action of these operators on hardy spaces and on l log l is discussed

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

nonparametric hierarchical bayes via sequential imputations we consider the empirical bayes estimation of a distribution using binary data via the dirichlet process let d denote a dirichlet process with being a finite measure on 0 1 instead of having direct samples from an unknown random distribution f from d we assume that only indirect binomial data are observable this paper presents a new interpretation of lo s formula and thereby relates the predictive density of the observations based on a dirichlet process model to likelihoods of much simpler models as a consequence the log likelihood surface as well as the maximum likelihood estimate of c 0 1 is found when the shape of is assumed known together with a formula for the fisher information evaluated at the estimate the sequential imputation method of kong liu and wong is recommended for overcoming computational difficulties commonly encountered in this area the related approximation formulas are provided an analysis of the tack data of beckett and diaconis which motivated this study is supplemented to illustrate our methods

sharp lorentz space estimates for rough operators we demonstrate the h 1 l 1 2 or l p l p 2 mapping properties of several rough operators in all cases these estimates are sharp in the sense that the lorentz exponent 2 cannot be replaced by any lower number