boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

branching index sum rules for simple lie algebras let l and l0 be a simple lie algebra and its sub lie algebra respectively then a given irreducible representation of l decomposes into a direct sum of irreducible components of l0 which is called the branching rule the general dynkin indices introduced earlier satisfy many sum rules for the branching rule these are found to be strong enough to uniquely determine the branching rule for many cases we have studied the sum rules are especially useful for cases of exceptional lie algebras

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

complete branching rules generating function for so 7 contains implies su 2 3 and polynomial basis states a branching rules generating function is given for so 7 contains implies su 2 3 along with its interpretation in terms of basis states and instructions for calculating generator matrix elements the generator matrix elements for the degenerate 0 0 c representations are calculated as an example of this procedure

a lower bound criterion for iterated commutators abstract we consider iterated commutators of multiplication by a symbol function and smooth calderon zygmund operators described by fourier multipliers of homogeneity 0 we establish a criterion for a collection of symbols so that the corresponding calderon zygmund operators characterize product bmo by means of iterated commutators we therefore extend in part the line of one parameter results following the work of uchiyama and li as well as the result in several parameters concerning commutators with riesz transforms by lacey petermichl pipher wick

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

multiple weighted estimates for commutators of multilinear singular integrals with non smooth kernels abstract multilinear commutators and iterated commutators generated by the multilinear singular integrals with non smooth kernels and bmo functions are studied by the weighted estimates of a class of new variant maximal operators and the sharp maximal functions the multiple weighted norm inequalities for such operators are obtained in particular some previous results in anh and duong 2010 1 lerner et al 2009 4 perez et al 2011 5 are improved or extended significantly

signal reconstruction from two close fractional fourier power spectra based on the definition of the instantaneous frequency signal phase derivative as a local moment of the wigner distribution we derive the relationship between the instantaneous frequency and the derivative of the squared modulus of the fractional fourier transform fractional fourier transform power spectrum with respect to the angle parameter we show that the angular derivative of the fractional power spectrum can be found from the knowledge of two close fractional power spectra it permits us to find the instantaneous frequency and to solve the phase retrieval problem up to a constant phase term if only two close fractional power spectra are known the proposed technique is noniterative and noninterferometric the efficiency of the method is demonstrated on several examples including monocomponent multicomponent and noisy signals it is shown that the proposed method works well for signal to noise ratios snrs higher than about 3 db the appropriate angular difference of the fractional power spectra used for phase retrieval depends on the complexity of the signal and can usually reach several degrees other applications of the angular derivative of the fractional power spectra for signal analysis are discussed the proposed technique can be applied for phase retrieval in optics where only the fractional power spectra associated with intensity distributions can be easily measured

fractional sampling theorem for alpha bandlimited random signals and its relation to the von neumann ergodic theorem considering that fractional correlation function and the fractional power spectral density for alpha stationary random signals form a fractional fourier transform pair we present an interpolation formula to estimate a random signal from a temporal random series based on the fractional sampling theorem for alpha bandlimited random signals furthermore by establishing the relationship between the sampling theorem and the von neumann ergodic theorem the estimation of the power spectral density of a random signal from one sample signal becomes a suitable approach thus the validity of the sampling theorem for random signals is closely linked to an ergodic hypothesis in the mean sense

time frequency analysis of the impulse response the impulse response is a key function for the analysis and design of systems therefore understanding its properties is a fundamental problem to investigate the nonstationary structure of the impulse response we derive its time frequency representation both in the wigner and smoothed wigner distribution domains first we consider the class of single input single output siso systems which models a wide variety of physical phenomena the obtained wigner distribution for the class of siso systems is made by frequency translated versions of the wigner distribution of the impulse response for a first order system a t shaped function of time and frequency plus the interference terms which are effectively filtered out in the smoothed wigner domain then we consider the case of multiple input multiple output mimo systems whose impulse response is the collection of the individual responses obtained by applying an impulse to every input separately also in this case the wigner distribution of the impulse response depends on the t function we apply our results to a two dimensional mimo harmonic oscillator and to a siso system with three resonances finally we propose the key idea of a system identification method that operates directly in the time frequency domain by fitting the smoothed t function we show that the method works for a nonlinear oscillator with a large cubic term as well as for a time varying harmonic oscillator and that it can tolerate the presence of both stationary and nonstationary noise

generating functions and elementary young tableaux when constructing polynomial bases for group representations one is led naturally to consider a stretched product of states elementary multiplets are states that cannot be expressed as the stretched product of any other two states and they can be identified with terms occurring in certain generating functions in this paper the authors define an analogous stretched product of young tableaux together with elementary tableaux these may be used in a manner similar to elementary multiplets to construct generating functions they consider several examples where this method can be applied and in particular give new large n rank free branching rule generating functions for the group subgroup pairs su n su m contained in implied by su n m first four labels non zero so n contained in implied by su n first three labels non zero and sp 2n contained in implied by su 2n first five labels non zero this method of finding these generating functions uses the theory of grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux

fractional power spectrum in this paper by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process we consider the case associated with the random process in an explicit manner the fractional power spectral relations for the fractional fourier domain filter are derived and the expression for the fractional power spectrum in terms of the fractional correlation is obtained in addition the definitions and the properties of the fractional white noise and the chirp stationary process are presented simulation results verify the theoretical derivations and demonstrate the potential applications such as detection and parameter estimation of chirp signals fractional power spectral estimation and system identification in the fractional fourier domain

multiple parameter discrete fractional transform and its applications in recent years several special multiple parameter discrete fractional transforms mpdfrts have been proposed and their advantages have been demonstrated in the fields of communication systems and information security however the general theoretical framework of mpdfrts has not yet been established in this paper we propose two separate theoretical frameworks called the type i and ii mpdfrt that can include existing multiple parameter transforms as special cases the properties of the type i and ii mpdfrt have been analyzed in detail and their high dimensional operators have been defined under the theoretical frameworks we can construct new types of transforms that may be useful in signal processing and information security finally we perform two applications about image encryption and image feature extraction in the type i and ii mpdfrt domain the simulation results demonstrate that the typical transforms constructed under the proposed theoretical frameworks yield promising results in these applications

two weight commutators in the dirichlet and neumann laplacian settings abstract in this paper we establish the characterization of the weighted bmo via two weight commutators in the settings of the neumann laplacian n on the upper half space r n and the reflection neumann laplacian n on r n with respect to the weights associated to n and n respectively this in turn yields a weak factorization for the corresponding weighted hardy spaces where in particular the weighted class associated to n is strictly larger than the muckenhoupt weighted class and contains non doubling weights in our study we also make contributions to the classical muckenhoupt wheeden weighted hardy space bmo space respectively by showing that it can be characterized via the area function carleson measure respectively involving the semigroup generated by the laplacian on r n and that the duality of these weighted hardy and bmo spaces holds for muckenhoupt a p weights with p 1 2 while the previously known related results cover only p 1 n 1 n we also point out that this two weight commutator theorem might not be true in the setting of general operators l and in particular we show that it is not true when l is the dirichlet laplacian d on r n

commutators in the two weight setting let r be the vector of riesz transforms on mathbb r n and let mu lambda in a p be two weights on mathbb r n 1 p infty the two weight norm inequality for the commutator b r l p mu rightarrow l p lambda is shown to be equivalent to the function b being in a bmo space adapted to mu and lambda this is a common extension of a result of coifman rochberg weiss in the case of both lambda and mu being lebesgue measure and bloom in the case of dimension one

fractional fourier analysis of random signals and the notion of spl alpha stationarity of the wigner ville distribution in this paper a generalized notion of wide sense stationarity for random signals is presented the notion of stationarity is fundamental in the fourier analysis of random signals for this purpose a definition of the fractional correlation between two random variables is introduced it is shown that for wide sense stationary random signals the fractional correlation and the fractional power spectral density functions form a fractional fourier transform pair thus the concept of stationarity plays an important role in the analysis of random signals through the fractional fourier transform for signals nonstationary in the standard formulation but stationary furthermore we define the wigner ville distribution in terms of the fractional correlation function in which the standard fourier analysis is the particular case for 2 and it leads to the wiener khinchin theorem

standard young tableaux and weight multiplicities of the classical lie groups by examining the branching rules for all irreducible representations of the classical groups u k su k so 2k 1 sp 2k and so 2k on restriction to u 1 u 1 u 1 standard young tableaux are specified for each of these groups it is shown that these tableaux determine the corresponding characters of the irreducible representations the rules for constructing these tableaux are derived and in this way the determination of weight multiplicities is reduced to a simple combinatorial exercise general formula for such weight multiplicities are given encompassing the most difficult case namely that of so 2k illustrative examples are provided including some yielding the explicit k dependence of weight multiplicities

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

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

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

fractional convolution and correlation via operator methods and an application to detection of linear fm signals using operator theory methods together with our previously introduced unitary fractional operator we derive explicit definitions of fractional convolution and correlation operations in a systematic and comprehensive manner via operator manipulations we also provide alternative formulations of those fractional operations that suggest efficient algorithms for discrete implementation through simulation examples we demonstrate how well the proposed efficient method approximates the continuous formulation of fractional autocorrelation it is also shown that the proposed fractional autocorrelation corresponds to radial slices of the ambiguity function af we also suggest an application of the fast fractional autocorrelation for detection and parameter estimation of linear fm signals

hardy sobolev spaces on product domains and application abstract we introduce the product hardy sobolev space h 1 p r n r m and obtain its characterization in terms of atomic decomposition as an application we give an endpoint version of multi parameter div curl lemma

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

multiparameter riesz commutators it is shown that product bmo of s y a chang and r fefferman defined on the space bbb r d 1 otimes cdots otimes bbb r d t can be characterized by the multiparameter commutators of riesz transforms this extends a classical one parameter result of r coifman r rochberg and g weiss and at the same time extends the work of m lacey and s ferguson and m lacey and e terwilleger on multiparameter commutators with hilbert transforms

product hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type the aim of this article is to develop the theory of product hardy spaces associated with operators which possess the weak assumption of davies gaffney heat kernel estimates in the setting of spaces of homogeneous type we also establish a calderon zygmund decomposition on product spaces which is of independent and use it to study the interpolation of these product hardy spaces we then show that under the assumption of generalized gaussian estimates the product hardy spaces coincide with the lebesgue spaces for an appropriate range of p

differentiation of the integral hardy spaces and calderon zygmund operators in the product setting abstract this article concerns strong differentiation and operators on product hardy spaces we first show that an example created by papoulis of a function on r 2 whose integral is strongly differentiable almost everywhere but the integral of its absolute value fails to be strongly differentiable on a set of positive measure belongs to the product hardy space h 1 r r the methods that we develop enable us to present a relaxed version of chang fefferman p atoms with a lower number of required vanishing moments and no smoothness needed on the elementary particles in analogy with the proof of this result we show a generalization of a theorem of r fefferman which concludes h p l p 0 p 1 boundedness of multiparameter operators from their behavior on rectangle atoms in addition we extend a result of pipher concerning boundedness of multiparameter calderon zygmund operators from h p to l p

relations between fractional operations and time frequency distributions and their applications the fractional fourier transform frft is a useful tool for signal processing it is the generalization of the fourier transform many fractional operations such as fractional convolution fractional correlation and the fractional hilbert transform are defined from it in fact the frft can be further generalized into the linear canonical transform lct and we can also use the lct to define several canonical operations in this paper we discuss the relations between the operations described above and some important time frequency distributions tfds such as the wigner distribution function wdf the ambiguity function af the signal correlation function and the spectrum correlation function first we systematically review the previous works in brief then some new relations are derived and listed in tables then we use these relations to analyze the applications of the frpt lct to fractional canonical filter design fractional canonical hilbert transform beam shaping and then we analyze the phase amplitude problems of the frft lct for phase amplitude problems we find as with the original fourier transform that in most cases the phase is more important than the amplitude for the frft lct we also use the wdf to explain why fractional canonical convolution can be used for space variant pattern recognition

fractional fourier transform wigner distribution and filter design for stationary and nonstationary random processes in this paper we derive the relationship among the fractional fourier transform frft the linear canonical transform lct and the stationary and nonstationary random processes we find many interesting properties for example if we perform the frft for a stationary process although the result is no longer stationary the amplitude of the autocorrelation function is still independent of time we also find that the lct of a white noise is still a white one for the frft of a stationary process the ambiguity function af is a tilted line and the wigner distribution function wdf is invariant along a certain direction we also define the fractional stationary random process and find that a nonstationary random process can be expressed by a summation of fractional stationary random processes in addition after performing the filter designed in the frft domain for a white noise we can use the segment length of the axis on the wdf plane to estimate the power of the noise and use the area circled by cutoff lines to estimate its energy thus in communication to reduce the effect of the white noise the area of the wdf of the transmitted signal should be as small as possible

on character generators for simple lie algebras we study character generating functions character generators of simple lie algebras the expression due to patera and sharp derived from the weyl character formula is first reviewed a new general formula is then found it makes clear the distinct roles of outside and inside elements of the integrity basis and helps determine their quadratic incompatibilities we review analyse and extend the results obtained by gaskell using the demazure character formulae we find that the fundamental generalized poset graphs underlying the character generators can be deduced from such calculations these graphs introduced by baclawski and towber can be simplified for the purposes of constructing the character generator the generating functions can be written easily using the simplified graphs and associated demazure expressions the rank two algebras are treated in detail but we believe our results are indicative of those for general simple lie algebras

higher order journe commutators and characterizations of multi parameter bmo abstract we characterize l p boundedness of iterated commutators of multiplication by a symbol function and tensor products of riesz and hilbert transforms we obtain a two sided norm estimate that shows that such operators are bounded on l p if and only if the symbol belongs to the appropriate multi parameter bmo class we extend our results to a much more intricate situation commutators of multiplication by a symbol function and paraproduct free journe operators we show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi parameter bmo class in this sense the tensor products of riesz transforms are a representative testing class for journe operators previous results in this direction do not apply to tensor products and only to journe operators which can be reduced to calderon zygmund operators upper norm estimates of journe commutators are new even in the case of no iterations lower norm estimates for iterated commutators only existed when no tensor products were present in the case of one dimension lower estimates were known for products of two hilbert transforms and without iterations new methods using journe operators are developed to obtain these lower norm estimates in the multi parameter real variable setting

standard young tableaux and character generators of classical lie groups character generators are derived for each of the classical groups su k so 2k 1 sp 2k and so 2k in order to do this use is made of various generalisations of standard young tableaux one set of these is in one to one correspondence with the contributions to the characters of each irreducible representation and the other involving shifted young diagrams provides a means of enumerating those maximal chains in the fundamental posets of each group which serve to generate via their descent subchains all possible multichains the character generators are then written down in terms of a suitable labelling of the poset elements illustrative examples are given

multi parameter local hardy spaces abstract though multi parameter hardy space theory has been well developed in the past half century not much has been studied for a local hardy space theory in the multi parameter settings such multi parameter local hardy spaces can play an important role in studying the boundedness of multi parameter pseudo differential operators multi parameter singular integrals of non convolution type and applications to partial differential equations etc by establishing a bi parameter local reproducing formula bi parameter local hardy space h p r n 1 r n 2 is introduced in this paper this space coincides with l p r n 1 n 2 when p 1 while p 1 this space is substantially different from the classical bi parameter hardy spaces h p r n 1 r n 2 we will establish its atomic decomposition of the bi parameter local hardy spaces and as an application we prove the boundedness from h p r n 1 r n 2 to l p r n 1 n 2 of the bi parameter singular integral operators in inhomogeneous journe class for p near 1 for the simplicity we have chosen to present all the results in the bi parameter setting nevertheless all of them hold for arbitrary number of parameters the multi parameter local theory developed in this paper can serve as a model case for similar theory in other multi parameter settings

character generators for unitary and symplectic groups combinatorial algorithms for computing the character generators of u n su n and sp 2n c are described these algorithms produce relatively compact nested expressions for the character generators moreover the terms appearing in these expressions all have positive coefficients this feature is not shared by the expression for the character generator which uses the weyl character formula

two weight characterization for commutators of bi parameter fractional integrals abstract let i be the bi parameter fractional integral operator on r n 1 r n 2 i f x r n 1 r n 2 f y 1 y 2 x 1 y 1 n 1 1 x 2 y 2 n 2 2 d y 0 i n i i 1 2 in this paper we give a characterization of two weight norm inequality for the commutator of i we show that for a p q r n b i l p p l q q b b m o where 1 and 1 p 1 q 1 n 1 1 n 2 it extends the recent one parameter theory to the bi parameter setting we use the modern dyadic methods in which the main idea is to represent continuous operators in terms of dyadic operators moreover by introducing some new full and mixed bi parameter dyadic paraproducts we write the commutator as a finite linear combination of them