on independence and conditioning on wiener space notant i f et i g des integrales de wiener ito multiples d ordre p et q respectivement on donne une condition necessaire et suffisante sur le couple f g sous laquelle les variables aleatoires i f et i g sont independantes ceci est base sur une caracterisation generale de l independance de variables aleatoires dans l espace de wiener dans le contexte d un calcul variationnel stochastique

properties of the voice transform of the blaschke group and connections with atomic decomposition results in the weighted bergman spaces abstract feichtinger and grochenig described a unified approach to atomic decomposition through integrable group representations in banach spaces studying the properties of a special voice transform of the blascke group outlined by the general theory developed by feichtinger and grochenig we obtain that every function from the minimal mobius invariant space will generate an atomic decomposition in the weighted bergman spaces in the unified approach of the atomic decomposition a useful tool is the q density the v separated property and the bounded uniform partitions of the unity of the locally compact group using the hyperbolic metric we can describe the q density from right and the separation from right in the blaschke group using this we can give an example of bounded uniform partitions of the unity from right in the general theory of atomic decomposition it is used the q density from the left this is the reason why we will make a small modification in the discretizing operator which corresponds to the q density from the right in order to obtain atomic decomposition in the weighted bergman spaces

frames arising from irreducible solvable actions i abstract in this work we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable lie groups in contrast to other well known techniques such as the coorbit theory the generalized coorbit theory and other discretization schemes we make no assumption on the integrability or square integrability of the representations of interest moreover our scheme produces explicit constructions of frames with precise frame bounds as an illustration of the scope of our results we highlight that a large class of representations which naturally occur in wavelet theory and time frequency analysis is handled by our scheme for example the affine group the generalized heisenberg groups the shearlet groups solvable extensions of vector groups and various solvable extensions of non commutative nilpotent lie groups are a few examples of groups whose irreducible representations are handled by our method the class of representations studied in this work is described as follows let g be a simply connected connected completely solvable lie group with lie algebra g p m next let be an infinite dimensional unitary irreducible representation of g obtained by inducing a character from a closed normal subgroup p exp p of g additionally we assume that g p m m exp m is a closed subgroup of g d m is a fixed haar measure on the solvable lie group m and there exists a linear functional p such that the representation ind p g is realized as acting in l 2 m d m making no assumption on the integrability of we describe explicitly a discrete subset of g and a vector f l 2 m d m such that f is a tight frame for l 2 m d m we also construct compactly supported smooth functions s and discrete subsets g such that s is a frame for l 2 m d m

generalized coorbit space theory and inhomogeneous function spaces of besov lizorkin triebel type coorbit space theory is an abstract approach to function spaces and their atomic decom positions the original theory developed by feichtinger and grochenig in the late 1980ies heavily uses integrable representations of locally compact groups their theory covers in particular homogeneous besov lizorkin triebel spaces modulation spaces bergman spaces and the recent shearlet spaces however inhomogeneous besov lizorkin triebel spaces cannot be covered by their group theoretical approach later it was recognized by fornasier and the first named author 24 that one may replace coherent states related to the group representation by more general abstract continuous frames in the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces a unified approach towards atomic decompositions and banach frames with new results for general coorbit spaces is presented in the second part we apply the abstract setting to a specific framework and study coorbits of what we call peetre spaces they allow to recover inhomogeneous besov lizorkin triebel spaces of var ious types of interest as coorbits we obtain several old and new wavelet characterizations based on explicit smoothness decay and vanishing moment assumptions of the respective wavelet as main examples we obtain results for weighted spaces muckenhoupt doubling general 2 microlocal spaces besov lizorkin triebel morrey spaces spaces of dominating mixed smoothness and even mixtures of the mentioned ones due to the generality of our approach there are many more examples of interest where the abstract coorbit space theory is applicable

frame transforms star products and quantum mechanics on phase space using the notions of frame transform and of square integrable projective representation of a locally compact group g we introduce a class of isometries tight frame transforms from the space of hilbert schmidt operators in the carrier hilbert space of the representation into the space of square integrable functions on the direct product group g g these transforms have remarkable properties in particular their ranges are reproducing kernel hilbert spaces endowed with a suitable star product which mimics at the level of functions the original product of operators a phase space formulation of quantum mechanics relying on the frame transforms introduced in the present paper and the link of these maps with both the wigner transform and the wavelet transform are discussed

continuous wavelet transform on a special homogeneous space we consider a semidirect product of two locally compact groups s and t with s abelian denoted by s t an action of s t on s is introduced to make s a homogeneous space of s t then we define a unitary representation from s t into the unitary group of l2 s which is our main tool for defining the continuous wavelet transform on l2 s also the main properties of the transform are discussed we prove the plancherel and inversion formulas and reproducing kernel s formula for this transform this is finally specialized to the case of the continuous wavelet transform on l2 rd

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

banach spaces of distributions having two module structures abstract after a discussion of a space of test functions and the corresponding space of distributions a family of banach spaces b b in standard situation is described these are spaces of distributions having a pointwise module structure and also a module structure with respect to convolution the main results concern relations between the different spaces associated to b established by means of well known methods from the theory of banach modules among them b0 and b the closure of the test functions in b and the weak relative completion of b respectively the latter is shown to be always a dual banach space the main diagram given in theorem 4 7 gives full information concerning inclusions between these spaces showing also a complete symmetry a great number of corresponding formulas is established how they can be applied is indicated by selected examples in particular by certain segal algebras and the ap algebras of herz various further applications are to be given elsewhere

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

some important classes of quantum measurements and their information gain the notions of ideal first kind and repeatable measurements are reformulated and their relations analyzed within a systematic theory of measurement in quantum mechanics this allows one to investigate new information theoretical characterizations of such measurements

geometric analysis of conditional independence on wiener space in this paper we study the conditional independence of sigma fields on the wiener space using the tools of the stochastic calculus of variations particular emphasis is given to the relation between the splitting of the random tangent spaces associated to the sigma fields and the conditional independence

wavelet frames and admissibility in higher dimensions this paper is concerned with the relations between discrete and continuous wavelet transforms on k dimensional euclidean space we start with the construction of continuous wavelet transforms with the help of square integrable representations of certain semidirect products thereby generalizing results of bernier and taylor we then turn to frames of l2 rk and to the question when the functions occurring in a given frame are admissible for a given continuous wavelet transform for certain frames we give a characterization which generalizes a result of daubechies to higher dimensions

gabor frames and time frequency analysis of distributions abstract this paper lays the foundation for a quantitative theory of gabor expansions f x k n c k n e 2 in x g x k in analogy to wavelet expansions of besov triebel lizorkin spaces we show that the correct class of spaces which can be characterized by the magnitude of the coefficients c k n is the class of modulation spaces to analyze the behavior of the coefficients it is necessary to invert the gabor frame operator on these spaces we show that the frame operator is invertible on modulation spaces if and only if it is invertible on l 2 and the atom g is in a suitable space of test functions a similar statement for wavelet theory is false the second part is devoted to gabor analysis on general time frequency lattices

wavelet transforms and discrete frames associated to semidirect products we consider a semidirect product g rn h and its unitary representations u of the form indg0g p0m where ind is the unitary induction p0 is in the dual group of rn g0 is the stability group of p0 and m is a unitary representation of g0 h we give sufficient conditions such that u defines a wavelet transform and a discrete frame

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

coorbit spaces and wavelet coefficient decay over general dilation groups we study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square integrable quasi regular representation on l 2 r d it turns out that these representations are integrable as well with respect to a wide variety of weights thus allowing to consistently quantify wavelet coefficient decay via coorbitspace norms we then show that these spaces always admit an atomic decomposition in terms of bandlimited schwartz wavelets we exhibit spaces of schwartz functions contained in all coorbit spaces and dense in most of them we also present an example showing that for a consistent definition of coorbit spaces the irreducibility requirement cannot be easily dispensed with we then address the question how to predict wavelet coefficient decay from vanishing moment assumptions to this end we introduce a new condition on the open dual orbit associated to a dilation group if the orbit is temperately embedded it is possible to derive rather general weighted l p q estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function these estimates have various applications they provide very explicit admissibility conditions for wavelets and integrable vectors as well as sufficient criteria for membership in coorbit spaces as a further consequence one obtains a transparent way of identifying elements of coorbit spaces with certain cosets of tempered distributions we then show that for every dilation group in dimension two the associated dual orbit is temperately embedded in particular the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces where they complement and generalize the known results

a transfer principle from wiener to poisson space and applications the aim of this work is to construct the stochastic calculus of variations on poisson space and some of its applications via the stochastic analysis on wiener space we define a new gradient operator on wiener space whose adjoint extends the poisson stochastic integral this yields a new decomposition of the ornstein uhlenbeck operator and a substructure of the standard dirichlet structure on wiener space with applications to stochastic analysis on poisson space and infinite dimensional analysis for the exponential density

coherent state lattices and square integrability of representations in this paper we discretize the continuous theory of coherent states on a general semidirect product group g v x s where v is a vector space and s subset of gl v is a semisimple connected lie group we show that it is always possible to construct a discrete frame associated with a unitary irreducible representation of g which is square integrable modulo an affine admissible section we also prove the converse result that the existence of a discrete frame associated with a unitary representation u of g indexed by a homogeneous space gamma implies the square integrability of u on gamma

squared chaotic random variables new moment inequalities with applications abstract we prove a new family of inequalities involving squares of random variables belonging to the wiener chaos associated with a given gaussian field our result provides a substantial generalization as well as a new analytical proof of an estimate by frenkel 2007 10 and also constitutes a natural real counterpart to an inequality established by arias de reyna 1998 2 in the framework of complex gaussian vectors we further show that our estimates can be used to deduce new lower bounds on homogeneous polynomials thus partially improving results by pinasco 2012 19 as well as to obtain a novel probabilistic representation of the remainder in hadamard inequality of matrix analysis

shear madness new orthonormal bases and frames using chirp functions the proportional bandwidth and constant bandwidth time frequency signal decompositions of the wavelet gabor and wilson orthonormal bases have attracted substantial interest for representing nonstationary signals however these representations are limited in that they are based on rectangular tessellations of the time frequency plane while much effort has gone into methods for designing nice wavelet and window functions for these frameworks little consideration has been given to methods for constructing orthonormal bases employing nonrectangular time frequency tilings the authors take a first step in this direction by deriving two new families of orthonormal bases and frames employing elements that shear or chirp in the time frequency plane in addition to translate and scale the new scale shear fan bases and shift shear chevron bases are obtained by operating on an existing wavelet gabor 1946 or wilson basis set with two special unitary warping transformations in addition to the theoretical benefit of broadening the class of valid time frequency plane tilings these new bases could possibly also be useful for representing certain types of signals such as chirping and dispersed signals

on the structure of independence on wiener space motives par des resultats de travaux anterieurs sur la caracterisation de l independance de variables aleatoires dans l espace de wiener les auteurs etudient ici la relation entre l independance de certaines classes de sous algebres et l orthogonalite des h derivees faibles des variables aleatoires qui engendrent ces sous algebres ou c h est l espace de wiener classique

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

sampling theorems on locally compact groups from oscillation estimates we present a general approach to derive sampling theorems on locally compact groups from oscillation estimates we focus on the l2 stability of the sampling operator by using notions from frame theory this approach yields particularly simple and transparent reconstruction procedures we then apply these methods to the discretization of discrete series representations and to paley wiener spaces on stratified lie groups

deformation of gabor systems abstract we introduce a new notion for the deformation of gabor systems such deformations are in general nonlinear and in particular include the standard jitter error and linear deformations of phase space with this new notion we prove a strong deformation result for gabor frames and gabor riesz sequences that covers the known perturbation and deformation results our proof of the deformation theorem requires a new characterization of gabor frames and gabor riesz sequences it is in the style of beurling s characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of gabor frames without inequalities from lattices to non uniform sets

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

unitary measurements of discrete quantities in quantum mechanics the pure measurements of discrete physical quantities are characterized within quantum theory of measurement and their unitary representations are given probabilistic aspects of measurements related to the so called strong correlation conditions and a probabilistic characterization of the first kind measurements are examined the problem of the objectification of the measurement result is analyzed in terms of a classical behavior of the measuring apparatus as a by product a generalization of the wigner araki yanase theorem is given

generalized metaplectic operators and the schrodinger equation with a potential in the sjostrand class it is well known that the matrix of a metaplectic operator with respect to phase space shifts is concentrated along the graph of a linear symplectic map we show that the algebra generated by metaplectic operators and by pseudodifferential operators in a sjostrand class enjoys the same decay properties we study the behavior of these generalized metaplectic operators and represent them by fourier integral operators our main result shows that the one parameter group generated by a hamiltonian operator with a potential in the sjostrand class consists of generalized metaplectic operators as a consequence the schrodinger equation preserves the phase space concentration as measured by modulation space norms

galilean wavelets coherent states of the affine galilei group we derive galilean wavelets by which we mean coherent states of the affine galilei group that is the galilei group extended by independent space and time dilations the construction follows a general method based on square integrable group representations possibly modulo a subgroup i e on a homogeneous space of the underlying group we also examine the restriction to the schrodinger subgroup which contains only dilations that leave invariant the schrodinger and the heat equations

functions of variable bandwidth via time frequency analysis tools abstract this paper is motivated by the apparent lack of a precise mathematical description of the very natural idea of variable bandwidth for a function defined on the real line different existing concepts suffer from serious shortcomings we will present a new and mathematically well justified function space model which is based on the theory of coorbit spaces in a time frequency context making use of the flexibility of this approach we define a family of hilbert spaces which can be viewed as generalised modulation spaces more precisely we define a family of banach spaces of functions with variable bandwidth vb functions by imposing a weighted mixed norm condition on the short time fourier transform of their elements punishing the contributions outside a strip of variable width described by the band width function b similar bandwidth functions define the same space with equivalent norms any good gabor frame e g with schwartz windows can be used to characterize the membership of a function in such a space moreover various classes of functions e g those obtained by a time variant filter are shown to belong to such a space in addition error estimates are given when approximating certain subclasses

some relations among classes of σ fields on wiener space in this work we study sigma fields and their tangent spaces on the wiener space which are invariant in some sense with respect to the basic operators of the malliavin calculus

global well posedness of the one dimensional cubic nonlinear schrodinger equation in almost critical spaces abstract in this paper we first introduce a new function space m h p whose norm is given by the l p sum of modulated h norms of a given function in particular when 1 2 we show that the space m h p agrees with the modulation space m 2 p r on the real line and the fourier lebesgue space f l p t on the circle we use this equivalence of the norms and the galilean symmetry to adapt the conserved quantities constructed by killip visan zhang to the modulation space and fourier lebesgue space setting by applying the scaling symmetry we then prove global well posedness of the one dimensional cubic nonlinear schrodinger equation nls in almost critical spaces more precisely we show that the cubic nls on r is globally well posed in m 2 p r for any p while the renormalized cubic nls on t is globally well posed in f l p t for any p in appendix we also establish analogous global in time bounds for the modified kdv equation mkdv in the modulation spaces on the real line and in the fourier lebesgue spaces on the circle an additional key ingredient of the proof in this case is a galilean transform which converts the mkdv to the mkdv nls equation

continuity properties of the gabor frame operator abstract for a gabor frame with small lattice it is shown that the frame operator is continuous and invertible on many banach spaces defined by smoothness and decay properties as a consequence it is shown that there exist gabor type frame dual frame pairs and tight frames with good decay and smoothness properties also it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly e g in lp and sobolev norms also explicit estimates on lattice sizes for which such results hold can be obtained

towards objectification of measurement in an orthodox but incomplete quantum mechanics the problem of objectification of measurement is investigated in orthodox quantum mechanics that is assuming the validity of the schrodinger equation and the other postulates for the composite object plus measuring instrument system it is argued that by giving physical meaning to basic entities the positivist and the realist positions must be explicitly distinguished it is shown that the objectification problem is satisfactorily solved within quantum mechanics for the positivist but not for the realist for the latter a partially satisfactory solution is obtained by assuming that the quantum mechanical description of individual systems is incomplete and by assuming further that the pointer observable has definite values prior to any measurement in any quantum state a so called beable

geometric dynamics of a harmonic oscillator arbitrary minimal uncertainty states and the smallest step 3 nilpotent lie group the paper presents a new method of geometric solution of a schrodinger equation by constructing an equivalent first order partial differential equation with a bigger number of variables the equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform the method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three lie group the group also known under many names e g quartic group we obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector in contrast it is shown that the well known fock segal bargmann transform and the heisenberg group require a specific fiducial vector to produce a geometric solution a technical aspect considered in this paper is that a certain modification of a coherent state transform is required although the irreducible representation of the group is square integrable modulo a subgroup the obtained dynamic is transverse to the homogeneous space