stochastic aspects of nonlinear refinement schemes this article develops a stochastic viewpoint on nonlinear refinement algorithms designed to construct continuous objects from discrete samples in a global npc nonpositive curvature space a particular focus lies on convergence properties of so called barycentric subdivision schemes a class of refinement algorithms acting on input data via weighted averages an essential observation is the characterization of these kinds of schemes as nonlinear markov semigroups exploiting this feature it is proven that convergence on arbitrary npc spaces is equivalent to convergence for real valued input data furthermore a strong law of large numbers leads to certain structure preserving properties for barycentric schemes on the space of positive definite matrices a concluding section addresses the relationship between the convergence properties of a scheme and its so called characteristic markov chain

asymptotic expansions for linear homogeneous divide and conquer recurrences algebraic and analytic approaches collated among all sequences that satisfy a divide and conquer recurrence those which are rational with respect to a numeration system are certainly the most basic and the most essential nevertheless until recently this specific class of sequences has not been systematically studied from the asymptotic standpoint we recall how a mechanical process designed by the author permits to compute their asymptotic expansions the process is based on linear algebra and involves computing jordan normal forms joint spectral radii and solving dilation equations the main contribution of the present article is the comparison between our algebraic method and the classical analytic number theory approach moreover we develop new ways to compute the fourier series of the periodic functions involved in the expansion the article comes with an extended bibliography

functions analytic on the half plane as quantum mechanical states a transform between the state space of one dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half plane is constructed it gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple generalizations to higher dimensions are also discussed

refinement type equations and grincevicjus series abstract we consider l 1 solutions of the following refinement type equations f x n z c n 1 f k x n n z c n 1 f k x n where k 2 is an integer and for all n z reals c n 1 c n 1 are non negative with n z c n 1 c n 1 k and n z log n c n 1 c n 1 necessary and sufficient conditions for the existence of non trivial l 1 solutions in several special cases are given

parabolic littlewood paley inequality for ϕ δ type operators and applications to stochastic integro differential equations abstract in this paper we prove a parabolic version of the littlewood paley inequality 1 4 for the operators of the type where is a bernstein function as an application we construct an l p theory for the stochastic integro differential equations of the type d u u f d t g d w t

sets of nonnegative matrices without positive products abstract for an arbitrary irreducible set of nonnegative d d matrices we consider the following problem does there exist a strictly positive product with repetitions permitted of those matrices under some general assumptions we prove that if it does not exist then there is a partition of the set of basis vectors of r d on which all given matrices act as permutations moreover there always exists a unique maximal partition with the maximal number of parts possessing this property and the number of parts is expressed by eigenvalues of matrices this generalizes well known results of perron frobenius theory on primitivity of one matrix to families of matrices we present a polynomial algorithm to decide the existence of a positive product for a given finite set of matrices and to build the maximal partition similar results are obtained for scrambling products applications to the study of lyapunov exponents inhomogeneous markov chains etc are discussed

matrix semigroups with constant spectral radius abstract multiplicative matrix semigroups with constant spectral radius c s r are studied and applied to several problems of algebra combinatorics functional equations and dynamical systems we show that all such semigroups are characterized by means of irreducible ones each irreducible c s r semigroup defines walks on euclidean sphere all its nonsingular elements are similar in the same basis to orthogonal we classify all nonnegative c s r semigroups and arbitrary low dimensional semigroups for higher dimensions we describe five classes and leave an open problem on completeness of that list the problem of algorithmic recognition of c s r property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones

computational approach to solvability of refinement equations the solvability and fredholm properties of refinement equations in spaces of square integrable functions are studied necessary and jointly necessary and sufficient conditions for the solvability of homogeneous and non homogeneous refinement equations are established it is shown that in the space l 2 r the kernel space of any homogeneous equation with a non trivial solution is infinite dimensional moreover the solvability problem is reduced to the study of singular values of certain matrix sequences these sequences arise from galerkin approximations of auxiliary linear operators the corresponding constructions use only the coefficients of refinement equations that generate multiresolution analysis and the coefficients of the refinement equation studied for the equations with polynomial symbols the most complete results are obtained if the corresponding operator is considered on an appropriate subspace of the space l 2

exact interpolation and iterative subdivision schemes we examine the circumstances under which a discrete time signal can be exactly interpolated given only every mth sample after pointing out the connection between designing an m fold interpolator and the construction of an m channel perfect reconstruction filter bank we derive necessary and sufficient conditions on the signal under which exact interpolation is possible bandlimited signals are one obvious example but numerous others exist we examine these and show how the interpolators may be constructed a main application is to iterative interpolation schemes used for the efficient generation of smooth curves we show that conventional bandlimited interpolators are not suitable in this context a better criterion is to use interpolators that are exact for polynomial functions we demonstrate that these interpolators converge when iterated and show how these may be designed for any polynomial degree n and any interpolation factor m this makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium

on l1 solutions of a two direction refinement equation abstract in this paper we examine l 1 solutions of the following two direction refinement equation f x n n n c n 1 f k x n n n n c n 1 f k x n we prove that the vector space of all l 1 solutions of the above equation is at most one dimensional and consists of compactly supported functions of constant sign we also show that in many interesting cases any l 1 solution of the two direction refinement equation is either positive or negative on its support next we present sufficient conditions easy for verification for the existence of nontrivial l 1 solutions of the two direction refinement equation as well as for the nonexistence of such solutions

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

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

the characterization of continuous four coefficient scaling functions and wavelets four coefficient dilation equations are examined and results converse to a theorem of daubechies lagarias are given these results complete the characterization of those four coefficient dilation equations having a continuous solution

chaos games for wavelet analysis and wavelet synthesis abstract in this paper we consider chaos gamesto generate wavelet decompositions and wavelet synthesis of functions the primary technique involved is mixing several chaos games for the individual wavelets in order to get the overall wavelet expansion generalizations to dilation factors other than two and to multidimensional wavelets are given

dilation equations and markov operators abstract using the theory of markov operators we give a new proof of the known fact saying that for every positive integers n and k 1 and for every nonnegative reals c 0 c n satisfying the first sum rule the dilation equation f x n 0 n c n f k x n has a nontrivial l 1 solution

bmo and morrey campanato estimates for stochastic convolutions and schauder estimates for stochastic parabolic equations abstract in this paper we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t x 2 u t x d t g t x d t u 0 0 t 0 t x g for a random field u t x 0 t g u t x u t x r where 2 2 0 2 is the fractional laplacian t 0 is arbitrarily fixed g r d is a bounded domain g 0 t g r is a joint measurable coefficient and t t 0 is either a brownian motion or a levy process on a given filtered probability space f p f t t 0 t to this end we derive the bmo estimates and morrey campanato estimates respectively for stochastic singular integral operators arising from the equations concerned then by utilizing the embedding theory between the campanato space and the holder space we establish the controllability of the norm of the space c 2 d where 0 d 0 t g with all these in hand we are able to show that the q th order bmo quasi norm of the q 0 order derivative of the solution u is controlled by the norm of g under the condition that t is a levy process finally we derive the schauder estimate for the p moments of the solution of the above stochastic fractional heat equations driven by levy noise

the optimal coefficients in daubechies wavelets abstract the construction of wavelets begins with a dilation equation x ck 2x k the choice of the coefficients ck controls the accuracy and orthogonality of wavelet expansions we show how the daubechies coefficients which give pth order accuracy with 2p coefficients the minimum number follow from orthogonality and accuracy conditions applied to the polynomial ckeik

parabolic bmo estimates for pseudo differential operators of arbitrary order abstract in this article we prove the bmo l estimate 2 u bmo r d 1 n t u a t u l r d 1 u c c r d 1 for a wide class of pseudo differential operators a t of order 0 the coefficients of a t are assumed to be merely measurable in time variable as an application to the equation t u a t u f t r we prove that for any u c c r d 1 u t l p r d 1 2 u l p r d 1 n u t a t u l p r d 1 where p 1 and the constant n is independent of u

on the least asymmetric wavelets the asymmetry of daubechies 1988 1992 scaling functions and wavelets can be diminished by minimizing a special second moment in time for the wavelet generating discrete time filter the moment is involved in an uncertainty relation for discrete time signals other measures of asymmetry are addressed as well and corresponding results are compared

vector subdivision schemes and multiple wavelets we consider solutions of a system of refinement equations written in the form formula math where the vector of functions 1 r t is in l p r r and a is a finitely supported sequence of r r matrices called the refinement mask associated with the mask a is a linear operator q a defined on l p r r by q a f z a f 2 this paper is concerned with the convergence of the subdivision scheme associated with a i e the convergence of the sequence q a n f n 1 2 in the l p norm our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask along the way properties of the joint spectral radius and its relation to the subdivision scheme are discussed in particular the l 2 convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace we analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations finally the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions this leads us to construct a class of continuous orthogonal double wavelets with symmetry

continuous and differentiable multidimensional iterative interpolation abstract we describe a multidimensional iterative interpolation process some examples of the two dimensional process are presented we give a sufficient condition for the continuity and the differentiability of the interpolation functions all the examples are continuous and differentiable

extremal lp norms of linear operators and self similar functions abstract we prove that for any p 1 a finite irreducible family of linear operators possesses an extremal norm corresponding to the p radius of these operators as a corollary we derive a criterion for the l p contractibility property of linear operators and estimate the asymptotic growth of orbits for any point these results are applied to the study of functional difference equations with linear contractions of the argument self similarity equations we obtain a sharp criterion for the existence and uniqueness of solutions in various functional spaces compute the exponents of regularity and estimate moduli of continuity this in particular gives a geometric interpretation of the p radius in terms of spectral radii of certain operators in the space l p 0 1

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

estimates for the joint spectral radius estimates for the computation of the joint spectral radius defined by a bounded collection of square matrices and of same size will be given in particular the computational result shows that for matrices constructed by four coefficient two scale dilation equation our approach leads to the exact value of this joint spectral radius

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

compactly supported solutions of two scale difference equations abstract we consider lebesgue integrable compactly supported solutions of two scale difference equations and investigate the relations between translates of these solutions a detailed study of corresponding invariant subspaces leads to new observations concerning the factorization of the refinement mask and certain spectral properties of corresponding coefficient matrices in particular new necessary conditions for the existence of integrable compactly supported solutions are derived

random affine iterated function systems curve generation and wavelets it is shown how refinement methods for smooth curve generation can be carried out efficiently through iterated function systems ifs affine transformations are constructed so that when composed randomly they generate the desired smooth curve underlying this random algorithm is the tree traversal property of ifs under a refinement method the points on the curve correspond to leaves on some n tree ifs theory enables one to generate all of these leaves through a single orbit of an appropriate markov chain applications include bezier curves splines wavelets and various interpolants

characterization of joint spectral radius via trace the joint spectral radius for a bounded collection of the square matrices with complex entries and of the same size is characterized by the trace of matrices this characterization allows us to give some estimates concerning the computation of the joint spectral radius

then metric completion of regular rings we have two main objectives in this paper let r be a unit regular ring the standard reference is g1 r admits a pseudo metric topology and the completion denoted s is called the n completion of r we study the relations between r and s complementary to a recent article of goodearl on n complete regular rings the rings r s are representable as rings of sections of a sheaf like object and we clarify and extend this notion the collection of pseudo rank functions of r terms not defined here will be found in g1 denoted ip r is a compact convex set with extreme boundary denoted either oelp r or e we define n r 0 1 via n r sup p r lpslp r then n induces a pseudo metric topology on r by means of d r s n r s this is a metric precisely when n r 0 implies r 0 r is n torsion free in gh2 p 208 r has a hausdorff family of pseudorank functions we shall consider only such rings chapter 1 deals with properties of the n completion of a general unit regular ring r called s our main result here which takes the bulk of the chapter to establish is that s possesses no new pseudo rank functions i e the natural map ip s ip r is an affine homeomorphism this was claimed in h1 proposition 15 but the injectivity part of the proof has a gap it follows that s is complete in its intrinsic n metric and so the recent results of goodearl i g2 apply to s when the appropriate identifications are made the sup norm completion of the image of ko r in afflp r the affine continuous functions on ip r is just ko s some consequences of these results include i if k is a metrizable choquet simplex there is an n complete regular ring s such that ko s is order isomorphic to aft k ii if g u is an unperforated interpolation group with order unit see ehs then the sup norm closure of its image in its natural representation as a group of affine functions on a simplex is an interpolation group

joint spectral radius dilation equations and asymptotic behavior of radix rational sequences abstract radix rational sequences are solutions of systems of recurrence equations based on the radix representation of the index for each radix rational sequence with complex values we provide an asymptotic expansion essentially in the scale n log l n the precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence of first order differences the coefficients are holderian functions obtained through some dilation equations which are usual in the domains of wavelets and refinement schemes the proofs are ultimately based on elementary linear algebra

c1 continuity of the generalized four point scheme abstract we show that the interpolatory four point scheme with tension parameter generates c 1 limit curves if and only if 0 where 0 19273 is the unique real solution of the cubic equation 32 3 4 1 0

quantum field theory in ninety minutes on presente divers aspects de la theorie quantique des champs la theorie p 2 a 2 dimensions la theorie 3 4 a 3 dimensions la theorie de yang mills a 4 dimensions

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

parabolic littlewood paley inequality for a class of time dependent pseudo differential operators of arbitrary order and applications to high order stochastic pde in this paper we prove a parabolic version of the littlewood paley inequality for a class of time dependent local and non local operators of arbitrary order and as an application we show that this inequality gives a fundamental estimate for an lp theory of high order stochastic partial differential equations

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

phase space localization theorem for ondelettes it is proven that an orthonormal basis of ondelettes can never have exponential localization in both position space and momentum space

ondelettes and phase cell cluster expansions a vindication y meyer has recently developed a particularly useful o n basis forl2 r d expansions using this basis i e expansions into ondelettes or wavelets have yielded important new results in soft and hard analysis the expansion into ondelettes of a boson scalar field naturally leads to phase cell cluster expansions a formalism already developed by the authors using other related bases adoption of ondelettes expansions into the phase cell program gives improvements of some extant results and excises an early error

metric completions of ordered groups and k0 of exchange rings we give a description of the closure of the natural affine continuous function representation of k0 r for any exchange ring r this goal is achieved by extending the results of goodearl and handelman about metric completions of dimension groups to a more general class of pre ordered groups which includes k0 of exchange rings as a consequence the results about k 0 of regular rings which the author gave in an earlier paper can be extended to a wider class of rings which includes c algebras of real rank zero among others also the framework of pre ordered groups developed here allows other potential applications

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

metrically complete regular rings this paper is concerned with the structure of those von neumann regular rings r which are complete with respect to the weakest metric derived from the pseudo rank functions on r known as the n metric it is proved that this class of regular rings includes all regular rings with bounded index of nilpotence and all 80 continuous regular rings the major tool of the investigation is the partially ordered grothendieck group ko r which is proved to be an archimedean normcomplete interpolation group such a group has a precise representation as affine continuous functions on a choquet simplex from earlier work of the author and d e handelman and additional aspects of its structure are derived here these results are then translated into ring theoretic results about the structure of r for instance it is proved that the simple homomorphic images of r are right and left self injective rings and r is a subdirect product of these simple self injective rings also the isomorphism classes of the finitely generated projective r modules are determined by the isomorphism classes modulo the maximal two sided ideals of r as another example of the results derived it is proved that if all simple artinian homomorphic images of r are n x n matrix rings for some fixed positive integer n then r is an n x n matrix ring all rings in this paper are associative with 1 and all modules are unital right modules for the overall theory of regular rings we refer the reader to 2 for the general development of ko of regular rings as partially ordered abelian groups and the theory of partially ordered abelian groups via their state spaces we refer the reader to 2 4 in particular these references should be consulted for more detail on definitions and concepts which are just sketched here i n completeness completeness of a regular ring with respect to a rank function or with respect to a family of pseudo rank functions implies that the ring is right and left self injective 2 theorems 19 7 and 20 8 hence a considerable amount of structure theory is available for such rings 2 chapters 9 12 the purpose of this paper is to investigate a much broader class of regular rings namely those which are complete with respect to the pseudo metric obtained from the supremum n of all pseudo rank functions on the ring in particular all regular rings complete with respect to a family of pseudo rank functions are n complete but also as we prove later in this section all regular rings with bounded index of nilpotence and all 8 0 continuous regular rings are n complete received by the editors january 13 1981 and in revised form april 3 1981 1980 mathematics subject classification primary 16a30 16a54 06f20

joint spectral radius and holder regularity of wavelets abstract we give preliminary results on the holder exponent of wavelets of compact support in particular we give a nearly complete map of this exponent for the family of four coefficient multiresolution analyses and determine the smoothest one this will resolve two conjectures by colella and heil wavelets of compact support can be generated via infinite products of certain matrices the rate of growth of these products determines the regularity of the wavelet this rate can be determined via joint generalized or common spectral radius of the given set of matrices we outline a method for calculating this radius for a given set of matrices the method relies on guessing the particular finite optimal product which satisfies the finiteness conjecture and exhibits the fastest growth then we generate an optimal unit ball by taking the convex hull of the action of the semigroup of matrices scaled by their joint radius on the invariant ball of the scaled optimal product if this process terminates in a finite number of steps and the convex hull does not grow then the guessed optimal product is confirmed and the joint radius is determined

holder continuous of the solutions to stochastic nonlocal heat equations abstract in this paper we aim to obtain the holder continuous of solutions to stochastic nonlocal heat equations by using campanato estimates and sobolev embedding theorem we first prove the holder continuous of the mild solution of stochastic nonlocal diffusion equations locally in r d in the sense that the solution u belongs to the space c d t l p then by using tail estimates we obtain the estimates of the mild solution in l p c d t lastly we prove the holder continuous of the mild solution on bounded domain

wavelet transform based adaptive filters analysis and new results in this paper the wavelet transform is used in an adaptive filtering structure the coefficients of the adaptive filter are updated by the help of the least mean square lms algorithm first the wavelet transform based adaptive filter wtaf is described and it is analyzed for its wiener optimal solution then the performance of the wtaf is studied by the help of learning curves for three different convergence factors 1 constant convergence factor 2 time varying convergence factor and 3 exponentially weighted convergence factor the exponentially weighted convergence factor is proposed to introduce scale based variation to the weight update equation it is shown for two different sets of data that the rate of convergence increases significantly for all three wtaf structures as compared to that of time domain lms the high convergence rates of the wtaf give us reason to expect that it will perform well in tracking rapid changes in a signal

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

optimal norms and the computation of joint spectral radius of matrices the notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix the finiteness conjecture fc claims that among the infinite products made from the elements of a given finite set of matrices there is a certain periodic product made from the repetition of the optimal product whose rate of growth is maximal fc has been disproved in this paper it is conjectured that fc is almost always true and an algorithm is presented to verify the optimality of a given product the algorithm uses optimal norms as a special subset of extremal norms several conjectures related to optimal norms and non decomposable sets of matrices are presented the algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi resolution analyses and wavelets the results of related numerical experiments are presented

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

a generalization of the littlewood paley inequality for the fractional laplacian δ α 2 abstract we prove a parabolic version of the littlewood paley inequality for the fractional laplacian 2 where 0 2