quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions this paper deals with typical problems that arise when using wavelets in numerical analysis applications the first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions several types of quadratures are discussed and compared for different classes of wavelets since their construction using monomials is ill conditioned also a modified well conditioned construction using chebyshev polynomials is presented the second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions they are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm this is illustrated with numerical examples

gaussian quadrature formulae of the third kind for cauchy principal value integrals basic properties and error estimates abstract let n 1 f be the polynomial of degree n 1 interpolating the function f at the points x1 x2 xn with pn xi 0 i e at the nodes of the classical gaussian quadrature formula for the numerical approximation of the cauchy principal value integral 1 1 x x 1 d x with 1 1 and f c1 1 1 we present the quadrature formula qn 1g3 given by q n 1 g3 1 n 1 x n 1 x d x 1 n 1 1 we show that this quadrature formula does not have the disadvantages of the other two well known quadrature formulae based on the same set of nodes in particular we prove that the sequence based on the same set of nodes in particular we prove that the sequence qn 1g3 f converges to the true value of the integral uniformly for all 1 1 we give estimates for the error term furthermore we state some relations connecting the present quadrature formula to the previously introduced formulae

a wavelet or lifting scheme based imputation method summary the paper proposes a new approach to imputation using the expected sparse representation of a surface in a wavelet or lifting scheme basis our method incorporates a bayesian mixture prior for these wavelet coefficients into a gibbs sampler to generate a complete posterior distribution for the variable of interest intuitively the estimator operates by borrowing strength from those observed neighbouring values to impute at the unobserved sites we demonstrate the strong performance of our estimator in both one and two dimensional imputation problems where we also compare its application with the standard imputation techniques of kriging and thin plate splines

markov chain algorithms for planar lattice structures consider the following markov chain whose states are all domino tilings of a 2n 2n chessboard starting from some arbitrary tiling pick a 2 2 window uniformly at random if the four squares appearing in this window are covered by two parallel dominoes rotate the dominoes 90 rm o in place repeat many times this process is used in practice to generate a random tiling and is a widely used tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics analogous markov chains are used to randomly generate other structures on various two dimensional lattices this paper presents techniques which prove for the first time that in many interesting cases a small number of random moves suffice to obtain a uniform distribution

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

asymptotic error expansion of wavelet approximations of smooth functions ii we generalize earlier results concerning an asymptotic error expansion of wavelet approximations the properties of the monowavelets which are the building blocks for the error expansion are studied in more detail and connections between spline wavelets and euler and bernoulli polynomials are pointed out the expansion is used to compare the error for different wavelet families we prove that the leading terms of the expansion only depend on the multiresolution subspaces v j and not on how the complementary subspaces w j are chosen consequently for a fixed set of subspaces v j the leading terms do not depend on the fact whether the wavelets are orthogonal or not we also show that daubechies orthogonal wavelets need in general one level more than spline wavelets to obtain an approximation with a prescribed accuracy these results are illustrated with numerical examples

iterated differentiation followed by composition from bloch type spaces to weighted bmoa spaces abstract let be a holomorphic self map of the open unit disk d and h d the class of holomorphic functions on d for n n 0 define an operator by c d n f f n f h d we characterize the boundedness and compactness of operator c d n from bloch type spaces to weighted bmoa spaces

on a product type operator from bloch spaces to weighted type spaces on the unit ball we characterize the boundedness and compactness of a product of radial derivative composition and multiplication operators from bloch spaces to weighted type spaces on the unit ball

on operator p g φ from the logarithmic bloch type space to the mixed norm space on the unit ball the boundedness and compactness of the following recently introduced by this author integral type operatorp f gf z email protected 0 1f f tz g tz dtt email protected b where f is a holomorphic self map of the unit ball b in c n and g is a holomorphic function on b such that g 0 0 from logarithmic bloch type spaces to mixed norm spaces on b are studied in this paper

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

composition operators from the hardy space to the nth weighted type space on the unit disk and the half plane the boundedness of the composition operator c ff z f f z from the hardy space h p x p 0 where x is the upper half plane p z c imz 0 or the unit disk d z c z 1 in the complex plane c to the nth weighted type space where f is an analytic self map of x is characterized

hitting matrix and domino tiling with diagonal impurities as a continuation to our previous work nakano and sadahiro in fundam inform 117 249 264 2012 nakano and sadahiro in j stat phys 139 4 565 597 2010 we consider the domino tiling problem with impurities 1 if we have more than two impurities on the boundary we can compute the number of corresponding perfect matchings by using the hitting matrix method fomin in trans am math soc 353 9 3563 3583 2001 2 we have an alternative proof of the main result in nakano and sadahiro fundam inform 117 249 264 2012 and result in 1 above using the formula by kenyon and wilson trans am math soc 363 3 1325 1364 2011 electron j comb 16 1 112 2009 of counting the number of groves on circular planar graphs 3 we study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity as well as the scaling limit of those

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

compact composition operators on the bloch space necessary and sufficient conditions are given for a composition operator cof f o q to be compact on the bloch space w and on the little bloch space o weakly compact composition operators on o are shown to be compact if q e o is a conformal mapping of the unit disk d into itself whose image q d approaches the unit circle t only in a finite number of nontangential cusps then co is compact on 7o on the other hand if there is a point of t n q d at which q d does not have a cusp then co is not compact

products of volterra type operator and composition operator from h and bloch spaces to zygmund spaces the boundedness and compactness of the products of volterra type operators and composition operators from the space of bounded analytic functions and the bloch space to the zygmund space are discussed in this paper

products of integral type operators and composition operators between bloch type spaces a complete picture on the boundedness and compactness of the products of integral type operators and composition operators between bloch type spaces of holomorphic functions on the unit disk is given in this paper

products of composition and differentiation operators from zygmund spaces to bloch spaces and bers spaces abstract the boundedness and compactness of the products of differentiation and composition operators from zygmund spaces to bloch spaces and bers spaces are discussed in this paper

generalized composition operators on zygmund spaces and bloch type spaces the boundedness and compactness of the generalized composition operator on zygmund spaces and bloch type spaces are investigated in this paper

on a new integral type operator from the bloch space to bloch type spaces on the unit ball abstract we introduce the following integral type operator on the space h b of all holomorphic functions on the unit ball b c n p g f z 0 1 f t z g t z d t t z b where g h b g 0 0 and is a holomorphic self map of b the boundedness and compactness of the operator from the bloch space b or the little bloch space b 0 to the bloch type space b or the little bloch type space b 0 are characterized in the main results we calculate the essential norm of the operators p g b or b 0 b or b 0 in an elegant way

weighted differentiation composition operators from h and bloch spaces to nth weighted type spaces on the unit disk we characterize the boundedness and compactness of weighted differentiation composition operators from the space of bounded analytic functions the bloch space and the little bloch space to nth weighted type spaces on the unit disk

a collocation scheme for a certain cauchy singular integral equation based on the superconvergence analysis in this paper we investigate the composite midpoint rule for the evaluation of cauchy principal value integral in an interval and place the key point on its pointwise superconvergence phenomenon the error expansion of the rule is obtained which shows that the superconvergence phenomenon occurs at the points of each subinterval whose local coordinate is the zeros of some function then by applying the midpoint rule to approximate the cauchy principal value integral and choosing the superconvergence points as the collocation points we obtain a collocation scheme for solving a certain cauchy singular integral equation the more interesting thing is that the coefficient matrix of the resulting linear system possesses some good properties from which we obtain an optimal error estimate finally some numerical examples are provided to validate the theoretical analysis

on an integral type operator from logarithmic bloch type spaces to mixed norm spaces on the unit ball let h b denote the space of all holomorphic functions on the open unit ball b of c n and email protected h b we characterize the boundedness and compactness of the following integral type operator l gf z email protected 0 1rf tz g tz dtt email protected b where rf is the radial derivative of function f from logarithmic bloch type spaces to mixed norm spaces on the unit ball

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

weighted radial operator from the mixed norm space to the nth weighted type space on the unit ball abstract the boundedness and compactness of the weighted radial operator from the mixed norm space to the n th weighted type space on the unit ball are characterized

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

characterizations of composition followed by differentiation between bloch type spaces abstract some results on the boundedness and compactness of composition followed by differentiation between bloch type spaces are refined in this paper

gauss type quadrature rule with complex nodes and weights for integrals involving daubechies scale functions and wavelets this paper deals with derivation of a gauss type quadrature rule named as gauss daubechies quadrature rule for numerical evaluation of integrals involving product of integrable function and daubechies scale functions wavelets some of the nodes and weights of the quadrature formula may be complex and appear with their conjugates this is in contrast with the standard gauss type quadrature rule for integrals involving products of integrable functions and non negative weight functions this quadrature rule has accuracy as good as the standard gauss type quadrature rule and is also found to be stable the efficacy of the quadrature rule derived here has been tested through some numerical examples

solution of second kind integral equation with cauchy type kernel using daubechies scale function a method based on the daubechies scale function has been devised to solve numerically a class of second kind integral equations with a cauchy type kernel the convergence associated to the method has been established the method is illustrated with integral equations in this class whose exact solutions are known the numerical results obtained by the present method are seen to differ negligibly from the exact results

dynamic multiscale spatiotemporal models for gaussian areal data summary we introduce a new class of dynamic multiscale models for spatiotemporal processes arising from gaussian areal data specifically we use nested geographical structures to decompose the original process into multiscale coefficients which evolve through time following state space equations our approach naturally accommodates data that are observed on irregular grids as well as heteroscedasticity moreover we propose a multiscale spatiotemporal clustering algorithm that facilitates estimation of the nested geographical multiscale structure in addition we present a singular forward filter backward sampler for efficient bayesian estimation our multiscale spatiotemporal methodology decomposes large data analysis problems into many smaller components and thus leads to scalable and highly efficient computational procedures finally we illustrate the utility and flexibility of our dynamic multiscale framework through two spatiotemporal applications the first example considers mortality ratios in the state of missouri whereas the second example examines agricultural production in espirito santo state brazil

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

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

generalized composition operators from f p q s spaces to bloch type spaces abstract let h b denote the space of all holomorphic functions on the unit ball b of c n let be a holomorphic self map of b and g h b such that g 0 0 in this paper we investigate the boundedness and compactness of the generalized composition operator c g f z 0 1 r f tz g tz dt t which map from f p q s and f0 p q s to bloch type space b in the unit ball

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

refinement of operator valued reproducing kernels this paper studies the construction of a refinement kernel for a given operator valued reproducing kernel such that the vector valued reproducing kernel hilbert space of the refinement kernel contains that of the given kernel as a subspace the study is motivated from the need of updating the current operator valued reproducing kernel in multi task learning when underfitting or overfitting occurs numerical simulations confirm that the established refinement kernel method is able to meet this need various characterizations are provided based on feature maps and vector valued integral representations of operator valued reproducing kernels concrete examples of refining translation invariant and finite hilbert schmidt operator valued reproducing kernels are provided other examples include refinement of hessian of scalar valued translation invariant kernels and transformation kernels existence and properties of operator valued reproducing kernels preserved during the refinement process are also investigated

on some integral type operators between a general space and bloch type spaces abstract the boundedness and compactness of the next integral operators p g f z 0 1 f tz g tz dt t and l g f z 0 1 r f tz g tz dt t between f p q s spaces and bloch type spaces on the unit ball of c n are studied among other results we solve some open problems and considerably extend results in the paper li and stevic compactness of riemann stieltjes operators between f p q s and bloch spaces publ math debrecen 72 1 2 2008 111 128

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

gauss quadrature rules for cauchy principal value integrals with wavelets gauss quadrature rules for cauchy principal value integrals with wavelets are established by the construction of orthogonal polynomials of wavelets and scaling functions as weight functions the wavelet gauss quadrature formulae for cauchy principal value integrals are introduced numerical examples are also presented

a bijection theorem for domino tilings with diagonal impurities we consider the dimer problem on a planar non bipartite graph g where there are two types of dimers one of which we regard as impurities computer simulations reveal a reminiscence of the cheerios effect that is impurities are attracted to the boundary which is the motivation to study this particular graph our main theorem is a variant of the temperley bijection a bijection between the set of dimer coverings and the set of spanning forests with certain conditions we further discuss some implications of this theorem 1 the local move connectedness yielding an ergodic markov chain on the set of all possible dimer coverings and 2 a rough bound for the number of dimer coverings and that for the probability of finding an impurity at a given edge which is an extension of a result in nakano and sadahiro in arxiv 0901 4824

a note on one point quadrature formula for daubechies scale function with partial support abstract this note is concerned with an efficient computation of integrals of products of a smooth function and daubechies scale function with partial support by using a one point quadrature rule the error estimate is obtained the rule is illustrated by considering an example from the literature

products of composition multiplication and radial derivative operators from logarithmic bloch spaces to weighted type spaces on the unit ball abstract let h b denote the space of all holomorphic functions on the unit ball b of c n h b and be a holomorphic self map of b let c m and r denote the composition multiplication and radial derivative operators respectively in this paper we characterize the boundedness and compactness of linear operators induced by products of these operators from logarithmic bloch spaces to weighted type spaces on the unit ball

products of multiplication composition and differentiation operators on weighted bergman spaces abstract let be a holomorphic function on the open unit disk d and a holomorphic self map of d let c m and d denote the composition multiplication and differentiation operator respectively we consider linear operators induced by products of these operators on weighted bergman spaces on d the boundedness is established by using carleson type measures