a q analogue of the weighted bergman space on the disk and associated second q bargmann transform abstract we provide a q analogue of the classical weighted bergman space on the complex unit disk and we give the explicit expression of the corresponding reproducing kernel function moreover we give a q analogue of the second bargmann integral transform introduced by v bargmann in 4 p 203 we show that it defines a unitary integral transform from the hilbert space on the nonnegative real half line spanned by the q laguerre polynomials with respect to the weight function x exp q 1 q x onto the considered weighted q bergman hilbert space

difference ladder operators for a harmonic schrodinger oscillator using unitary linear lattices the formalism of raising and lowering operators is developed for the difference operator analogue of a quantum harmonic oscillator which acts on functions on a discrete support the grid under consideration is a mixed version of an equidistant lattice and a q linear grid several properties of the grid are described the grids under consideration are referred to by the name unitary linear lattices the ladder difference operators are derived and compared with the continuum situation the arising spectral problems for these operators are dealt by using the theory of bilateral jacobi operators in weighted l2 z spaces eventual applications to mathematical physics and numerical schrodinger theory are briefly discussed

antenna characterization in the time domain a theory for a complete far field transmit receive system characterization of short pulse antennas is derived in the time domain the transmit receive antenna system is characterized by a set of cascaded operators which transform the source waveform and power into similar quantities at the receiving antenna terminals two such sets are defined the first one is phrased in terms of the wave type time dependent effective height operator while the second one is defined in terms of the energy type gain operator both definitions fit within a complete transmit receive system description the latter being equivalent to the frequency domain friis equation however these operators are derived entirely in the context of the time domain field equation the starting point in the time domain analysis of the effective height is the slant stack transform sst of the time dependent current distribution in a manner equivalent to the spatial fourier transform used in the frequency domain the vector autocorrelation of the transmitting effective height is then used to define the time dependent gain operator under impulsive source excitation time domain reciprocity leads to the definitions of antenna parameters under receiving conditions and the corresponding equivalent circuit the parameters defined in this way fit within a consistent transmit receive convolution operator operating on the autocorrelation of the input signal this independent time domain representation is thus similar to the frequency domain representation however unlike the conventional frequency domain circuit parameters which relate voltage and current amplitudes the time domain circuit representation is based on incident and reflected wave type constituents in addition the use of appropriate norms facilitates the transformation of our operators to stand alone figures of merits the general concepts developed are demonstrated for a short dipole antenna

formulation of probe corrected planar near field scanning in the time domain probe corrected planar near field formulas in the time domain are derived for both acoustic and electromagnetic fields so that a single set of near field measurements in the time domain yields the fields of the test antenna directly in the time domain the time domain probe corrected formulas are first derived by taking the inverse fourier transform of the corresponding frequency domain formulas and then by using a time domain expansion for the fields of the test antenna and a time domain receiving characteristic of the probe because these general formulas which involve a double integral over the scan plane and an infinite time convolution integral are rather complicated we consider a special probe whose output due to an incoming time domain plane wave is proportional to the time derivative of the field of that plane wave for this special d dot probe the probe corrected formulas simplify to give the time domain far held pattern as a double spatial integral of the time domain output of the probe over the scan plane multiplied by the angular dependence of the inverse receiving characteristic of the probe time domain reciprocity relations are derived for reciprocal probes and their time domain receiving characteristics are related to their far fields finally a time domain sampling theorem is derived and a numerical example illustrates the use of the time domain probe corrected formulas

antenna coupling and near field sampling in plane polar coordinates a probe corrected vector transmission formula and a rigorous sampling reconstruction theorem for near field antenna measurements in plane polar coordinates are derived from three fundamental theorems of antenna theory the mutual coupling function between two antennas satisfies the homogeneous wave equation a receiving antenna can be represented as a differentiator of the incident field and the mutual coupling function is virtually bandlimited the rigorous sampling equations are applied to compute the far fields of a circular aperture antenna sampled in the near field at half wavelength radial spacing

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

on some indeterminate moment problems for measures on a geometric progression abstract we consider some indeterminate moment problems which all have a discrete solution concentrated on geometric progressions of the form cq k k z or cq k k z it turns out to be possible that such a moment problem has infinitely many solutions concentrated on the geometric progression

asymptotics of the largest zeros of some orthogonal polynomials we study the asymptotics of the largest zeros of the wilson hermite and q laguerre polynomials using two distinct techniques the first is based on the coulomb fluid technique developed in a previous paper where the primary input is the weight function while the second uses the method of chain sequences which supplies inequalities for the largest zeros using the recurrence coefficients we also investigate the asymptotics of the largest zeros of the polynomials orthogonal to the weight functions for c 0 and m a positive even integer

inner products involving q differences the little q laguerre sobolev polynomials in this paper polynomials which are orthogonal with respect to the inner product p r s k 0 p qk r qk aq k aq q q q k k 0 dqp qk dqr qk aq k aq q q q k where dq is the q difference operator 0 0 q 1 and 0 aq 1 are studied for these polynomials algebraic properties and q difference equations are obtained as well as their relation with the monic little q laguerre polynomials some properties about the zeros of these polynomials are also deduced finally the relative asymptotics qn x pn x a q n on compact subsets of c 0 1 is given where qn x is the nth degree monic orthogonal polynomial with respect to the above inner product and pn x a q denotes the monic little q laguerre polynomial of degree n

planar near field scanning in the time domain 2 sampling theorems and computation schemes for part 1 see ibid vol 47 no 9 p 1280 1994 two computation schemes for calculating the far field pattern in the time domain from sampled near field data are developed and applied the sampled near field data consists of the values of the field on the scan plane measured at discrete times and at discrete points on the scan plane the first computation scheme is based on a frequency domain near field to far field formula and applies frequency domain sampling theorems to the computed frequency domain near field the second computation scheme is based on a time domain near field to far field formula and computes the time domain far field directly from the time domain near field a time domain sampling theorem is derived to determine the spacing between sample points on the scan plane the computer time for each of the two schemes is determined and numerical examples illustrate the use and the general properties of the schemes for large antennas the frequency domain computation scheme takes less time to compute the full far field than the time domain computation scheme however the time domain computation scheme is simpler more direct and easier to program it is also found that planar time domain near field antenna measurements unlike single frequency near field measurements have the capability of eliminating the error caused by the finite scan plane and thus can be applied to broadbeam antennas

theq laguerre polynomials and related moment problems abstract we study two indeterminate hamburger moment problems associated with q laguerre polynomials the coefficients in their recurrence relations are of exponential growth this completes earlier work started by d moak some of the measures that the polynomials are orthogonal with respect to are found and lead to the evaluation of certain integrals a symmetric set of orthogonal polynomials is also considered

planar near field scanning in the time domain 1 formulation time domain planar near field measurement techniques are formulated for acoustic and electromagnetic fields probe correction is ignored in that it is assumed that the probe measures the exact values of the field on the scan plane two fundamentally different approaches are used in deriving three sets of formulas that give the fields in the source free half space z z sub 0 in terms of their values on the scan plane z z sub 0 in the first approach the time domain formulas are obtained by inverse fourier transforming the corresponding frequency domain formulas in the second approach the time domain formulas are derived directly in the time domain by working with time domain green s functions

the receiving antenna as a linear differential operator application to spherical near field scanning the general receiving antenna is represented as a linear differential operator converting the incident field and its spatial derivatives at a single point in space to an output voltage the differential operator is specified explicitly in terms of the multipole coefficients of the antenna s complex receiving pattern when the linear operator representation is applied to the special probes used in spherical near field measurements a probe corrected spherical transmission formula is revealed that retains the form applicability and simplicity of the nonprobe corrected equations the new spherical transmission formula is shown to be consistent with the previous transmission formula derived from the rotational and translational addition theorems for spherical waves

asymptotics of basic bessel functions and q laguerre polynomials abstract we establish a large n complete asymptotic expansion for q laguerre polynomials and a complete asymptotic expansion for a q bessel function of large argument these expansions are needed in our study of an exactly solvable random transfer matrix model for disordered electronic systems we also give a new derivation of an asymptotic formula due to littlewood 1907

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

accelerating l1 l2 deblurring using wavelet expansions of operators abstract image deblurring is a fundamental problem in imaging usually solved with computationally intensive optimization procedures the goal of this paper is to provide new efficient strategies to reduce computing times for simple deblurring models regularized using orthogonal wavelet transforms we show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis the proposed methodology consists of three ingredients i a sparse approximation of the blur operator in wavelet bases ii a diagonal preconditioner and iii an implementation on massively parallel architectures combining the three ingredients leads to acceleration factors ranging from 4 to 250 on a typical workstation for instance a 1024 1024 image can be deblurred in 0 15 s

q laguerre and wall polynomials are related by the fourier gauss transform it is shown that the q laguerre and wall or little q laguerre polynomials are interrelated by the fourier gauss transform in the limit when the degree of these polynomials tends to infinity this integral transform provides the relation between jackson s second and third q bessel functions

near field scanning in the time domain on a spherical surface a formulation using the free space green s function two formulations for determining the characteristics of an unknown source of acoustic waves using the measurement of its field at its near zone are presented the measurement in both cases is to be performed on a spherical scan surface which encapsulates the source the first is for an ideal probe which measures the field at its location the knowledge of the field is sufficient its normal derivative is not required in the second formulation a realistic probe is considered this time it is required only that the probe has an axially symmetric receiving characteristic with this formulation the time functions which characterize the source are found using only the signal at the output of the probe both formulations are such that they are not specific to the scan surface radius furthermore they are entirely in the time domain requiring no inverse fourier transformations left to be performed

nonlinear approximation and the space bv r2 given a function 2 0 q 0 l 2 and a real number t 0 let u f t infg bv g 11 ll vgig where the infimum is taken over all functions g g bv of bounded variation on this and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation as well as in digital image processing techniques for finding minimizers g for u f t based on variational calculus and nonlinear partial differential equations have been put forward by several authors dms ro ms cl the main disadvantage of these approaches is that they are numerically intensive on the other hand it is well known that more elementary methods based on wavelet shrinkage solve related extremal problems for example the above problem with bv replaced by the besov space b l i see e g cdll however since bv has no simple description in terms of wavelet coefficients it is not clear that minimizers for u f t can be realized in this way we shall show in this paper that simple methods based on haar thresholding provide near minimizers for u f t our analysis of this extremal problem brings forward many interesting relations between haar decompositions and the space bv 1 introduction nonlinear approximation has recently played an impor tant role in several problems of image processing including compression noise removal and feature extraction we have in mind techniques such as wavelet compression djl wavelet shrinkage or thresholding djkp1 wavelet packets cw and greedy algorithms mz dt there has also been an impressive contribution of techniques based on variational calculus and nonlinear partial dif ferential equations see e g dms ro ms cl especially to the problems of noise removal and image segmentation the common point between these two approaches is their ability to adapt to the composite nature of images edge tex tures and smooth regions should be treated adaptively a requirement which is certainly not fulfilled by the classical linear filtering techniques

forward fourier wavelet regularized deconvolution for ill conditioned systems we propose an efficient hybrid fourier wavelet regularized deconvolution forward algorithm that performs noise regularization via scalar shrinkage in both the fourier and wavelet domains the fourier shrinkage exploits the fourier transform s economical representation of the colored noise inherent in deconvolution whereas the wavelet shrinkage exploits the wavelet domain s economical representation of piecewise smooth signals and images we derive the optimal balance between the amount of fourier and wavelet regularization by optimizing an approximate mean squared error mse metric and find that signals with more economical wavelet representations require less fourier shrinkage forward is applicable to all ill conditioned deconvolution problems unlike the purely wavelet based wavelet vaguelette deconvolution wvd moreover its estimate features minimal ringing unlike the purely fourier based wiener deconvolution even in problems for which the wvd was designed we prove that forward s mse decays with the optimal wvd rate as the number of samples increases further we demonstrate that over a wide range of practical sample lengths forward improves on wvd s performance

fast space varying convolution using matrix source coding with applications to camera stray light reduction many imaging applications require the implementation of space varying convolution for accurate restoration and reconstruction of images here we use the term space varying convolution to refer to linear operators whose impulse response has slow spatial variation in addition these space varying convolution operators are often dense so direct implementation of the convolution operator is typically computationally impractical one such example is the problem of stray light reduction in digital cameras which requires the implementation of a dense space varying deconvolution operator however other inverse problems such as iterative tomographic reconstruction can also depend on the implementation of dense space varying convolution while space invariant convolution can be efficiently implemented with the fast fourier transform this approach does not work for space varying operators so direct convolution is often the only option for implementing space varying convolution in this paper we develop a general approach to the efficient implementation of space varying convolution and demonstrate its use in the application of stray light reduction our approach which we call matrix source coding is based on lossy source coding of the dense space varying convolution matrix importantly by coding the transformation matrix we not only reduce the memory required to store it we also dramatically reduce the computation required to implement matrix vector products our algorithm is able to reduce computation by approximately factoring the dense space varying convolution operator into a product of sparse transforms experimental results show that our method can dramatically reduce the computation required for stray light reduction while maintaining high accuracy

a fast multilevel algorithm for wavelet regularized image restoration we present a multilevel extension of the popular ldquothresholded landweberrdquo algorithm for wavelet regularized image restoration that yields an order of magnitude speed improvement over the standard fixed scale implementation the method is generic and targeted towards large scale linear inverse problems such as 3 d deconvolution microscopy the algorithm is derived within the framework of bound optimization the key idea is to successively update the coefficients in the various wavelet channels using fixed subband adapted iteration parameters step sizes and threshold levels the optimization problem is solved efficiently via a proper chaining of basic iteration modules the higher level description of the algorithm is similar to that of a multigrid solver for pdes but there is one fundamental difference the latter iterates though a sequence of multiresolution versions of the original problem while in our case we cycle through the wavelet subspaces corresponding to the difference between successive approximations this strategy is motivated by the special structure of the problem and the preconditioning properties of the wavelet representation we establish that the solution of the restoration problem corresponds to a fixed point of our multilevel optimizer we also provide experimental evidence that the improvement in convergence rate is essentially determined by the unconstrained linear part of the algorithm irrespective of the type of wavelet finally we illustrate the technique with some image deconvolution examples including some real 3 d fluorescence microscopy data

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

deconvolution a wavelet frame approach this paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches which has already appeared in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b as wavelet frame based high resolution image reconstruction methods we first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation which completes the formulation given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b this formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties these missing coefficients are recovered iteratively together with a built in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating this approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b however an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers this paper is to establish the theoretical foundation of this wavelet frame approach in particular a proof of convergence an analysis of the stability of algorithms and a study of the minimization property of solutions are given

generalized q hermite polynomials we consider two operators a and a in a hilbert space of functions on the exponential lattice where 0 q 1 the operators are formal adjoints of each other and depend on a real parameter we show how these operators lead to an essentially unique symmetric ground state 0 and that a and a are ladder operators for the sequence the sequence n 0 is shown to be a family of orthogonal polynomials which we identify as symmetrized q laguerre polynomials we obtain in this way a new proof of the orthogonality for these polynomials when 0 the polynomials are the discrete q hermite polynomials of type ii studied in several papers on q quantum mechanics

nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes abstract we consider two classes of birth and death processes which lead to indeterminate stieltjes moment problems and we calculate the entire functions c and d from the corresponding nevanlinna matrix this permits a direct computation of a nevanlinna extremal orthogonality measure for each of the processes

indeterminate symmetric moment problems let n x be a sequence of orthogonal polynomials with respect to a real distribution according to a classical theorem of reisz 191 x x is complete in l if and only if w is either the solution of a determined hamburger moment problem or an extremal solution of an indeterminate moment problem in the special case where the zeros of the orthogonal polynomials have a finite infimum a and the hamburger moment problem is indeterminate there is an extremal solution which is essentially the only solution whose spectrum is a subset of a co 4 this solution is in a sense the best solution from the viewpoint of orthogonal polynomials since its spectrum consists of the limit points of the zeros of the polynomials askey remarked to the author that in view of riesz theorem it is surprising that so little attention has been paid to the question of extremal solutions moak 8 has recently studied the extremal solutions associated with the q laguerre nee generalized stieltjes wigert polynomials askey suggested it would be desirable to investigate the existence of best extremal solutions in the general case or as a first step even in the symmetric case moments of odd order are 0 in this paper we will consider this special case and the related systems of orthogonal polynomials in particular we will show that an indeterminate symmetric moment problem has precisely two symmetric extremal solutions this result which we found surprizing at first appears after reflection quite natural and perhaps obvious since the two solutions are the symmetrizations of the best extremal solutions of the related moment problems on 0 co

a fast thresholded landweber algorithm for wavelet regularized multidimensional deconvolution we present a fast variational deconvolution algorithm that minimizes a quadratic data term subject to a regularization on the norm of the wavelet coefficients of the solution previously available methods have essentially consisted in alternating between a landweber iteration and a wavelet domain soft thresholding operation while having the advantage of simplicity they are known to converge slowly by expressing the cost functional in a shannon wavelet basis we are able to decompose the problem into a series of subband dependent minimizations in particular this allows for larger subband dependent step sizes and threshold levels than the previous method this improves the convergence properties of the algorithm significantly we demonstrate a speed up of one order of magnitude in practical situations this makes wavelet regularized deconvolution more widely accessible even for applications with a strong limitation on computational complexity we present promising results in 3 d deconvolution microscopy where the size of typical data sets does not permit more than a few tens of iterations

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

solving constrained total variation image restoration and reconstruction problems via alternating direction methods in this paper we study alternating direction methods for solving constrained total variation image restoration and reconstruction problems alternating direction methods can be implementable variants of the classical augmented lagrangian method for optimization problems with separable structures and linear constraints the proposed framework allows us to solve problems of image restoration impulse noise removal inpainting and image cartoon texture decomposition as the constrained model is employed we need only to input the noise level and the estimation of the regularization parameter is not required in these imaging problems experimental results for such imaging problems are presented to illustrate the effectiveness of the proposed method we show that the alternating direction method is very efficient for solving image restoration and reconstruction problems