properties of the solutions of the fourth order bessel type differential equation the structured bessel type functions of arbitrary even order were introduced by everitt and markett in 1994 these functions satisfy linear ordinary differential equations of the same even order the differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity they are all natural extensions of the classical second order bessel differential equation further these differential equations have real valued coefficients on the positive real half line of the plane and can be written in lagrange symmetric formally self adjoint form in the fourth order case the lagrange symmetric differential expression generates self adjoint unbounded operators in certain hilbert function spaces these results are recorded in many of the papers here given as references it is shown in the original paper of 1994 that in this fourth order case one solution exists which can be represented in terms of the classical bessel functions of order 0 and 1 the existence of this solution further aided by computer programs in maple led to the existence of a linearly independent basis of solutions of the differential equation in this paper a new proof of the existence of this solution base is given on using the advanced theory of special functions in the complex plane the methods lead to the development of analytical properties of these solutions in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane

vector quantizers with direct sum codebooks the use of direct sum codebooks to minimize the memory requirements of vector quantizers is investigated assuming arbitrary fixed partitions necessary conditions for minimum distortion codebooks are derived first for scalar codebooks assuming mean squared error distortion and then for vector codebooks and a broader class of distortion measures an iterative procedure is described for designing locally optimal direct sum codebooks both optimal and computationally efficient suboptimal encoding schemes are considered it is shown that although an optimal encoding can be implemented by a sequential encoder the complexity of implementing optimal stagewise partitions generally exceeds the complexity of an exhaustive search of the direct sum codebook it is also shown that sequential nearest neighbor encoders can be extremely inefficient the m search method is explored as one method of improving the effectiveness of suboptimal sequential encoders representative results for simulated direct sum quantizers are presented

advances in residual vector quantization a review advances in residual vector quantization rvq are surveyed definitions of joint encoder optimality and joint decoder optimality are discussed design techniques for rvqs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended fixed rate rvqs and variable rate rvqs that employ entropy coding are examined predictive and finite state rvqs designed and integrated into neural network based source coding structures are revisited successive approximation rvqs that achieve embedded and refinable coding are reviewed a new type of successive approximation rvq that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system

variable length constrained storage tree structured vector quantization constrained storage vector quantization csvq introduced by chan and gersho 1990 1991 allows for the stagewise design of balanced tree structured residual vector quantization codebooks with low encoding and storage complexities on the other hand it has been established by makhoul et al 1985 riskin et al 1991 and by mahesh et al see ieee trans inform theory vol 41 p 917 30 1995 that variable length tree structured vector quantizer vltsvq yields better coding performance than a balanced tree structured vector quantizer and may even outperform a full search vector quantizer due to the nonuniform distribution of rate among the subsets of its input space the variable length constrained storage tree structured vector quantization vlcs tsvq algorithm presented in this paper utilizes the codebook sharing by multiple vector sources concept as in csvq to greedily grow an unbalanced tree structured residual vector quantizer with constrained storage it is demonstrated by simulations on test sets from various synthetic one dimensional 1 d sources and real world images that the performance of vlcs tsvq whose codebook storage complexity varies linearly with rate can come very close to the performance of greedy growth vltsvq of riskin et al and mahesh et al the dramatically reduced size of the overall codebook allows the transmission of the code vector probabilities as side information for source adaptive entropy coding

alphabet constrained vector quantization alphabet constrained rate distortion theory is extended to coding of sources with memory two different cases are considered when only the size of the codebook is constrained and when the codevector values are also held fixed for both cases nth order constrained alphabet rate distortion functions are defined and a convergent algorithm for their evaluation is presented specific simulations using ar 1 sources show that performance near the rate distortion bound is possible using a reproduction alphabet consisting of a small number of codevectors it is also shown that the additional constraint of holding the codevector values fixed does not degrade performance of the coder in relation to the size only constrained case this observation motivates the development of a fixed codebook vector quantizer called the alphabet and entropy constrained vector quantizer the performance of which is comparable to the entropy constrained vector quantizer a number of examples using an ar 1 and a speech source are presented to corroborate the theory

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

successive refinement of information characterization of the achievable rates let r spl middot be the rate distortion function assume that we want to describe a source with distortion no larger than spl delta sub 1 from the rate distortion theory we know that we need to do so at a rate r sub 1 no smaller than r spl delta sub 1 spl lsqb bits symbol spl rsqb if it turns out that a more accurate description at distortion spl delta sub 2 spl delta sub 2

adaptive entropy coded subband coding of images the authors describe a design approach called 2 d entropy constrained subband coding ecsbc based upon recently developed 2 d entropy constrained vector quantization ecvq schemes the output indexes of the embedded quantizers are further compressed by use of noiseless entropy coding schemes such as huffman or arithmetic codes resulting in variable rate outputs depending upon the specific configurations of the ecvq and the ecpvq over the subbands many different types of sbc schemes can be derived within the generic 2 d ecsbc framework among these the authors concentrate on three representative types of 2 d ecsbc schemes and provide relative performance evaluations they also describe an adaptive buffer instrumented version of 2 d ecsbc called 2 d ecsbc aec for use with fixed rate channels which completely eliminates buffer overflow underflow problems this adaptive scheme achieves performance quite close to the corresponding ideal 2 d ecsbc system

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

alphabet constrained data compression the optimal data compression problem is posed in terms of an alphabet constraint rather than an entropy constraint solving the optimal alphabet constrained data compression problem yields explicit source encoder decoder designs which is in sharp contrast to other approaches the alphabet constrained approach is shown to have the additional advantages that 1 classical waveform encoding schemes such as pulse code modulation pcm differential pulse code modulation dpcm and delta modulation dm as well as rate distortion theory motivated tree trellis coders fit within this theory 2 the concept of preposterior analysis in data compression is introduced yielding a rich new class of coders and 3 it provides a conceptual framework for the design of joint source channel coders for noisy channel applications examples are presented of single path differential encoding delayed or tree encoding preposterior analysis and source coding over noisy channels

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

vector quantization by deterministic annealing a deterministic annealing approach is suggested to search for the optimal vector quantizer given a set of training data the problem is reformulated within a probabilistic framework no prior knowledge is assumed on the source density and the principle of maximum entropy is used to obtain the association probabilities at a given average distortion the corresponding lagrange multiplier is inversely related to the temperature and is used to control the annealing process in this process as the temperature is lowered the system undergoes a sequence of phase transitions when existing clusters split naturally without use of heuristics the resulting codebook is independent of the codebook used to initialize the iterations

progressive image transmission using vector quantization on images in pyramid form a progressive image transmission scheme in which vector quantization is applied to images represented by pyramids is proposed a mean pyramid representation of an image is first built up by forming a sequence of reduced size images by averaging over blocks of 2 2 pixels a difference pyramid is then built up by taking the differences between successive levels in the mean pyramid progressive transmission is achieved by sending all the nodes in the difference pyramid starting from the top level and ending at the bottom level the kth approximate image can be formed by adding the information of level k to the previously reproduced k 1 st approximation to gain efficiency vector quantization is applied to the difference pyramid of the image on a level by level basis if the errors due to quantization at level k are properly delivered and included in the next level k 1 then it is demonstrated that the original image can be reconstructed an entropy coder is used to encode the final residual error image losslessly thus ensuring perfect reproduction of the original image the experiments demonstrate that it is possible to achieve simultaneously lossless and progressive transmission with compression at the intermediate level the use of vector quantization results in a coding gain over that obtained using only a huffman coder excellent reproduction is achieved at a bit rate of only 0 06 bits pixel

vector quantization of image subbands a survey subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels the concentration of energy in a few coefficients their multirate multiresolution framework and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system vector quantization vq provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces since 1988 a growing body of research has examined the use of vq for subband wavelet transform coefficients we present a survey of these methods

unrestricted multistage vector quantizers a low complexity high performance vector quantization technique for medium to high bit rate encoding based on multistage residual vq is introduced the design a variable rate version of multistage vq is the vector extension of the piecewise uniform scalar quantizers of p f swaszek and j b thomas 1984 and is related to the piecewise uniform vqs of f kuhlmann and j a buckview 1988 examples for the independent gaussian and laplacian sources are included

2 step scalar deadzone quantization for bitplane image coding modern lossy image coding systems generate a quality progressive codestream that truncated at increasing rates produces an image with decreasing distortion quality progressivity is commonly provided by an embedded quantizer that employs uniform scalar deadzone quantization usdq together with a bitplane coding strategy this paper introduces a 2 step scalar deadzone quantization 2sdq scheme that achieves same coding performance as that of usdq while reducing the coding passes and the emitted symbols of the bitplane coding engine this serves to reduce the computational costs of the codec and or to code high dynamic range images the main insights behind 2sdq are the use of two quantization step sizes that approximate wavelet coefficients with more or less precision depending on their density and a rate distortion optimization technique that adjusts the distortion decreases produced when coding 2sdq indexes the integration of 2sdq in current codecs is straightforward the applicability and efficiency of 2sdq are demonstrated within the framework of jpeg2000

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

variable rate vector quantization for speech image and video compression the performance of a vector quantizer can be improved by using a variable rate code three variable rate vector quantization systems are applied to speech image and video sources and compared to standard vector quantization and noiseless variable rate coding approaches the systems range from a simple and flexible tree based vector quantizer to a high performance but complex jointly optimized vector quantizer and noiseless code the systems provide significant performance improvements for subband speech coding predictive image coding and motion compensated video but provide only marginal improvements for vector quantization of linear predictive coefficients in speech and direct vector quantization of images criteria are suggested for determining when variable rate vector quantization may provide significant performance improvement over standard approaches

predictive residual vector quantization image coding this paper presents a new vector quantization technique called predictive residual vector quantization prvq it combines the concepts of predictive vector quantization pvq and residual vector quantization rvq to implement a high performance vq scheme with low search complexity the proposed prvq consists of a vector predictor designed by a multilayer perceptron and an rvq that is designed by a multilayer competitive neural network a major task in our proposed prvq design is the joint optimization of the vector predictor and the rvq codebooks in order to achieve this a new design based on the neural network learning algorithm is introduced this technique is basically a nonlinear constrained optimization where each constituent component of the prvq scheme is optimized by minimizing an appropriate stage error function with a constraint on the overall error this technique makes use of a lagrangian formulation and iteratively solves a lagrangian error function to obtain a locally optimal solution this approach is then compared to a jointly designed and a closed loop design approach in the jointly designed approach the predictor and quantizers are jointly optimized by minimizing only the overall error in the closed loop design however a predictor is first implemented then the stage quantizers are optimized for this predictor in a stage by stage fashion simulation results show that the proposed prvq scheme outperforms the equivalent rvq operating at the same bit rate and the unconstrained vq by 2 and 1 7 db respectively furthermore the proposed prvq outperforms the pvq in the rate distortion sense with significantly lower codebook search complexity

progressive image transmission by transform coefficient residual error quantization a progressive image transmission scheme based on iterative transform coding structure is proposed for application in interactive image communication over low bandwidth channels the scheme not only provides progressive transmission but also guarantees lossless reproduction combined with a degree of compression the image to be transmitted undergoes an orthogonal transform and the transform coefficients are quantized scalar or vector before transmission the novelty is that the residual error array due to quantization is iteratively fedback and requantized scalar or vector the coded residual error information is progressively transmitted and utilized in reconstructing the successive approximations it is shown that the average reconstruction error variance converges to zero as the number of iterative stages approaches infinity in practice lossless reproduction can be achieved with a small number of iterations by using an entropy coder on the final residual error image computer simulation results demonstrate the effectiveness of the technique

constrained storage quantization of multiple vector sources by codebook sharing a codebook sharing technique called constrained storage vector quantization csvq is introduced this technique offers a convenient and optimal way of trading off performance against storage the technique can be used in conjunction with tree structured vector quantization vq and other structured vq techniques that alleviate the search complexity obstacle the effectiveness of csvq is illustrated for coding transform coefficients of audio signals with multistage vq

new representation and factorizations of the higher order ultraspherical type differential equations abstract the paper deals with the class of linear differential equations of any even order 2 4 n 0 which are associated with the so called ultraspherical type polynomials these polynomials form an orthogonal system on the interval 1 1 with respect to the ultraspherical weight function 1 x 2 and additional point masses of equal size at the two endpoints the differential equations of ultraspherical type were developed by r koekoek in 1994 by utilizing special function methods in the present paper a new and completely elementary representation of these higher order differential equations is presented this result is used to deduce the orthogonality relation of the ultraspherical type polynomials directly from the differential equation property moreover we introduce two types of factorizations of the corresponding differential operators of order 2 4 into a product of 2 linear second order operators

necessary conditions for the optimality of variable rate residual vector quantizers necessary conditions for the optimality of variable rate residual vector quantizers are derived and an iterative descent algorithm based on a lagrangian formulation is introduced for designing residual vector quantizers having minimum average distortion subject to an entropy constraint simulation results for entropy constrained residual vector quantizers are presented for memoryless gaussian laplacian and uniform sources a gauss markov source is also considered the rate distortion performance is shown to be competitive with that of entropy constrained vector quantization and entropy constrained trellis coded quantization

layered video coding based on displaced frame difference prediction and multiresolution block matching this paper presents a new video coding technique based on displaced frame difference dfd prediction and multiresolution block matching for layered transmission starting from the base layer the algorithm encodes one layer at a time until the design of the top layer is completed for the encoding of each layer the dfd prediction and multistage motion vector quantization msmvq techniques are employed so that the results of dfd encoding and block matching at previous layers can be used to enhance the coding efficiency at that layer numerical results show that the algorithm is effective for the applications where high rate distortion performance and low computational complexity for layered video transmission are required

side match and overlap match vector quantizers for images a class of vector quantizers with memory that are known as finite state vector quantizers fsvqs in the image coding framework is investigated two fsvq designs namely side match vector quantizers smvqs and overlap match vector quantizers omvqs are introduced these designs take advantage of the 2 d spatial contiguity of pixel vectors as well as the high spatial correlation of pixels in typical gray level images smvq and omvq try to minimize the granular noise that causes visible pixel block boundaries in ordinary vq for 512 by 512 gray level images smvq and omvq can achieve communication quality reproduction at an average of 1 2 b pixel per image frame and acceptable quality reproduction because block boundaries are less visible the perceived improvement in quality over ordinary vq is even greater owing to the structure of smvq and omvq simple variable length noiseless codes can achieve as much as 60 bit rate reduction over fixed length noiseless codes

enhanced multistage vector quantization by joint codebook design multistage vector quantization msvq can achieve very low encoding and storage complexity in comparison to unstructured vector quantization however the conventional stage by stage design of the codebooks in msvq is suboptimal with respect to the overall performance measure the authors introduce an algorithm for the joint design of the stage codebooks to optimize the overall performance the performance improvement although modest is achieved with no effect on encoding or storage complexity and only a slight increase in design effort

gradient algorithms for designing predictive vector quantizers a predictive vector quantizer pvq is a vector extension of a predictive quantizer it consists of two parts a conventional memoryless vector quantizer vq and a vector predictor two gradient algorithms for designing a pvq are developed in this paper the steepest descent sd algorithm and the stochastic gradient sg algorithm both have the property of improving the quantizer and the predictor in the sense of minimizing the distortion as measured by the average mean squared error the differences between the two design approaches are the period and the step size used in each iteration to update the codebook and predictor the sg algorithm updates once for each input training vector and uses a small step size while the sd updates only once for a long period possibly one pass over the entire training sequence and uses a relatively large step size code designs and tests are simulated for both gauss markov sources and for sampled speech waveforms and the results are compared to codes designed using techniques that attempt to optimize only the quantizer for the predictor and not vice versa

image coding using vector quantization a review a review of vector quantization techniques used for encoding digital images is presented first the concept of vector quantization is introduced then its application to digital images is explained spatial predictive transform hybrid binary and subband vector quantizers are reviewed the emphasis is on the usefulness of the vector quantization when it is combined with conventional image coding techniques or when it is used in different domains

a unified approach to tree structured and multistage vector quantization for noisy channels the large encoding complexity and sensitivity to channel errors of vector quantization vq are discussed the performance of two low complexity vqs the tree structured vq tsvq and the multistage vq msvq when used over noisy channels are analyzed an algorithm is developed for the design of channel matched tsvq cm tsvq and channel matched msvq cm msvq under the squared error criterion extensive numerical results are given for the correlation coefficient 0 9 comparisons with the ordinary tsvq and msvq designed for the noiseless channel show substantial improvements when the channel is very noisy the cm msvq which can be regarded as a block structured combined source channel coding scheme is compared with a block structured tandem source channel coding scheme with the same block length as the cm msvq for the gauss markov source the cm msvq outperforms the tandem scheme in all cases that the authors have considered it is demonstrated that the cm msvq is fairly robust to channel mismatch

an improvement of the minimum distortion encoding algorithm for vector quantization in this note we present a very simple method for improving the efficiency of minimum distortion encoding for vector quantization simulations indicates a reduction of up to 70 percent in the number of multiplications for a full search vector quantizer with a large number of codewords and about 25 40 percent for a tree search vector quantizer similar improvement can be achieved in other vector quantization systems

block transform image coding by multistage vector quantization with optimal bit allocation a multistage vector quantization with optimal bit allocation mvq oba in the transform domain is presented a set of bit allocation planes is first obtained by slicing a scalar optimal bit allocation map where the number of bits assigned to each coefficient is proportional to the coefficient variance the set of bit allocation planes determines the coefficients to be used and the codebook size at each stage the vector dimensionalities are restricted to small values and relatively small codebooks are used thus reducing both the overhead required for transmitting the codebooks and the complexity in codebook design the computer simulation results demonstrate that mvq oba is competitive with many other transform coding techniques including variable length transform coding mvq oba is well suited for progressive transmission

product code vector quantizers for waveform and voice coding memory and computation requirements imply fundamental limitations on the quality that can be achieved in vector quantization systems used for speech waveform coding and linear predictive voice coding lpc one approach to reducing storage and computation requirements is to organize the set of reproduction vectors as the cartesian product of a vector codebook describing the shape of each reproduction vector and a scalar codebook describing the gain or energy such shape gain vector quantizers can be applied both to waveform coding using a quadratic error distortion measure and to voice coding using an itakura saito distortion measure in each case the minimum distortion reproduction vector can be found by first selecting a shape code word and then based on that choice selecting a gain codeword several algorithms are presented for the design of shape gain vector quantizers based on a traning sequence of data or a probabilistic model the algorithms are used to design shape gain vector quantizers for both the waveform coding and voice coding application the quantizers are simulated and their performance is compared to that of previously reported vector quantization systems

image coding using entropy constrained residual vector quantization an entropy constrained residual vector quantization design algorithm is used to design codebooks for image coding entropy constrained residual vector quantization has several important advantages it can outperform entropy constrained vector quantization in terms of rate distortion performance memory and computation requirements it can also be used to design vector quantizers with relatively large vector sizes and high output rates experimental results indicate that good image reproduction quality can be achieved at relatively low bit rates for example a peak signal to noise ratio of 30 09 db is obtained for the 512 spl times 512 lena image at a bit rate of 0 145 b p

on a generalization of bessel functions satisfying higher order differential equations bessel type functions j m x 0 with two parameters 12 and m 0 which include the classical normalized bessel function for m 0 are introduced as a certain confluent limit of koornwinder s laguerre type polynomials l nn x n n0 for any n0 they arise as solutions of a spectral dependent differential equation of order 2 4 a necessary and sufficient condition to transform the differential equation into symmetric form is given in terms of an overdetermined system of linear equations it is shown that for 0 1 2 a solution of this problem exists and leads to a symmetric fourth sixth and eighth order differential equation respectively for any 12 we also derive a second order differential equation but with coefficients depending nonlinearly on the eigenvalue parameter finally two differential expressions of order 2 2 n0 are constructed which map the classical bessel functions onto their nonclassical counterparts and vice versa this result may be used to establish an orthogonality relation for the bessel type functions in a distributional sense with respect to the distributional weight function 2 1 x2 1 m x denoting the point mass centered at the origin

finite difference approximations and superconvergence for the discrete ordinate equations in slab geometry a unified framework is developed for calculating the order of the error for a class of finite difference approximations to the monoenergetic linear transport equation in slab geometry in particular the global discretization errors for the step characteristic diamond and linear discontinuous methods are shown to be of order two while those for the linear moments and linear characteristic methods are of order three and that for the quadratic method is of order four a superconvergence result is obtained for the three linear methods in the sense that the cell averaged flux approximations are shown to converge at one order higher than the global errors