anti symmetric multivariate exponential functions and corresponding fourier transforms we define and study symmetrized and antisymmetrized multivariate exponential functions they are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable these functions are eigenfunctions of the laplace operator on the corresponding fundamental domains satisfying certain boundary conditions to symmetric and antisymmetric multivariate exponential functions there correspond fourier transforms there are three types of such fourier transforms expansions into the corresponding fourier series integral fourier transforms and multivariate finite fourier transforms eigenfunctions of the integral fourier transforms are found

algebraic signal processing theory sampling for infinite and finite 1 d space we derive a signal processing framework called space signal processing that parallels time signal processing as such it comes in four versions continuous discrete infinite finite each with its own notion of convolution and fourier transform as in time these versions are connected by sampling theorems that we derive in contrast to time however space signal processing is based on a different notion of shift called space shift which operates symmetrically our work rigorously connects known and novel concepts into a coherent framework most importantly it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete fourier transform and hence can be derived by sampling the platform for our work is the algebraic signal processing theory an axiomatic approach and generalization of linear signal processing that we recently introduced

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

discrete and continuous exponential transforms of simple lie groups of rank 2 we develop and describe continuous and discrete transforms of class functions on compact simple lie group g as their expansions into series of uncommon special functions called here e functions in recognition of the fact that the functions generalize common exponential functions the rank of g is the number of variables in the e functions a uniform discretization of the decomposition problem is described on lattices of any density and symmetry admissible for the lie group g

some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

corrigendum to mixed interpolating smoothing splines and the v spline j math anal appl 322 2006 28 40 abstract in 6 the problem of mixed interpolation and smoothing by splines was generalized and applied to certain practical problems in particular conditions for the existence of solutions were provided in theorem 1 in proposition 6 of 8 it was shown that solutions may not always exist under the conditions of theorem 1 therefore it is the purpose of this paper to rectify this theorem we do so with an additional condition on finiteness of the kernel of an operator while the original proof holds with this finiteness assumption we provide a revised proof that more clearly shows where this assumption is used this extra assumption of finiteness has been assumed in all examples by the author in 6

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

multiplicity and concentration results for fractional schrodinger system with steep potential wells abstract this paper is concerned with the fractional coupled schrodinger system by using the nehari manifold and fibering map we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells where some new estimates will be established in particular although there exist concave convex nonlinearities in the coupled system it is not necessary to assume that the corresponding lebesgue norms of the weight functions of the convex terms need to be small enough

anomalous diffusion and charge relaxation on comb model exact solutions abstract the random walks on the comb structure are considered it is shown that due to fingers a diffusion has an anomalous character that is an r m s displacement depends on time by a power way with exponent 1 2 the generalized diffusion equation for an anomalous case is deduced it essentially differs from a usual diffusion equation in the continuity equation form instead of the first time derivative the time derivative of fractal order 1 2 appears in the second part the charge relaxation on the comb structure is studied a non maxwell character is established the reason is that the electric field has three components but a charge may relax only along some conducting lines

semiclassical states for weakly coupled fractional schrodinger systems abstract in this paper we consider the nonlinear system of two weakly coupled fractional schrodinger equations with nonconstant potentials e 2 s s u p 1 x u u 2 p b u p 1 v p 1 u x r n e 2 s s v p 2 x v v 2 p b v p 1 u p 1 v x r n we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials

generalized langevin equation with a three parameter mittag leffler noise abstract the relaxation functions for a given generalized langevin equation in the presence of a three parameter mittag leffler noise are studied analytically the results are represented by three parameter mittag leffler functions exact results for the velocity and displacement correlation functions of a diffusing particle are obtained by using the laplace transform method the asymptotic behavior of the particle in the short and long time limits are found by using the tauberian theorems it is shown that for large times the particle motion is subdiffusive for 1 and superdiffusive for many previously obtained results are recovered due to the many parameters contained in the noise term the model considered in this work may be used to improve the description of data and to model anomalous diffusive processes in complex media

unified continuum description for sub diffusion random walks on multi dimensional comb model abstract the anisotropic sub diffusion random walks on multi dimensional comb structure model have been studied in the continuum approximation the problem is that in the considered model mean square displacements on different directions have different power temporal dependencies so this case essentially differs from usual anisotropic diffusion case and it is not obviously how to describe such a diffusion by a unified way nevertheless the unified continuum description for such random walks has been developed as result the new generalization of fick s law for diffusion current has been obtained for anisotropic anomalous sub diffusion random walks in multi dimensional case namely it has been shown that the effective diffusion coefficient in the generalized fick s law has new operator form components of diffusion tensor instead of constant values have the fractional order temporal derivatives

fractional time quantum dynamics application of the fractional calculus to quantum processes is presented in particular the quantum dynamics is considered in the framework of the fractional time schr odinger equation se which differs from the standard se by the fractional time derivative ensuremath partial ensuremath partial t ensuremath rightarrow ensuremath partial ensuremath alpha ensuremath partial t ensuremath alpha it is shown that for ensuremath alpha 1 2 the fractional se is isospectral to a comb model an analytical expression for the green s functions of the systems are obtained the semiclassical limit is discussed

time dependent schrodinger like equation with nonlocal term we investigate a time dependent schrodinger like equation in presence of a nonlocal term by using the method of variable separation and the green function approach we analyze the green function for different forms of the memory kernel and the nonlocal term results for delta potential energy function are presented distributed order memory kernels are also considered and the asymptotic behaviors of the green function are derived by using tauberian theorem the obtained results for the green function for the considered schrodinger like equation may be transformed to those for the probability distribution function of a diffusion like equation with memory kernel and can be used to explain various anomalous diffusive behaviors

time dependent solutions for a fractional schrodinger equation with delta potentials we investigate for an arbitrary initial condition the time dependent solutions for a fractional schrodinger equation in the presence of delta potentials by using the green function approach the solutions obtained show an anomalous spreading asymptotically characterized by a power law behavior which is governed by the order of the fractional spatial operator present in the schrodinger equation

discrete and continuous sine transform generalized to semisimple lie groups of rank two a new continuous group transform together with its discretization over a lattice of any density and admissible symmetry is defined and described for the four compact semisimple lie groups of rank 2 in the case of rank 1 the discrete version is the sine transform properties of the expansion functions of the transform called s functions are studied digital data processing is our motivating application

how the site degree influences quantum probability on inhomogeneous substrates we investigate the effect of the node degree and energy e on the electronic wave function for regular and irregular structures namely regular lattices disordered percolation clusters and complex networks we evaluate the dependency of the quantum probability for each site on its degree for a class of biregular structures formed by two disjoint subsets of sites sharing the same degree the probability p k e of finding the electron on any site with k neighbors is independent of e 0 a consequence of an exact analytical result that we prove for any bipartite lattice for more general nonbipartite structures p k e may depend on e as illustrated by an exact evaluation of a one dimensional semiregular lattice p k e is large for small values of e when k is also small and its maximum values shift towards large values of e with increasing k numerical evaluations of p k e for two different types of percolation clusters and the apollonian network suggest that this observed feature might be generally valid

anti symmetric multivariate trigonometric functions and corresponding fourier transforms four families of special functions depending on n variables are studied we call them symmetric and antisymmetric multivariate sine and cosine functions they are given as determinants or antideterminants of matrices whose matrix elements are sine or cosine functions of one variable each these functions are eigenfunctions of the laplace operator satisfying specific conditions at the boundary of a certain domain f of the n dimensional euclidean space discrete and continuous orthogonality on f of the functions within each family allows one to introduce symmetrized and antisymmetrized multivariate fourier like transforms involving the symmetric and antisymmetric multivariate sine and cosine functions

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

diffusion on random comb structure effective medium approximation abstract the random walks on comb model of percolation clusters are studied the generalized diffusion equations in the form of fractal temporal derivatives are deduced the random distributions of fingers over lengths gaussian and power are studied in the effective medium approximation

computation of character decompositions of class functions on compact semisimple lie groups a new algorithm is described for splitting class functions of an arbitrary semisimple compact lie group k into sums of irreducible characters the method is based on the use of elements of finite order efo in k and is applicable to a number of problems including decompositions of tensor products and various symmetry classes of tensors as well as branching rules in group subgroup reductions the main feature is the construction of a decomposition matrix d computed once and for all for a given range of problems and for a given k which then reduces any particular splitting to a simple matrix multiplication determination of d requires selection of a suitable set s of conjugacy classes of efo representing a finite subgroup of a maximal torus t of k and the evaluation of weyl group orbit sums on s in fact the evaluation of d can be coupled with the evaluation of the orbit sums in such a way as to greatly enhance the efficiency of the latter the use of the method is illustrated by some extensive examples of tensor product decompositions in e6 modular arithmetic allows all computations to be performed exactly

existence of solutions for a bi nonlocal fractional p kirchhoff type problem in this paper we are concerned with the existence of nonnegative solutions for a p kirchhoff type problem driven by a non local integro differential operator with homogeneous dirichlet boundary data as a particular case we study the following problem m x u s p p p s u f x u u s p p in u 0 in r n u s p p r 2 n u x u y p x y n p s d x d y where p s is a fractional p laplace operator is an open bounded subset of r n with lipschitz boundary m r 0 r is a continuous function and f r r 0 r is a continuous function satisfying the ambrosetti rabinowitz type condition the existence of nonnegative solutions is obtained by using the mountain pass theorem and an iterative scheme the main feature of this paper lies in the fact that the kirchhoff function m depends on x and the nonlinearity f depends on the energy of solutions

properties of continuous fourier extension of the discrete cosine transform and its multidimensional generalization a versatile method is described for the practical computation of the exact discrete fourier transforms dft both the direct and the inverse ones of a continuous function g given by its values gj at the points of a uniform grid fn generated by conjugacy classes of elements of finite adjoint order n in the fundamental region f of compact semisimple lie groups the present implementation of the method is for the groups su 2 when f is reduced to a one dimensional segment and for su 2 su 2 su 2 in multidimensional cases this simplest case turns out to be a version of the discrete cosine transform dct implementations abbreviated as dgt for discrete group transform based on simple lie groups of higher ranks are to be considered separately dct is often taken to be simply a specific type of the standard dft here we show that the dct is very different from the standard dft when the properties of the continuous extensions of the two inverse discrete transforms are studied the following propertie

alternating multivariate trigonometric functions and corresponding fourier transforms we define and study multivariate sine and cosine functions symmetric with respect to the alternating group an which is a subgroup of the permutation symmetric group sn these functions are eigenfunctions of the laplace operator they determine fourier type transforms there exist three types of such transforms expansions into corresponding sine fourier and cosine fourier series integral sine fourier and cosine fourier transforms and multivariate finite sine and cosine transforms in all these transforms alternating multivariate sine and cosine functions are used as a kernel

time fractional schrodinger equation fox s h functions and the effective potential after introducing the formalism of the general space and time fractional schrodinger equation we concentrate on the time fractional schrodinger equation and present new results via the elegant language of fox s h functions we show that the general time dependent part of the wave function for the separable solutions of the time fractional schrodinger equation is the mittag leffler function with an imaginary argument by two different methods after separating the mittag leffler function into its real and imaginary parts in contrast to existing works we show that the total probability is 1 and decays with time introducing the effective potential approach we also write the mittag leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts in the light of these we reconsider the simple box problem

exact solutions of fractional schrodinger like equation with a nonlocal term we study the time space fractional schrodinger equation with a nonlocal potential by the method of fourier transform and laplace transform the green function and hence the wave function is expressed in terms of h functions graphical analysis demonstrates that the influence of both the space fractal parameter and the nonlocal parameter on the fractional quantum system is strong indeed the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative

generalized fractional diffusion equations for accelerating subdiffusion and truncated levy flights fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time for processes lacking such scaling the corresponding description may be given by diffusion equations with fractional derivatives of distributed order such equations were introduced in a v chechkin r gorenflo and i sokolov phys rev e 66 046129 2002 for the description of the processes getting more anomalous in the course of time decelerating subdiffusion and accelerating superdiffusion here we discuss the properties of diffusion equations with fractional derivatives of the distributed order for the description of anomalous relaxation and diffusion phenomena getting less anomalous in the course of time which we call respectively accelerating subdiffusion and decelerating superdiffusion for the former process by taking a relatively simple particular example with two fixed anomalous diffusion exponents we show that the proposed equation effectively describes the subdiffusion phenomenon with diffusion exponent varying in time for the latter process we demonstrate by a particular example how the power law truncated levy stable distribution evolves in time to the distribution with power law asymptotics and gaussian shape in the central part the special case of two different orders is characteristic for the general situation in which the extreme orders dominate the asymptotics

green s function for the time dependent scattering problem in the fractional quantum mechanics we study the green s function of the time dependent 3d space fractional schrodinger equation for the scattering problem in the fractional quantum mechanics the green s function is expressed in terms of fox s h function and in a computable series form we get the asymptotic formula of the green s function and apply it to obtain the approximate wave function for the fractional quantum scattering problem

levy processes on a generalized fractal comb comb geometry constituted of a backbone and fingers is one of the most simple paradigm of a two dimensional structure where anomalous diffusion can be realized in the framework of markov processes however the intrinsic properties of the structure can destroy this markovian transport these effects can be described by the memory and spatial kernels in particular the fractal structure of the fingers which is controlled by the spatial kernel in both the real and the fourier spaces leads to the levy processes levy flights and superdiffusion this generalization of the fractional diffusion is described by the riesz space fractional derivative in the framework of this generalized fractal comb model levy processes are considered and exact solutions for the probability distribution functions are obtained in terms of the fox h function for a variety of the memory kernels and the rate of the superdiffusive spreading is studied by calculating the fractional moments for a special form of the memory kernels we also observed a competition between long rests and long jumps finally we considered the fractal structure of the fingers controlled by a weierstrass function which leads to the power law kernel in the fourier space this is a special case when the second moment exists for superdiffusion in this competition between long rests and long jumps

generalized time dependent schrodinger equation in two dimensions under constraints we investigate a generalized two dimensional time dependent schrodinger equation on a comb with a memory kernel a dirac delta term is introduced in the schrodinger equation so that the quantum motion along the x direction is constrained at y 0 the wave function is analyzed by using green s function approach for several forms of the memory kernel which are of particular interest closed form solutions for the cases of dirac delta and power law memory kernels in terms of fox h function as well as for a distributed order memory kernel are obtained further a nonlocal term is also introduced and investigated analytically it is shown that the solution for such a case can be represented in terms of infinite series in fox h functions green s functions for each of the considered cases are analyzed and plotted for the most representative ones anomalous diffusion signatures are evident from the presence of the power law tails the normalized green s functions obtained in this work are of broader interest

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

continuous time random walk as a guide to fractional schrodinger equation we argue that the continuous time random walk approach may be a useful guide to extend the schrodinger equation in order to incorporate nonlocal effects avoiding the inconsistencies raised by jeng et al j math phys 51 062102 2010 as an application we work out a free particle in a half space obtaining the time dependent solution by considering an arbitrary initial condition

field induced drift and trapping in percolation networks the authors study the effects of a bias producing external field on a random walk on the infinite cluster in the percolation problem there are two competing physical effects drift and trapping which result in a drift velocity nu which rises and then falls as the field increases they study these effects on a one dimensional lattice with random length branches and on the diluted bethe lattice they calculate nu in the first model with a maximum allowed length for each branch in the limit that this cutoff length becomes infinite they find that the velocity vanishes identically above a finite critical value of the field for the bethe lattice they derive an upper bound on the critical field which varies as p pc 1 2 as the percolation concentration pc is approached in the one dimensional model they also investigate the anomalous regime in which the velocity vanishes they discuss the distribution of steady state times required to traverse n sites and find that it can be described in terms of a stable distribution of index x with superimposed oscillations the index x of the stable distribution is given by l xi where xi is a characteristic branch length and l is a bias induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch

discrete and continuous cosine transform generalized to lie groups su 3 and g 2 in the paper we complete the development and description of the four variants of two dimensional generalization of the cosine transform started in patera and zaratsyan j math phys 46 053514 2005 each variant is based on a compact semisimple lie group g of rank 2 here the groups are su 3 and g 2 the cosines are generalized as the corresponding c functions of the lie group a c function is the contribution to an irreducible character from one orbit of the appropriate weyl group an explicit description is provided for expansions of functions given on the fundamental region f of the two compact simple lie groups into series of c functions the fundamental region f is an equilateral triangle for su 3 and half of such a triangle for g 2 expansion coefficients are calculated using orthogonality of c functions on f discrete expansions are set up on a grid fm f the grid is defined group theoretically for all positive integers m it consists of points in f that represent conjugacy classes of ele

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

scaling properties of diffusion on comb like structures the authors study the scaling properties of the probability density of random walks on comb like structures they find that in contrast to the first passage time in which several exponents are needed to characterise their distribution the probability density can be described by a single exponent

eigenvalue problems for fractional differential equations with right and left fractional derivatives this paper studies the eigenvalue problem of a class of fractional differential equations with right and left fractional derivatives with the aid of the spectral theory of compact self adjoint operators in hilbert spaces we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system furthermore the lower bound of the eigenvalues is obtained

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

mixed interpolating smoothing splines and the v spline abstract in their monograph bezhaev and vasilenko have characterized the mixed interpolating smoothing spline in the abstract setting of a hilbert space in this paper we derive a similar characterization under slightly more general conditions this is specialized to the finite dimensional case and applied to a few well known problems including the spline a piecewise polynomial spline in tension and near interpolation as well as interpolation and smoothing in particular one of the main objectives in this paper is to show that the spline is actually a mixed spline an observation that we believe was not known prior to this work we also show that the spline is a limiting case of smoothing splines as certain weights increase to infinity and a limiting case of near interpolants as certain tolerances decrease to zero we conclude with an iteration used to construct curvature bounded spline curves

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

solutions for a schrodinger equation with a nonlocal term we obtain time dependent solutions for a schrondiger equation in the presence of a nonlocal term by using the green function approach these solutions are compared with recent results obtained for the fractional schrodinger equation as well as for the usual one the nonlocal term incorporated in the schrodinger equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution

the global solution for a class of systems of fractional nonlinear schrodinger equations with periodic boundary condition in this paper we consider a class of systems of fractional nonlinear schrodinger equations we prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the faedo galerkin method

on the problems of smoothing and near interpolation in the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions in particular we show that near interpolants are smoothing splines with weights that arise as lagrange multipliers corresponding to the constraints in the problem of near interpolation in the second part of this paper we apply certain fixed point iterations to compute these weights a similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing

best near interpolation by curves existence the conditions derived in k scherer and p w smith siam j numer anal 20 1989 pp 160 168 for the existence of minimizers to the nonlinear problem of best interpolation by curves are extended to the problem of best near interpolation by curves that meet arbitrary sets such as closed balls as in s kersey best near interpolation by curves optimality conditions technical report 99 05 center for the mathematical sciences university of wisconsin madison wi 1999 the minimizers are spline curves with breakpoints at the data sites at which the curves meet the sets and the nonlinearities arise as these data sites vary from curve to curve the results here apply to hermite type interpolation conditions with the possibility of repeated data sites

stability of standing waves for the fractional schrodinger hartree equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger hartree equation by using the profile decomposition theory and variational methods we firstly obtain the best constant of a generalized gagliardo nirenberg inequality and then prove that the standing waves are orbitally stable

fractional diffusion on a fractal grid comb a grid comb model is a generalization of the well known comb model and it consists of n backbones for n 1 the system reduces to the comb model where subdiffusion takes place with the transport exponent 1 2 we present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones we show that for an arbitrarily large but finite number of backbones the transport exponent does not change contrary to that for an infinite number of backbones the transport exponent depends on the fractal dimension of the backbone structure

constrained quantum motion in δ potential and application of a generalized integral operator abstract a quantum motion of a particle along the x direction constrained at y 0 in presence of the dirac potential is considered the green s function approach is applied for solving the corresponding two dimensional schrodinger equation the exact results for the green s function are presented in terms of the fox h function and a generalized integral operator which contains the mittag leffler function in the kernel

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices