digital computation of linear canonical transforms we deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform lct of a function from the samples of the original function two approaches are presented and compared the first is based on decomposition of the lct into chirp multiplication fourier transformation and scaling operations the second is based on decomposition of the lct into a fractional fourier transform followed by scaling and chirp multiplication both algorithms take n log n time where n is the time bandwidth product of the signals the only essential deviation from exactness arises from the approximation of a continuous fourier transform with the discrete fourier transform thus the algorithms compute lcts with a performance similar to that of the fast fourier transform algorithm in computing the fourier transform both in terms of speed and accuracy

error analysis of reconstruction from linear canonical transform based sampling in this paper we consider the performance of sampling associated with the linear canonical transform lct which generalizes a large number of classical integral transforms and fundamental operations linked to signal processing and optics first we revisit sampling approximation in the lct domain to introduce a generalized approximation operator then we derive an exact closed form expression for the integrated squared error that occurs when a signal is approximated by a basis of shifted scaled and chirp modulated versions of a generating function in the lct domain several basic properties of the approximation error are presented the derived results can be applied to a wide variety of sampling approximation schemes in the lct domain finally experimental examples are given to illustrate the theoretical derivations

the geometric approach to quantum correlations computability versus reliability we propose a modified metric based on the hilbert schmidt norm and adopt it to define a rescaled version of the geometric measure of quantum discord such a measure is found not to suffer from pathological dependence on state purity although the employed metric is still non contractive under quantum operations we show that the resulting indicator of quantum correlations is in agreement with other bona fide discord measures in a number of physical examples we present a critical assessment of the requirements of reliability versus computability when approaching the task of quantifying or measuring general quantum correlations in a bipartite state

approximate signal reconstruction using nonuniform samples in fractional fourier and linear canonical transform domains approximate signal reconstruction formulas for the class of l 2 r signals in the fractional fourier and linear canonical transform lct domains are presented the results make use of the finite number of nonuniform samples of the signal in fractional fourier or lct domains taken at the positions determined by the zeros of the hermite polynomials the results provide exact representation for the class of signals that can be expressed in the form of polynomials of some finite order the truncation error bounds in the presented results are also discussed simulation results for some of the proposed theorems are also presented

quantum mass correction for the twisted kink we present an analytic result for the 1 loop quantum mass correction in semiclassical quantization for the twisted 4 kink on s1 without explicit knowledge of the fluctuation spectrum for this purpose we use the contour integral representation of the spectral zeta function by solving the bethe ansatz equations for the n 2 lam equation we obtain an analytic expression for the corresponding spectral discriminant we discuss the renormalization issues of this model an energetically preferred size for the compact space is finally obtained

on the zeta function regularization of a two dimensional series of epstein hurwitz type as a further step in the general program of zeta function regularization of multiseries expressions some original formulas are provided for the analytic continuation to any value of s of two dimensional series of epstein hurwitz type namely n1 n2 0 a1 n1 c1 2 a2 n2 c2 2 s where the aj are positive reals and the cj are not simultaneously nonpositive integers they come out from a generalization to hurwitz functions of the zeta function regularization theorem of the author and romeo phys rev d 40 436 1989 for ordinary zeta functions for s k 0 2 with k 1 2 3 the final results are in fact expressed in terms of hurwitz zeta functions only for general s they also involve bessel functions a partial numerical investigation of the different terms of the exact algebraic equations is also carried out as a by product the series n 0exp a n c 2 a c 0 is conveniently calculated in terms of them

uncertainty principles for linear canonical transform this correspondence investigates the uncertainty principles under the linear canonical transform lct first a lower bound on the uncertainty product of signal representations in two lct domains for complex signals is derived which can be achieved by a complex chirp signal with gaussian envelope then the tighter lower bound for real signals in two lct domains proposed by sharma and joshi is also proven to hold for arbitrary lct parameters based on the properties of moments for the lct the uncertainty principle for the fractional fourier transform is a special case of the achieved results

on the quantum zeta function it is remarkable that the quantum zeta function defined as a sum over energy eigenvalues e admits of exact evaluation in some situations for which not a single e be known herein we show how to evaluate instances of z s and of an associated parity zeta function y s for various quantum systems for some systems both z n y n can be evaluated for infinitely many integers n such z y values can be used for example to effect sharp numerical estimates of a system s ground energy the difficult problem of evaluating the analytic continuation z s for arbitrary complex s is discussed within the contexts of perturbation expansions path integration and quantum chaos

xi and eta function resummation of finite series general case given a spectrum of positive numbers lambda m from which a xi function z s sigma m lambda sm can be constructed the reorganization of series of the type f s t identical to m lambda smf lambda mt into power series in t is examined in detail using the method of xi function resummation for summand functions f lambda mt having power series expansions in lambda mt with infinite radius of convergence and which satisfy other conditions of a rather general nature the author finds that f s t can be reorganized to f s t n antbn k cktdk in t r s t where r s t vanishes exponentially as t to 0 the numbers an bn cn dn can all be computed in terms of the xi function z s r s t is difficult to evaluate but important general features of this function can be determined the power series expansion of f s t can be regarded as a generalization of the heat kernel expansion for which f lambda mt exp lambda mt and s 0 to nonzero complex variable s which is useful and to many other summand functions f lambda mt remarkably the xi function resummation method can be applied as easily to divergent series f s t as it can to convergent ones the method is therefore both a rearrangement procedure for convergent series and a summation prescription for divergent series

some analytic continuations of the barnes zeta function in two and higher dimensions abstract formulas for the analytic continuation of the barnes zeta function and some affine extensions thereof in two and more dimensions are constructed the expressions are used to deal with determinants of multidimensional harmonic oscillators an example is therewith obtained of the multiplicative anomaly or defect associated with the most common definition due to ray and singer d b ray reidemeister torsion and the laplacian on lens spaces adv math 4 1970 109 126 d b ray i m singer r torsion and the laplacian on riemannian manifolds adv math 7 1971 145 201 d b ray i m singer analytic torsion for complex manifolds ann math 98 1973 154 177 of determinant of a pseudodifferential operator admitting a zeta function m kontsevich s vishik geometry of determinants of elliptic operators in functional analysis on the eve of the 21st century vol 1 1995 pp 173 197

explicit zeta functions for bosonic and fermionic fields on a non commutative toroidal spacetime explicit formulae for the zeta functions s corresponding to bosonic 2 and to fermionic 3 quantum fields living on a non commutative partially toroidal spacetime are derived formulae for the most general case of the zeta function associated with a quadratic linear constant form in z are obtained they provide the analytical continuation of the zeta functions in relation to the whole complex s plane in terms of series of bessel functions of fast exponential convergence thus being extended chowla selberg formulae as is well known this is the most convenient expression that can be found for the analytical continuation of a zeta function in particular the residua of the poles and their finite parts are explicitly given an important novelty is the fact that simple poles show up at s 0 as well as in other places simple or double depending on the number of compactified non compactified and non commutative dimensions of the spacetime where they had never appeared before this poses a challenge to the zeta function regularization procedure

extrapolation of bandlimited signals in linear canonical transform domain the linear canonical transform lct has been shown to be a powerful analyzing tool in signal processing many results of this transform are already known including bandlimited extrapolation the existing algorithm for solving the problem of lct bandlimited extrapolation is based on signal expansion into a series of generalized prolate spheroidal wave functions gpswfs however the requirement to compute and store the gpswfs and the errors due to the series truncation render this algorithm ill suitable for a practical implementation in this correspondence we first propose a new formulation of the gerchberg papoulis gp algorithm for lct bandlimited extrapolation then we present a fast convergence algorithm for the new formulation the classical gp algorithm related to fourier bandlimited signals is noted as a special case moreover the comparison between the proposed algorithm and the one based on gpswfs expansion is provided and the validity of the theoretical derivations is demonstrated via simulations several potential applications of the achieved theory are also presented

eigenfunctions of linear canonical transform the linear canonical transform the lct is a useful tool for optical system analysis and signal processing it is parameterized by a 2 spl times 2 matrix a b c d many operations such as the fourier transform ft fractional fourier transform frft fresnel transform and scaling operations are all the special cases of the lct we discuss the eigenfunctions of the lct the eigenfunctions of the ft frft fresnel transform and scaling operations have been known and we derive the eigenfunctions of the lct based on the eigenfunctions of these operations we find for different cases that the eigenfunctions of the lct also have different forms when a d 2 the eigenfunctions become the chirp multiplication and chirp convolution of self similar functions fractals besides since many optical systems can be represented by the lct we can thus use the eigenfunctions of the lct derived in this paper to discuss the self imaging phenomena in optics we show that there are usually varieties of input functions that can cause the self imaging phenomena for an optical system

analysis of an inhomogeneous generalized epstein hurwitz zeta function with physical applications the inhomogeneous generalized epstein hurwitz like multidimensional series em s a1 am c1 cm c2 n1 nm 0 a1 n1 c1 2 am nm cm 2 c2 s is investigated by means of a nontrivial asymptotic recurrence it is reduced to the one dimensional case f s a b2 n 0 n a 2 b2 s which is then studied in full detail in particular asymptotic expansions for f and its derivatives f s and f a together with analytical continuations of the same in the variable s are explicitly obtained using zeta function techniques several plots and tables of the numerical results are given some explicit applications to the regularization by means of hurwitz zeta functions of different problems that have appeared recently in the physical literature are considered

zeta function regularization is uniquely defined and well hawking s zeta function regularization procedure 1977 is shown to be rigorously and uniquely defined thus putting an end to the spreading lore about different difficulties associated with it basic misconceptions misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method and other analytical procedures are clarified and corrected

uncertainty principle for real signals in the linear canonical transform domains the linear canonical transform lct is a generalization of the fractional fourier transform frft having applications in several areas of signal processing and optics in this paper we extend the uncertainty principle for real signals in the fractional fourier domains to the linear canonical transform domains giving us the tighter lower bound on the product of the spreads of the signal in two specific lct domains than the existing lower bounds in the lct domains it is seen that this lower bound can be achieved by a gaussian signal the effect of time shifting and scaling the signal on the uncertainty principle is also discussed it is shown here that a signal bandlimited in one lct domain can be bandlimited in some other lct domains also the exceptions to the uncertainty principle in the lct domains arising out of this are also discussed

relations between fractional operations and time frequency distributions and their applications the fractional fourier transform frft is a useful tool for signal processing it is the generalization of the fourier transform many fractional operations such as fractional convolution fractional correlation and the fractional hilbert transform are defined from it in fact the frft can be further generalized into the linear canonical transform lct and we can also use the lct to define several canonical operations in this paper we discuss the relations between the operations described above and some important time frequency distributions tfds such as the wigner distribution function wdf the ambiguity function af the signal correlation function and the spectrum correlation function first we systematically review the previous works in brief then some new relations are derived and listed in tables then we use these relations to analyze the applications of the frpt lct to fractional canonical filter design fractional canonical hilbert transform beam shaping and then we analyze the phase amplitude problems of the frft lct for phase amplitude problems we find as with the original fourier transform that in most cases the phase is more important than the amplitude for the frft lct we also use the wdf to explain why fractional canonical convolution can be used for space variant pattern recognition

on the issue of imposing boundary conditions on quantum fields an interesting example of the deep interrelation between physics and mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields this procedure has recently been re examined with care comments on that and previous analysis are provided here together with considerations on the results of the purely mathematical zeta function method in an attempt at clarifying the issue hadamard regularization is invoked in order to fill the gap between the infinities appearing in the qft renormalized results and the finite values obtained in the literature with other procedures

generalized prolate spheroidal wave functions associated with linear canonical transform time limited and a b c d band limited signals are of great interest not only in theory but also in real applications in this paper we use the sampling theorem associated with linear canonical transform to investigate an operator whose effect on a signal is to produce its first time limited then b b c d band limited version first the eigenvalue problem for the operator is shown to be equivalent to a discrete eigenvalue problem for an infinite matrix then the eigenfunctions of the operator which are referred to as generalized prolate spheroidal wave functions gpswfs are shown to be first orthogonal over finite as well as infinite intervals and second complete over l2 l l and the class of a b c d band limited signals a simple method based on sampling theorem for computing gpswfs is presented and the definite parity of gpswfs is also given finally based on the dual orthogonality and completeness of gpswfs several applications of gpswfs to the representation of time limited and a b c d band limited signals are presented

on sampling of band limited signals associated with the linear canonical transform sampling is one of the fundamental topics in the signal processing community theorems proposed under this topic form the bridge between the continuous time signals and discrete time signals several sampling theorems which aid in the reconstruction of signals in the linear canonical transform lct domain have been proposed in the literature however two main practical issues associated with the sampling of the lct still remain unresolved the first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically focusing on these issues this paper seeks to address the above from the lct point of view first we extend several previously developed theorems for signals band limited in the fourier domain to signals band limited in the lct domain followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the lct simulation results and the potential applications of the theorem are also proposed