fast algorithms for discrete polynomial transforms on arbitrary grids abstract consider the vandermonde like matrix p p k x m l l k 0 m n where the polynomials p k satisfy a three term recurrence relation and x m l 1 1 are arbitrary nodes if p k are the chebyshev polynomials t k then p coincides with a t k x m l l 0 k 0 m n this paper presents a fast algorithm for the computation of the matrix vector product p a in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces p a with a a and a fast cosine transform on arbitrary nodes ndct since the first part of the algorithm was considered in math comp 67 1998 1577 we focus on approximative algorithms for the ndct our considerations are completed by numerical tests

computing with expansions in gegenbauer polynomials we develop fast algorithms for computations involving finite expansions in gegenbauer polynomials a method is described to convert any finite expansion between different families of gegenbauer polynomials for a degree n expansion the computational cost is mathcal o n log 1 varepsilon alpha beta where varepsilon is the prescribed accuracy and alpha and beta are the respective gegenbauer indices special cases involving chebyshev polynomials of first kind are particularly important in combination with nonequispaced discrete cosine transforms we obtain efficient methods for the evaluation of expansions at prescribed nodes and for the projection onto a sequence of gegenbauer polynomials from given function values respectively

fast scattered data approximation with neumann and other boundary conditions abstract an important problem in applications such as signal and image procesing is the approximation of a function f from a finite set of randomly scattered data f x j a common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials e 2 i kx this leads to fast numerical algorithms but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data which is rarely satisfied in practice to overcome this drawback we impose neumann boundary conditions on the data this implies the use of cosine polynomials cos kx as basis functions we show that using cosine polynomials leads to a least squares problem involving certain toeplitz plus hankel matrices and derive estimates on the condition number of these matrices unlike other toeplitz plus hankel matrices these matrices cannot be diagonalized by the discrete cosine transform dct but they still allow a fast matrix vector multiplication via dct which gives rise to fast conjugate gradient type algorithms we show how the results can be generalized to higher dimensions we also consider anti symmetric boundary conditions leading to sine polynomials as proper trigonometric basis finally we demonstrate the performance of the proposed methods by an application to a two dimensional geophysical scattered data problem

strong and ratio asymptotics for laguerre polynomials revisited abstract in this paper we consider the strong asymptotic behavior of laguerre polynomials in the complex plane the leading behavior is well known from perron and mehler heine formulas but higher order coefficients which are important in the context of krall laguerre or laguerre sobolev type orthogonal polynomials are notoriously difficult to compute in this paper we propose the use of an alternative expansion due to buchholz in terms of bessel functions of the first kind the coefficients in this expansion can be obtained in a straightforward way using symbolic computation as an application we derive extra terms in the asymptotic expansion of ratios of laguerre polynomials in c 0

nonuniform fast fourier transform the nonuniform discrete fourier transform ndft can be computed with a fast algorithm referred to as the nonuniform fast fourier transform nfft in l dimensions the nfft requires o n lne l x1d4c1 1lm x1d4c1 x1d4c1 1llogm x1d4c1 operations where m x1d4c1 is the number of fourier components along dimension x1d4c1 n is the number of irregularly spaced samples and e is the required accuracy this is a dramatic improvement over the o n x1d4c1 1lm x1d4c1 operations required for the direct evaluation ndft the performance of the nfft depends on the lowpass filter used in the algorithm a truncated gauss pulse proposed in the literature is optimized a newly proposed filter a gauss pulse tapered with a hanning window performs better than the truncated gauss pulse and the b spline also proposed in the literature for small filter length a numerically optimized filter shows the best results numerical experiments for 1 d and 2 d implementations confirm the theoretically predicted accuracy and efficiency properties of the algorithm

asymptotics for laguerre sobolev type orthogonal polynomials modified within their oscillatory regime abstract in this paper we consider sequences of polynomials orthogonal with respect to the discrete sobolev inner product f g s 0 f x g x x e x dx f c a g c t 1 where f and g are polynomials with real coefficients a r 2 2 and the vectors f c g c are a m 0 0 n f c f c f c and g c g c g c respectively with m n r and the mass point c is located inside the oscillatory region for the classical laguerre polynomials we focus our attention on the representation of these polynomials in terms of classical laguerre polynomials and we analyze the behavior of the coefficients of the corresponding five term recurrence relation when the degree of the polynomials is large enough also the outer relative asymptotics of the laguerre sobolev type with respect to the laguerre polynomials is analyzed

an improved fast multipole algorithm for potential fields on the line a new version of the fast multipole method fmm for the evaluation of potential fields on the line is introduced it uses a new representation of potential fields based on generalized gaussian quadratures j ma v rokhlin and s wandzura siam j numer anal 33 1996 pp 971 996 n yarvin and v rokhlin siam j sci comput 20 1999 pp 699 718 in this representation most translation operators are diagonal to efficiently incorporate this representation into the fmm an apparatus is introduced for transforming between different types of expansions this apparatus is highly general and is based on formulae for the least squares approximation of linear operators the performance of the method is illustrated with several numerical examples it is roughly twice as fast as previously published algorithms

fast polynomial multiplication and convolutions related to the discrete cosine transform abstract a classical scheme for multiplying polynomials is given by the cauchy product formula faster methods for computing this product have been developed using circular convolution and fast fourier transform algorithms from the numerical point of view the chebyshev expansion of polynomials is preferred to the monomial form we develop a direct scheme for multiplication of polynomials in chebyshev form as well as a fast algorithm using discrete cosine transforms this approach leads to a new convolution operation and a new type of circulant matrices both related to the discrete cosine transform extensions to bivariate polynomial products are also discussed

asymptotic behaviour of laguerre sobolev type orthogonal polynomials a nondiagonal case in this paper we study the asymptotic behaviour of polynomials orthogonal with respect to a sobolev type inner product s 0 p x q x x ae xdx p 0 taq 0 a 1 where p and q are polynomials with real coefficients a m 0 l lm 1 p 0 p 0 p 0 q 0 q 0 q 0 and a is a positive semidefinite matrix we will focus our attention on their outer relative asymptotics with respect to the standard laguerre polynomials as well as on an analog of the mehler heine formula for the rescaled polynomials

fast algorithms for discrete polynomial transforms consider the vandermonde like matrix p p k cos j n j k 0 n where the polynomials p k satisfy a three term recurrence relation if p k are the chebyshev polynomials t k then p coincides with c n 1 cos jk n j k 0 n this paper presents a new fast algorithm for the computation of the matrix vector product pa in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces pa with c n 1 and a subsequent fast cosine transform the first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications numerical tests demonstrate that our fast polynomial transform realizes pa with almost the same precision as the clenshaw algorithm but is much faster for n 128

asymptotics of polynomials orthogonal with respect to a discrete complex sobolev inner product let be a finite positive borel measure supported on a compact set of the real line and introduce the discrete sobolev type inner product f g f x g x d x k 1k sigma i 0nk mk if i ck g i ck where the mass points ck belong to supp and mk i are complex numbers such that mk nk 0 in this paper we investigate the asymptotics of the polynomials orthogonal with this product when the mass points ck belong to c supp the problem was solved in a paper by g lopez et al constr approx 11 1995 107 137 and for mass points in supp 1 1 the solution was given by i a rocha et al j approx theory 121 2003 336 356 provided that x 0 a e x 1 1 and mk i are nonnegative constants if m 0 1 the possibility ck supp 1 1 must be considered here we solve this last case with complex constants mk i

the regular fourier matrices and nonuniform fast fourier transforms for any triple of positive integers m n q the matrix f m n q called the m n q regular fourier matrix is defined the regular fourier matrices f m n q are then applied to set up new algorithms for nonuniform fast fourier transforms numerical results show that the accuracies obtained by our algorithms are much better than previously reported results with the same computation complexity the algorithms require o n cdot log 2n arithmetic operations where n is the number of data points

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

fast approximate fourier transforms for irregularly spaced data several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed the algorithms can be viewed as approximate generalizations of the fast fourier transform fft and they are compared with regard to their accuracy and their computational efficiency

asymptotic behavior of the partial derivatives of laguerre kernels and some applications abstract the aim of this paper is to present some new results about the asymptotic behavior of the partial derivatives of the kernel polynomials associated with the gamma distribution we also show how these results can be used in order to obtain the inner relative asymptotics for certain laguerre sobolev type polynomials

fast algorithms for polynomial interpolation integration and differentiation for functions tabulated at chebyshev nodes on an interval spectral interpolation and indefinite integration can be performed stably and efficiently via the fast fourier transform for many other sets of nodes such as the gaussian nodes on an interval the classical interpolation and indefinite integration schemes are stable but slow requiring o n 2 arithmetic operations with n the number of nodes in the discretization of the interval in this paper a group of algorithms is presented for the efficient evaluation of lagrange polynomial interpolants at multiple points on the line and for the rapid indefinite integration and differentiation of functions tabulated at nodes other than chebyshev the interpolation scheme requires o n cdot log 1 varepsilon arithmetic operations and o n cdot log n n cdot log 1 varepsilon operations are required for the integration and differentiation schemes where varepsilon is the precision of computations and n is the number of nodes t

systems of multidimensional volterra integral equations and inequalities on obtient des bornes superieures explicites pour les solutions u x d equations integrales de volterra lineaires ou non lineaires on demontre des theoremes d existence globale

on w1 p convergence of fourier sobolev expansions abstract let q n n 0 be the sequence of polynomials orthonormal with respect to the sobolev inner product f g s 1 1 f x g x w 0 x d x 1 1 f x g x w 1 x d x where w 0 l 1 1 and w 1 is a weight of kufner opic type we study necessary and or sufficient conditions for the convergence in the w 1 p 1 1 w 0 w 1 norm of the fourier expansion in terms of q n n 0 with 1 p

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

direct fourier reconstruction in computer tomography direct fourier transform ft reconstruction of images in computerized tomography ct is not widely used because of the difficulty of precisely interpolating from polar to cartesian samples in this paper an exact interpolation scheme is proposed which in practice can be approached with arbitrary accuracy using well conditioned algorithms several features of the direct ft method are discussed a method that allows angular band limiting of the data before processing to avoid angular aliasing artifacts in the reconstructed image is discussed and experimentally verified the experimental results demonstrate the feasibility of direct ft reconstruction of ct data