gaussian quadrature for splines via homotopy continuation we introduce a new concept for generating optimal quadrature rules for splines to generate an optimal quadrature rule in a given target spline space we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one while preserving the number of quadrature points and therefore optimality the quadrature nodes and weights are considered as a higher dimensional point a zero of a particular system of polynomial equations as the space is continuously deformed by changing the source knot vector the quadrature rule gets updated using polynomial homotopy continuation for example starting with c 1 cubic splines with uniform knot sequences we demonstrate the methodology by deriving the optimal rules for uniform c 2 cubic spline spaces where the rule was only conjectured to date we validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen

on a class of modified newton cotes quadrature formulae based upon mixed type interpolation abstract we present quadrature rules which integrate exactly not only polynomials up to a certain degree but also the functions sin kx and cos kx where k is a free parameter the formulae we obtain are modified newton cotes formulae they are derived by replacing the integrand by an interpolation function of the form a cos kx b sin kx n 2j 0cjxj based on equally spaced nodes the total truncation error of the modified quadrature formulae is discussed and a rigorous proof of the error term is given for the modified simpson s 3 8 rule numerical examples show the efficiency of the modified rules and the importance of the error term

a three point formula for numerical quadrature of oscillatory integrals with variable frequency abstract a simple three point formula is constructed for the evaluation of general oscillatory integrals a rigorous derivation of the local error term is presented and the implications to high frequency oscillations are discussed simple examples given include integrals with variable frequency for which the usual filon formula would be inappropriate for cases where filon s formula is appropriate the new formula appears to be computationally more efficient the main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach comparison with exact results is possible from the work of stoker 16 for a case which whilst special in the physical sense does not simplify the integral involved in all cases the implementation of the formula is as straightforward as the implementation of the ordinary simpson rule

on a new type of mixed interpolation abstract we approximate every function f by a function f n x of the form a cos kx b sin kx n 2 i 0 c i x i so that f jh f n jh for the n 1 equidistant points jh j 0 n that interpolation function f n x is proved to be unique and can be written as the sum of the n th degree interpolation polynomial based on the same points and two correction terms the error term is also discussed the results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0 this new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations

the role of restriction theorems in harmonic analysis on harmonic na groups we shall obtain inequalities for fourier transform via moduli of continuity on na groups these results in particular settle the conjecture posed in a recent paper by w o bray and m pinsky in the context of noncompact rank one symmetric spaces these problems naturally demand versions of fourier restriction theorem on these spaces which we shall prove we shall also elaborate on the connection between the restriction theorem and the kunze stein phenomena on na groups for noncompact riemannian symmetric spaces of rank one analogues of all the results follow the same way

an endpoint estimate for the kunze stein phenomenon and related maximal operators one of the purposes of this paper is to prove that if g is a noncompact connected semisimple lie group of real rank one with flnite center then l 2 1 g l 2 1 g l 2 1 g let k be a maximal compact subgroup of g and x g k a symmetric space of real rank one we will also prove that the noncentered maximal operator m2f z sup2b 1 jbj z b jf z 0 jdz 0 is bounded from l 2 1 x t ol 2 1 x and from l p x t ol p x in the sharp range of exponents p2 2 1 the supremum in the deflnition ofm2f z i s taken over all balls containing the point z

on the error and its control in a two parameter generalised newton cotes rule a quadrature formula containing two free phase parameters k k and recently written by van daele et al 13 is rederived using an extension of lagrange s identity by using this method a closed form expression is determined for the local error term e f and the relevant peano kernel is given explicitly sufficient conditions are established under which this kernel remains definite thus allowing a particularly simple expression to be deduced for e f which reduces to the classical result for boole s rule in the limit k k 0 recent work on optimisation of global error is extended to include the new rule it is shown that this error can be reduced by a factor o h 2 on a certain curve in the phase space a further reduction by a factor o h 2 is sometimes possible by choosing the phase parameters on the intersection of this curve and another such when these curves intersect only at complex values the reduction is still achieved a closed form expression for the global error is also derived under these circumstances and this is seen to be asymptotically o h 1 0 as h 0 a second limiting form of the five point formula is found to reduce to the generalised boole s rule written by vanden berghe et al 14 and a third limiting form is also written this third rule which is a special case of a one parameter family of generalised boole s rules is seen to perform better than the other two in the two examples studied several numerical examples are given with extensive diagrams to illustrate all uses of the techniques proposed

the numerical inversion of two classes of kontorovich lebedev transform by direct quadrature abstract the inversion of the conventional kontorovich lebedev transform with the macdonald function as kernel and also with a kelvin function as kernel is considered strategies are presented first for computing values of these functions for purely imaginary order the interior strategy and these are tested in the subsequent inversion routines the exterior strategy which are developed for use where high speed mainframe or workstation computing is available both the interior and exterior strategies can exploit the w transformation and the w algorithm pioneered by sidi it is shown that conventional nag quadrature software is often incapable of carrying out the inversion without an implementation of this type examples considered show that relative error can generally be kept to o 1 d 8 although for work requiring hundreds of full inversion computations this would have to be relaxed somewhat to obviate the need for overnight computer runs application to a divergent inversion integral summable in the sense of abel is made with satisfactory results a case with weaker summability is also considered with somewhat inferior results

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

on the error estimation for a mixed type of interpolation we analyse the error induced by the approximation of a function by an interpolation function which is a combination of algebraic and first order trigonometric polynomials we prove that under certain conditions this error can be expressed in a similar form as in the purely polynomial case as an application we establish in closed form the local truncation error for a class of extended linear multistep methods of the adams type

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

some function spaces on symmetric spaces related to convolution operators soit g un groupe de lie semi simple connexe non compact de rang 1 une mesure positive k droite invariante a support compact soit 1 p 2 alors definit un operateur de convolution borne sur l p g si et seulement si il y a une fonction h dans l p 1 g telle que pour toute fonction c sur g k u avec u 4p 2 pp u g u g o d g g u g o h g dg la norme convolution de est equivalente a la norme l p 1 de h

an lp fourier analysis on symmetric spaces spherical fourier transforms of lp 1 p 2 functions on a riemannian symmetric space are studied it is proved that the spherical transform of such a function extends holomorphically to the same domain of the dual space as the one where the spherical transform of a schwartz function of lp type extends holomorphically and is rapidly decreasing cf 11 and also it is proved that a hausdorff young inequality and a riemann lebesgue lemma hold in the domain

growth properties of fourier transforms via moduli of continuity abstract we obtain new inequalities for the fourier transform both on euclidean space and on non compact rank one symmetric spaces in both cases these are expressed as a gauge on the size of the transform in terms of a suitable integral modulus of continuity of the function in all settings the results present a natural corollary a quantitative form of the riemann lebesgue lemma a prototype is given in one dimensional fourier analysis

fourier restriction theorem and characterization of weak l2 eigenfunctions of the laplace beltrami operator abstract in this paper we prove that for r 0 the poisson transform p f of f l 2 k m belongs to l 2 g k this complements an earlier result of lohoue and rychener using this we then prove an analogue of tomas stein restriction theorem for riemannian symmetric spaces of noncompact type with real rank one by using this estimate of the poisson transform we also characterize all weak l 2 eigenfunction of the laplace beltrami operator of riemannian symmetric spaces of noncompact type with real rank one and eigenvalue 2 2 for r 0

a note on a recent study of oscillatory integration rules abstract in a recent comparison of methods of highly oscillatory numerical integration the authors appear to have overlooked some works which seem to compete effectively with several of those described it is the intention briefly to illuminate this with a number of examples reworked in so doing it is found that three commonly used fortran compilers give curiously differing results on a number of occasions

on the error of parameter dependent compound quadrature formulas abstract ehrenmark this journal 1988 and vanden berghe et al this journal 1990 introduced quadrature formulas of a compound type which are obtained by interpolation with the null space of the linear differential operator d r f 2 i d r 2 f where i may be chosen differently for each subinterval in this paper the asymptotic behaviour of the error of this type of compound quadrature rule is determined for smooth functions it is shown that an improvement of the rate of convergence by a factor h 2 can be obtained by choosing i opt for all i and n this optimal can be determined explicitly if the derivatives of f are available and approximations to opt can be obtained numerically

fourier and radon transform on harmonic na groups in this article we study the fourier and the horocyclic radon transform on harmonic n a groups also known as damek ricci spaces we consider the geometric fourier transform for functions on l p spaces and prove an analogue of the l 2 restriction theorem we also prove some mixed norm estimates for the fourier transform generalizing the hausdorff young and hardy littlewood paley inequalities unlike euclidean spaces the domains of the fourier transforms are various strips in the complex plane all the theorems are considered on these entire domains of the fourier transforms finally we deal with the existence of the radon transform on l p spaces and obtain its continuity property

on computing uniformly valid approximations for viscous waves on a plane beach abstract existing expansions for the near and far field flow of linear waves in a viscous liquid on a plane beach are supplemented by an approximation in the intermediate region which enables the flow to be matched from the shore line to infinity methods of computation are discussed and incorporated are recent improvements on the performance of numerical computation of oscillatory integrals sidi s user friendly w transform 1988 and kohler s parameter optimisation this journal 1993 for generalised newton cotes quadrature are found to be significant elements particularly with regard to the inversion of the kontorovich lebedev transform comprehensive computations are undertaken and diagrams shown to display the behaviour of the stream function for all depths an attempt is also made to construct a composite expansion of the vorticity and this is computed on a ray bisecting the wedge

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

optimally blended spectral finite element scheme for wave propagation and nonstandard reduced integration we study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent finite element mass matrix and lumped spectral element mass matrix for the small wave number limit we find and prove that for the optimum blending the resulting scheme a provides 2p 4 order accuracy for p th order method two orders more accurate compared with finite and spectral element schemes b has an absolute accuracy which is mathcal o p 3 and mathcal o p 2 times better than that of the pure finite and spectral element schemes respectively c tends to exhibit phase lag moreover we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived

a characterization of the l2 range of the poisson transform related to strichartz conjecture on symmetric spaces of noncompact type abstract the main purpose of this paper is to solve strichartz conjecture concerning an image characterization of the poisson transform in the l 2 theory on symmetric spaces of noncompact type we prove that the poisson transform provides an isomorphism between the l 2 space on the boundary and a certain weighted l 2 space consisting of joint eigenfunctions on symmetric spaces of noncompact type our approach is based on the scattering theory for the schrodinger operator moreover we give a fourier restriction estimate and an asymptotic formula for the poisson transform