calculating the hilbert transform on spaces with energy concentration convergence and divergence regions in many different applications it is important to determine the hilbert transform of a given function however it is generally impossible to calculate it in closed form therefore hilbert transform approximations are used this paper studies the convergence and divergence behavior of general classes of such approximation methods these classes are characterized by two very natural axioms and they include basically all known traditional numerical algorithms the convergence of these methods is investigated on a family of signal spaces of continuous functions with finite energy these spaces are parametrized by a number which measures the energy concentration in the low frequency components of the signal it is shown that stable methods only exist on signal spaces with a sufficient energy concentration and this paper gives some explicit examples of convergent methods on all other spaces in the family of signal spaces every sampling based hilbert transform approximation shows a blowup behavior of its peak value i e on these spaces every sampling based hilbert transform approximation diverges

bifurcations with o 3 symmetry including applications to the benard problem this paper has two purposes to study from the singularity theory point of view certain bifurcation problems which commute with the five dimensional irreducible representation of the orthogonal group o 3 and to give some implications of this study for the benard problem in spherical geometry we shall now describe in general terms our results and compare them with the work of chossat 3 whose paper motivated our own interest in the problem the benard problem is concerned with convection in a viscous fluid when it is heated from below the fluid is assumed to be confined in a spherical shell of outer radius r and inner radius qro where q is near 0 3 this choice of q is partially motivated by considering convection within the molten layer of the core of the earth see busse l for further discussion we consider the binard problem in the boussinesq approximation in this model there is a trivial solution representing pure heat conduction radially outward as the temperature on the inner sphere that is the rayleigh number r is increased this trivial solution loses stability say at r r the benard problem is the study of the resulting bifurcation chossat studies this problem using the lyapunov schmidt reduction in the boussinesq approximation the fluid is driven from the pure conduction state by a term in the momentum equation involving the gravity vector g r and a term in the temperature equation involving the gradient of the conduction temperature vt assuming that the production of heat and the density are uniform throughout the shell these vectors have the form

an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

analysis of dynamics in magnetotelluric data by using the fisher shannon method abstract the fisher shannon information fs plane defined by the fisher information measure and the shannon entropy power is used to investigate the complex dynamics of magnetotelluric data of three stations in taiwan in the fs plane the electric and magnetic components are significantly separated characterized by different degrees of order further investigation shows that signals measured in areas with very high level of seismic activity are well discriminated

on the convergence of spline product quadratures for cauchy principal value integrals abstract in a recent paper this journal 1990 the authors proposed product integration formulas for the numerical evaluation of the cauchy principal value integral f1 1u x f x x dx based on cubic spline interpolation of f and obtained convergence results for functions f ck 1 1 k 1 2 or 3 in this report the same rules are considered and their convergence is investigated for a larger class of functions f an error bound and some uniform convergence results are established in the case of equally spaced quadrature nodes for functions f satisfying a holder condition of order on 1 1 0

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series we distinguish between two dimensions of a dynamical system given by experimental time series statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe tje attractor up to the accuracy of the data taking into account sampling and noise problems the dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data singular spectrum analysis ssa provides estimates of the statistical dimension ssa also describes the main physical phenomena reflected by the data it gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data we apply ssa to four paleoclimatic records the principal climatic oscillations and the regime changes in their amplitude are detected about 10 degrees of freedom are statistically significant in the data large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension

spline approximation for cauchy principal value integrals abstract in this paper we prove the uniform convergence of some quadrature formulas based on spline approximation for cauchy principal value integrals of the type f b a w x f k x x y d x k 0 1 and we present some numerical applications in particular we apply our rules to the well known prandtl s integral equation

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

a quadrature rule of interpolatory type for cauchy integrals abstract we construct a quadrature rule of interpolatory type based on a trigonometric interpolation for cauchy integrals we obtain the same error bound o ann as hasegawa and torii math comp 56 1991 741 754 numerical examples are included to substantiate the performance of the present method

the superconvergence of the newton cotes rule for cauchy principal value integrals we consider the general composite newton cotes method for the computation of cauchy principal value integrals and focus on its pointwise superconvergence phenomenon which means that the rate of convergence of the newton cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point the necessary and sufficient conditions satisfied by the superconvergence point are given moreover the superconvergence estimate is obtained and the properties of the superconvergence points are investigated finally some numerical examples are provided to validate the theoretical results

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

numerical integration based on approximating splines abstract convergence results are proved for product integration rules based on approximating splines these results are both for bounded and unbounded integrands pointwise and uniform convergence results are proved for sequences of cauchy principal values of these approximating splines

on the evaluation of cauchy principal value integrals by rules based on quasi interpolating splines abstract in this paper we consider the numerical evaluation of one dimensional cauchy principal value integrals of the form 1 1 k x f x x d x 1 by rules obtained by subtracting out the singularity and then applying quadrature formulas based on quasi interpolating splines

the cauchy principal value and the hadamard finite part integral as values of absolutely convergent integrals the divergent integral abf x x x0 n 1dx for a x0 b and n 0 1 2 is assigned under certain conditions the value equal to the simple average of the contour integrals c f z z x0 n 1dz where c c is a path that starts from a and ends at b and which passes above below the pole at x0 it is shown that this value which we refer to as the analytic principal value is equal to the cauchy principal value for n 0 and to the hadamard finite part of the divergent integral for positive integer n this implies that where the conditions apply the cauchy principal value and the hadamard finite part integral are in fact values of absolutely convergent integrals moreover it leads to the replacement of the boundary values in the sokhotski plemelj fox theorem with integrals along some arbitrary paths the utility of the analytic principal value in the numerical analytical and asymptotic evaluations of the principal value and the finite part integral is discussed and demonstrated

fisher information and nonlinear dynamics abstract the past decade has witnessed an impressive growth in the scope and variety of physical applications of fisher s information measure fim as a consequence fim is widely regarded today as providing one with a unifying principle for physics in the present paper we investigate its usefulness in connection with the description of important aspects of the behavior of dynamical systems and find that the unification referred to above can be extended to the field of these systems

an error embedded method based on generalized chebyshev polynomials in this paper we develop an error embedded method based on generalized chebyshev polynomials for solving stiff initial value problems the solution and the error at each integration step are calculated by generalized chebyshev polynomials of two consecutive degrees having overlapping zeros which enables us to minimize overall computational costs further the errors at each integration step are embedded in the algorithm itself in terms of concrete convergence and stability analysis the constructed algorithm turns out to have the 6th order convergence and an almost l stability we assess the proposed method with several numerical results showing that it uses larger time step sizes and is numerically more efficient

indefinite integration of oscillatory functions by the chebyshev series expansion abstract an automatic quadrature scheme is presented for evaluating the indefinite integral of oscillatory function x 0 t e i t dt 0 x 1 of a given function t which is usually assumed to be smooth the function t is expanded in the chebyshev series to make an efficient evaluation of the indefinite integral combining the automatic quadrature method obtained and sidi s extrapolation method makes an effective quadrature scheme for oscillatory infinite integral a x cos x d x for which numerical examples are also presented

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

a new interpretation of cauchy type singular integrals with an application to singular integral equations abstract cauchy type integrals were given the interpretation of the principal value for points inside the integration interval here this interpretation is modified and generalized in a very simple manner the new interpretation in general is not equivalent to the classical one the relationship between the new interpretation and the classical one is investigated and various applications of the new interpretation to the plemelj formulas the riemann hilbert boundary value problem singular integral equations the inversion formula quadrature rules and interface crack problems are presented

a practical algorithm for computing cauchy principal value integrals of oscillatory functions abstract a new automatic quadrature scheme is proposed for evaluating cauchy principal value integrals of oscillatory functions 1 1 f x exp i x x 1 dx 1 1 r the desired approximation is obtained by expanding the function f in the series of chebyshev polynomials of the first kind and then by constructing the indefinite integral for a properly modified integrand to overcome the singularity the method is proved to converge uniformly with respect to both and for any function f satisfying max 1 x 1 f x

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

evaluation of hypersingular integrals in the boundary element method by complex variable techniques strongly singular and hypersingular integrals that occur in many two dimensional applications of the boundary element method e g potential theory and liner elasticity can be treated in a unified manner by complex analysis this paper presents an elegant approach for the evaluation of such hypersingular integrals the limit to the boundary method where a source point is moved from the inside or outside of a body to its boundary is carried out using complex variables the connection of this limit with the hadamard finite part of hypersingular integrals and principal value integrals is established many examples are worked out throughout the paper to illustrate the theory

hilbert and hadamard transforms by generalized chebyshev expansion abstract an automatic quadrature is presented for approximating hadamard finite part fp integrals of a smooth function with a double pole singularity within the range of integration the quadrature rule is derived from the differentiation of an approximation to a cauchy principal value integral or the hilbert transform the approximation to the fp integral is represented as a function of the value of pole by using chebyshev polynomials of the second kind since the error can be estimated independently of the value of pole a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance with the same number of function evaluations numerical examples are also included to illustrate the performance of the methods

cubic splines and approximate solution of singular integral equations of concern here is the numerical solution of singular integral equations of cauchy type i e equations involving principal value integrals the unknown function is expressed as the product of an appropriate weight function and a cubic spline the problem is reduced to a system of linear algebraic equations which is solved for the approximate values of the function at the knots an estimate is provided for the maximum error of the approximate solution numerical results from the spline method are compared with those obtained using other methods

on a numerical liapunov schmidt spectral method and its application to biological pattern formation spectral expansions are used to provide a basis which preserves continuous symmetries we show that spectral methods satisfy the conditions for convergence of numerical liapunov schmidt methods an explicit algorithm for the calculation of stationary bifurcation scenarios near primary instabilities in general continuous symmetric equations is given the above convergence is extended to gamma equivalent discrete and original bifurcation scenarios the method is applied to a biologically motivated reaction diffusion system with spherical symmetry forming patterns a specific singularity of a generic steady state bifurcation is investigated in detail

generalized chebyshev interpolation and its application to automatic quadrature a generalized chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed it is incorporated into an efficient automatic quadrature scheme of clenshaw curtis tvpe numerical examples indicate that the present method is efficient not onlv for well behaved functions but for those w ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points

on the convergence of interpolatory type quadrature rules for evaluating cauchy integrals the aim of this work is to analyse the stability and the convergence for the quadrature rule of interpolatory type based on the trigonometric approximation for the discretization of the cauchy principal value integrals 11 f t d we prove that the quadrature rule has almost optimal stability property behaving in the form o logn 1 sin2x x cos t using this result we show that the rule has an exponential convergence rate when the function f is differentiable enough when f possesses continuous derivatives up to order p 0 and the derivative f p t satisfies holder continuity of order we can also prove that the rule has the convergence rate of the form o a b log n n2 np p where is as small as we like a and b are constants depending only on x

two trigonometric quadrature formulae for evaluating hypersingular integrals two trigonometric quadrature formulae one of non interpolatory type and one of interpolatory type for computing the hypersingular integral are developed on the basis of trigonometric quadrature formulae for cauchy principal value integrals the formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae fast three term recurrence relations for evaluating the quadrature weights are derived numerical tests are carried out using the current formula as applications two simple crack problems are considered one is a semi infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions it is found that the present method generally gives superior results copyright 2002 john wiley sons ltd

generalized mangler type principal value integrals with an application to fracture mechanics abstract a generalized interpretation of the classical mangler type principal value integrals is proposed this new interpretation in general is not equivalent to the classical one several formulae for this new class of integrals are proved concerning their relationship to cauchy type generalized cauchy type and ordinary mangler type principal value integrals moreover the plemelj formulae and quadrature rules are considered for the same class of integrals finally the present results are applied to the hypersingular integrodifferential equation of an interface crack in the classical theory of plane elasticity which takes now the form of a simple hypersingular integral equation

near onset convection patterns in the spherical benard problem with l 2 critical abstract bifurcation of convective patterns in a spherical shell of boussinesq fluid is investigated using center manifold reduction at the onset of the instability of the l 2 spherical harmonics the existence and stability of patterns with d 2 and o 2 symmetry are determined using results from singularity theory and a complete classification is given as a function of aspect ratio and prandtl number for both the degenerate self adjoint case and for small non self adjoint perturbations the results are compared to those of a previous study for l 4 critical

singular spectrum analysis and fisher shannon analysis of spring flow time series an application to anjar spring lebanon abstract in this study the time dynamics of water flow from anjar spring was investigated which is one of the major issuing springs in the central part of lebanon likewise many water sources in lebanon this spring has no continuous records for the discharge and this would prevent the application of standard time series analysis tools furthermore the highly nonstationary character of the series implies that suited methodologies can be employed to get insight into its dynamical features therefore the singular spectrum analysis ssa and fisher shannon fs method which are useful methods to disclose dynamical features in noisy nonstationary time series with gaps are jointly applied to analyze the anjar spring water flow series the ssa revealed that the series can be considered as the superposition of meteo climatic periodic components low frequency trend and noise like high frequency fluctuations the fs method allowed to extract and to identify among all the ssa reconstructed components the long term trend of the series the long term trend is characterized by higher fisher information measure fim and lower shannon entropy and thus represents the main informative component of the whole series generally water discharge time series presents very complex time structure therefore the joint application of the ssa and the fs method would be very useful in disclosing the main informative part of such kind of data series in the view of existing climatic variability and or anthropogenic challenges

on the chebyshev quadrature rule of finite part integrals the stability and the convergence of the chebyshev quadrature rule of one sided finite part integrals j approx theory 111 2001 196 219 is considered with some numerical examples the three term recurrence relation is suggested for computing the quadrature weights

quadrature formula for approximating the singular integral of cauchy type with unbounded weight function on the edges new quadrature formulas qfs for evaluating the singular integral si of cauchy type with unbounded weight function on the edges is constructed the construction of the qfs is based on the modification of discrete vortices method mmdv and linear spline interpolation over the finite interval 1 1 it is proved that the constructed qfs converge for any singular point x not coinciding with the end points of the interval 1 1 numerical results are given to validate the accuracy of the qfs the error bounds are found to be of order o h a lnh and o h lnh in the classes of functions h a 1 1 and c 1 1 1 respectively

application of quadrature rules for cauchy type integrals to the generalized poincare bertrand formula the classical poincare bertrand transposition formula for the inversion of the order of integration in repeated cauchy type integrals is generalized in accordance with a new interpretation of cauchy type integrals next the gauss jacobi quadrature rule is applied in a particular case of the generalized poincare bertrand formula to both members of this formula and it is proved that this formula still remains valid after the approximation of the integrals by quadrature sums two simple applications of this result one concerning the convergence of a quadrature rule for repeated cauchy type integrals and the other the numerical solution of singular integral equations are made further generalizations and applications of the present results follow easily

a piecewise linear quadrature of cauchy singular integrals the aim of this paper is to develop an algorithm for the quadrature weights induced by the piecewise polynomial quadrature of cauchy singular integrals we present a simple calculation scheme for the quadrature weights some numerical test problems are given

convergence results for piecewise linear quadratures for cauchy principal value integrals etablissement des conditions de convergence simple et uniforme des quadratures lineaires par morceaux pour les integrales a valeur principale de cauchy

the use of piecewise quadratic polynomials for the solution of singular integral equations of cauchy type abstract a method for the numerical solution of singular integral equations of cauchy type is developed the unknown function is expressed as a product of a weight function and a continuous function t the continuous function t is approximated by piecewise quadratic polynomials and the singular integral equation is reduced to a linear algebraic system numerical examples are given and comparisons are made with the widely used gauss type methods

product integration of logarithmic singular integrands based on cubic splines abstract this paper is concerned with the practical evaluation of the product integral 1 1 f x k x d x for the case when k x in x 1 1 and f is bounded in 1 1 the approximation is a quadrature rule where the weights w n n i are chosen to be exact when f is given by a linear combination of a chosen set of functions n j in this paper the functions n j are chosen to be cubic b splines an error bound for product quadrature rules based on cubic splines is provided examples that test the performance of the product quadrature rules for different choices of the function are given a comparison is made with product quadrature rules based on first kind chebyshev polynomials

a quadrature rule for weighted cauchy integrals abstract using a variable transformation t cos y for weighted cauchy integrals we develop an automatic quadrature method based on trigonometric interpolation we show that error bounds are independent of the location of poles

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method