asymptotic expansion of a bessel function integral using hypergeometric functions abstract generalized hypergeometric functions are used to extend simplify and complete the analysis of stoyanov and farrel math comput 49 1987 275 279 and of wong math comput 50 1998 229 234 as well as putting their considerations within a wider framework the integral 0 2 sin a cos b j sin j sin d where j is the bessel function of order is analyzed in terms of generalized hypergeometric functions and a complete asymptotic expansion is given for 0 2 j sin j sin d

szlenk and w dentability indices of the banach spaces c 0 α abstract let be an infinite ordinal and the unique ordinal satisfying 1 we show that the banach space c 0 of all continuous scalar valued functions on the compact ordinal interval 0 has szlenk index equal to 1 and w dentability index equal to 1 1

nonlinear stability of θ methods for neutral differential equations in banach space abstract this paper is concerned with the stability properties of theoretical solution and numerical solution to nonlinear neutral differential equations with two delays in banach space some sufficient conditions for the stability and asymptotic stability of the systems are given the numerical stability of methods when applied to the systems is also considered a series of new stability results of methods are obtained

green s function for an unbounded biaxial medium in cylindrical coordinates the dyadic green s function for an unbounded biaxial medium is treated analytically in the fourier domain the green s function is initially expressed as a triple fourier integral which is next reduced to a double one by performing the integration over the longitudinal fourier variable a delta type source term is extracted which is dependent on the particular coordinate system

mellin transform methods applied to integral evaluation taylor series and asymptotic approximations mellin transform methods have been used to obtain taylor series or asymptotic approximations to definite integrals whose integrands consist of the product of two generalized hypergeometric functions this approach has proved extremely useful in many applied physics problems including the analysis of electromagnetic propagation in a turbulent medium the main result in this article is a generalization of the method to integrands containing the product of an arbitrary number of hypergeometric functions providing a very powerful and general technique for integral evaluation with many applications in physics

c α preserving operators on separable banach spaces abstract let t be a bounded linear operator from a separable banach space x to a banach space y a necessary and sufficient condition on t for the existence of a subspace z of x such that z is isomorphic to c and the restriction of t to z is an isomorphism is given the special case where x is the disc algebra is then considered and results similar to those previously obtained by the author for c k spaces are obtained for the disc algebra finally some additional results of the same type are proved for subspaces of c k with small annihilator

nonlinear stability of runge kutta methods for neutral delay differential equations this paper deals with stability properties of runge kutta methods for the initial value problem in nonlinear neutral delay differential equationsy t f t y t y t t y t t t 0 y t f t t t 0 the new concepts of gs l stability gas l stability and weak gas l stability are introduced and it is shown that k l algebraically stable runge kutta methods with piecewise linear interpolation are gs l gas l and weakly gas l stable two numerical examples are given in the end of this paper which confirm our results

asymptotic expansions of integrals of two bessel functions via the generalized hypergeometric and meijer functions abstract asymptotic expansions of certain finite and infinite integrals involving products of two bessel functions of the first kind are obtained by using the generalized hypergeometric and meijer functions the bessel functions involved are of arbitrary generally different orders but of the same argument containing a parameter which tends to infinity these types of integrals arise in various contexts including wave scattering and crystallography and are of general mathematical interest being related to the riemann liouville and hankel integrals the results complete the asymptotic expansions derived previously by two different methods a straightforward approach and the mellin transform technique these asymptotic expansions supply practical algorithms for computing the integrals the leading terms explicitly provide valuable analytical insight into the high frequency behavior of the solutions to the wave scattering problems

preserving operators on spaces let a c k x be a bounded linear operator where if is a compact hausdorff space and x is a separable banach space sufficient conditions are given for a to be an isomorphism into when restricted to a subspace y of c k such that y is isometrically isomorphic to a space c a of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal a 1 for a compact hausdorff space k c k will denote the banach space of continuous real valued functions on k with the supremum norm the greek letters a s and y will be reserved for countable ordinal numbers we consider the set 0 a y 0 x 0 for each countable ordinal a we will define a family pa x b of open subsets g and associated measures ju g g b let p0 x b g jttg g is an open set in k ixg g b and g g x if a s 1 then pa x b g nc g jiig g p0 x b and there is a disjoint sequence g of open subsets of g and associated measures xg such that g ftg g ps x b ju gn jug and ung c g if a is a limit ordinal then x b g xg g p0 x b there is a disjoint sequence of open subsets gn of g with associated measures xg such that fic nc ung c g and g jug g pa x b where an a the notation a a means that a is an increasing sequence of ordinals with a sup an a simple example illustrating the sets pa x b is contained in 4 received by the editors march 20 1979 and in revised form july 30 1981 1980 mathematics subject classification primary 46e15 research on this paper was partially supported by the national science foundation grant number mcs 77 01690 1982 american mathematical society o0o2 9939 82 oo0o o726 so5 oo

properties of the dyadic green s function for an unbounded anisotropic medium radiation in an unbounded anisotropic medium is treated analytically by studying the dyadic green s function of the problem initially expressed as a triple fourier integral which is next reduced to a double one under certain conditions the existence of incoming waves is verified it is also found that exponentially decaying waves are possible in such media finally the existence of branch points in the remaining integrand function is investigated and appropriate branch cuts are proposed

salvatore pincherle the pioneer of the mellin barnes integrals the 1888 paper by salvatore pincherle professor of mathematics at the university of bologna on generalized hypergeometric functions is revisited we point out the pioneering contribution of the italian mathematician towards the mellin barnes integrals based on the duality principle between linear differential equations and linear difference equation with rational coefficients by extending the original arguments used by pincherle we also show how to formally derive the linear differential equation and the mellin barnes integral representation of the meijer g functions

operators on c k spaces preserving copies of schreier spaces it is proved that an operator t c k x k compact metrizable x a separable banach space for which the e szlenk index of t bx is greater than or equal to 1 is an isomorphism on a subspace of c k isomorphic to x the schreier space of order as a corollary one obtains that a complemented subspace of c k with szlenk index equal to 1 contains a subspace isomorphic to x

a frequency domain approach to eigenvalue based detection with diversity reception and spectrum estimation in this paper we investigate a frequency domain approach for eigenvalue based detection of a primary user based on equal gain combining egc and spectrum estimation with bartlett s method this paper considers two techniques for eigenvalue detection which are maximum eigenvalue detection med and the maximum minimum eigenvalue mme detector we exploit the eigenvalues that are associated with the hermitian form representation of bartlett s estimate to assess the performance of the aforementioned eigenvalue techniques in the frequency domain for each case we quantify the performance based on the probabilities of false alarm and missed detection over rayleigh and rician fading a bivariate mellin transform approach is employed to obtain the probability distribution function for the ratio of the extreme eigenvalues under each hypothesis all obtained formulas are validated via monte carlo simulations and the results give a clear insight into the performance of the investigated methods in frequency domain med outperforms both the mme detector and periodogram based energy detection even in a worst case scenario of noise uncertainty while the mme detector exhibits heavy tailed statistical characteristics and thus its receiver operating characteristics tend to stay on the line of no discrimination the performance of med is further enhanced by careful choice of combinations of the total length of the sensing frame and number of sub slots

approximation theory and functional analysis part 1 research and survey articles on bernstein durrineyer polynomials with jacob weights h berens and y xu an overview of wavelets c chui a note on weak inequalities in orlicz and lorentz spaces g a edgar and l sucheston bivariate birkhoff interpolation a survey r a lorentz linear approximations of functions with several restricted derivatives y makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle k pan and e b saff box splines cardinal series and wavelets s d reimenschneider and z shen some aspects of the subspace structure of infinite dimensional banach space h rosenthal fairness and monotone curvature j roulier et al projections on 2 dimensional spaces n tomczak jacgermann real versus complex best rational approximation r s varga and a ruttan

on the use of nonsingular kernels in certain integral equations for thin wire circular loop antennas fundamental properties of certain popular integral equations for thin wire circular loop antennas with nonsingular kernels are studied the cornerstone of our study is a large asymptotic formula for the n th fourier coefficient of the nonsingular kernel four different methods of driving the circular loop are considered namely the delta function generator the finite gap generator a certain type of frill generator and the case of plane wave incidence the excitation model is crucial to our discussions since it determines the behavior of the solution convergence also discussed are associated difficulties that may arise when moment methods are applied to the aforementioned equations as well as a simple improvement to the fourier series method for determining the current in the case of the frill generator the main results herein closely parallel recent results for the case of the straight antenna and similarities and differences between the straight and circular cases are discussed

comments on asymptotic expansion of a bessel function integral using hypergeometric functions by l j landau and n j luswili in a recent paper landau and luswili j comput appl math 132 2001 387 used generalized hypergeometric functions to obtain a complete asymptotic expansion for the integral 0 2j sin jv sin d where j is the th order bessel function of the first kind and is a large parameter tending to infinity the purpose of this note is to point out that the same complete asymptotic expansion for this integral as well as another one for a hankel type integral has previously been obtained by stoyanov et al j comput appl math 50 1994 533 by using the same method in addition an alternative simpler representation of the algebraic series contribution to the asymptotic expansion is provided a few errors are also corrected and additional relevant references indicated

formulas for the force dipole interaction of surface line defects in homoepitaxy using asymptotics we derive explicit simplified formulas for integrals representing the force dipole interaction energy per unit length between line defects steps of the same sign that form perturbations of circles in homoepitaxy our starting point is continuum linear elasticity in accordance with the classic model by marchenko and parshin 1981 sov phys jetp 52 129 in the case with concentric circular steps we define a small geometric parameter 2 which expresses the smallness of interstep distance relative to the circle radii we invoke the mellin transform with respect to 2 and derive systematically an approximation for the requisite integral this technique offers an alternative to an exact evaluation in terms of elliptic integrals we then demonstrate the use of the mellin transform when calculating the force dipole interaction energy between smoothly slowly varying steps that form perturbations of circles we discuss the implications of our results for small amplitude modulations of circular step profiles

proximity to l1and distortion in asymptotic l1spaces abstract for an asymptotic l 1 space x with a basis x i certain asymptotic l 1 constants x are defined for 1 x measures the equivalence between all normalized block bases y i k i 1 of x i which are s admissible with respect to x i s is the th schreier class of sets and the unit vector basis of l k 1 this leads to the concept of the delta spectrum of x x which reflects the behavior of stabilized limits of x the analogues of these constants under all renormings of x are also defined and studied we investigate x both in general and for spaces of bounded distortion we also prove several results on distorting the classical tsirelson s space t and its relatives

optimal sequences of continuous functions converging to a baire 1 function a proof of rosenthal s co index conjecture is given our approach uses optimal sequences of continuous functions converging to a baire 1 function their existence is obtained by the optimal sequences theorem stated and proved here for a sequence of functions and a countable ordinal the variation is also introduced if pointwise converges to f the relation between and is completely clarified finally optimal sequences associated to a baire 1 function are defined and their existence for every baire 1 function is provided

integral evaluation using the mellin transform and generalized hypergeometric functions tutorial and applications to antenna problems this is a tutorial presentation of the mellin transform mt method for the exact calculation of one dimensional definite integrals and an illustration of the application of this method to antenna electromagnetics problems once the basics have been mastered one quickly realizes that the mt method is extremely powerful often yielding closed form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals yet as opposed to other methods the mt method is very straightforward to apply it usually requires laborious calculations but little ingenuity in fact the mt method is used by mathematica to symbolically calculate definite integrals the first part of this paper is a step by step tutorial proceeding from first principles it includes basic information on mellin barnes integrals and generalized hypergeometric functions and summarizes the key ideas of the mt method in the remaining parts the mt method is applied to three examples from the antenna area the results here are believed to be new at least in the antenna electromagnetics literature in our first example we obtain a closed form expression as a generalized hypergeometric function for the power radiated by a constant current circular loop antenna this quantity has been extensively discussed recently our second example concerns the admittance of a 2 d slot antenna in both these examples the exact closed form expressions are applied to improve upon existing formulas in standard antenna textbooks in our third example finally we obtain a very simple expression for an integral arising in recent unpublished studies of unbounded biaxially anisotropic media

szlenk indices and uniform homeomorphisms we prove some rather precise renorming theorems for banach spaces with szlenk index wo we use these theorems to show the invariance of certain quantitative szlenk type indices under uniform homeomorphisms

two dimensional highly directive currents on large circular loops properties of idealized two dimensional current distributions on circular loops are investigated analytically via the solution of a constrained optimization problem the directivity in the far field is maximized under a fixed c n t where n is the integral of the square of the current magnitude and t is the total radiated power c enters the ensuing fourier series for the current implicitly through a lagrange multiplier for non negative and large electrical radius ka the directivity and the current are evaluated approximately via combined use of the poisson summation formula and the mellin transform technique as a result a geometrical ray representation for the current is derived for the case of directivities that are slightly larger than that of the uniform distribution the analysis indicates certain advantages of large radiating structures for moderate values of the constraint c in the limit c of oseen s einstein needle radiation an asymptotic formula for the directivity is obtained pos

comment on two dimensional highly directive currents on large circular loops j math phys 41 6130 2000 margetis and fikioris apparently unaware of the earlier results available in the literature used an elaborate mellin transform technique to obtain an asymptotic approximation for a particular case of a bessel function integral here the earlier more general results are pointed out

nonlinear stability of explicit and diagonally implicit runge kutta methods for neutral delay differential equations in banach space abstract this paper is devoted to the investigations into the nonlinear stability properties of explicit and diagonally implicit runge kutta methods for solving neutral delay differential equations in banach space two approaches to numerically treating neutral term are considered which allow us to prove several results on numerical stability and conditional contractivity of explicit and diagonally implicit runge kutta methods in particular some new results on stiff odes and stiff ddes which can be regarded as the generalizations of previous results in the literature are obtained

convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods we have recently developed a generic approach to solving retarded and neutral delay differential equations ddes the approach is based on the use of an explicit continuous runge kutta formula and employs defect control the approach can be applied to problems with state dependent delays and vanishing delays in this paper we determine convergence properties for the numerical solution associated with methods that implement this approach we first analyze such properties for retarded ddes and then the analysis is extended to the case of neutral ddes ndes such an extension is particularly important since ndes have received little attention in the literature the main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect our analysis can also be applied directly to other numerical methods which use defect error control based on runge kutta formulas for ddes

operators whose dual has non separable range let x and y be separable banach spaces and t x y be a bounded linear operator we characterize the non separability of t y by means of fixing properties of the operator t