A direct approach to the mellin transform

The aim of this paper is to present an approach to the Mellin transform that is fully independent of Laplace or Fourier transform theory, in a systematic, unified form, containing the basic properties and major results under natural, minimal hypotheses upon the functions in questions. Cornerstones of the approach are two definitions of the transform, a local and global Mellin transform, the Mellin translation and convolution structure, in particular approximation-theoretical methods connected with the Mellin convolution singular integral enabling one to establish the Mellin inversion theory. Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame. These two operators are different than those considered thus far and more general. They are of particular importance in solving differential and integral equations. As applications, the wave equation onℝ ₊ × ℝ₊ and the heat equation in a semi-infinite rod are considered in detail. The paper is written in part from an historical, survey-type perspective.     

The GL(3) mellin transform for twisted non-cuspidal forms of higher level

The main result in this paper is the explicit computation of the functional equation satisfied by the GL(3) Mellin transform of a twisted non-cuspidal metaplectic form of non-trivial level. For concreteness, we work with one particular metaplectic form, automorphic under Λ(3), although our methods extend without change to any form automorphic with respect to Λ(p),p an odd prime. We clearly show the computations one must undertake in order to determine the pole locations of this transform (which depend on the form in question), and carry out those straightforward computations in the case of our one specific form. This particular form, first considered by Bump and Hoffstein [BH], is the maximal parabolic Eisenstein series on the cubic cover of GL(3) (induced from the theta function on the cubic cover of GL(2)), which has the remarkable property that its Fourier coefficients are essentially Hecke cubicL-series. In joint work with Hoffstein, the authors have applied the main theorems in this paper to compute the average values of these Hecke cubicL-series, a result of great arithmetic interest. Both authors are partially supported by the National Science Foundation. Both authors wish to thank the Mathematical Sciences Research Institute for its hospitality during part of the time this research was conducted. In addition, the authors wish to thank Jeffrey Hoffstein for many helpful conversations on this and related problems, and the referee for several helpful comments.     

Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces

This paper studies the asymptotic expansions of spherical functions on symmetric spaces and Fourier transform of rapidly decreasing functions of L^p type (0 < p ? 2) on these spaces.     

Evaluation of integrals by decomposition

It is shown that in addition to its advantages for nonlinear and/or stochastic differential equations [1,2], the decomposition method may be preferable even for equations, such as linear deterministic ordinary differential equations which are easily solvable by well-known methods in integral form because the evaluations of the integrals is easier. It is also shown that since solutions of differential equations are easily obtained by decomposition, it can be convenient to change a difficult integration problem to an easily solved differential equation and consequently evaluate the integral in an easily computed convergent series.     

Pseudodifferential operators of mellin type on sobolev spaces

Sunto Proposito di questa nota è di introdurre una classe di operatori pseudodifferenziali che agiscono su sottospazi chiusi di W^p,k(R ₊), caratterizzati da opportune condizioni di compatibilità. Per tali operatori vengono studiate proprietà di Fredholm e di indice.

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Research partially supported by C.N.R. Gruppo G.N.A.F.A.     

The boundedness of maximal operators and singular integrals via Fourier transform estimates

In this paper, the author studies the mapping properties for some general maximal operators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.     

Evaluation of mean and variance integrals without integration

The mean and variance of some continuous distributions, in particular the exponentially decreasing probability distribution and the normal distribution, are considered. Since they involve integration by parts, many students do not feel comfortable. In this note, a technique is demonstrated for deriving mean and variance through differential calculus. The general nature of the technique has potential for wider applications.     

Evaluation of a class of definite integrals

An attempt is made to put the notion of sample quartiles on a mathematical footing in the light of ranks of observations, and equisegmentation property that the quartiles divide ordered sample observations into four segments leaving the same number of observations in each if all the observations are distinct. Sample quartiles provided by the proposed halving method, based on hinges, does satisfy the property.     

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.     

Evaluation of Fresnel integrals based on the continued fractions method

We present a simple algorithm for evaluating Fresnel integrals based on the continued fractions method: we use the relation between these integrals and first-kind Bessel functions of fractional order, and we apply a fast code to calculate them based on the continued fractions method. This latter code is especially useful for evaluating high order Bessel functions because it does not require recalculations using normalization relations. Comments on the same procedure but using Miller’s algorithm to evaluate the required Bessel functions are presented and a comparison with a standard code for evaluating Fresnel integrals (Numerical Recipes program FRENEL) is provided.     

Evaluation formulas for conditional abstract wiener integrals

In this paper we consider abstract Wiener space version of conditional Wiener integrals and establish formulas for evaluating conditional abstract Wiener integrals for various classes of functions on an abstract Wiener space. We then apply our formulas to evaluate certain Wiener integrals and conditional Wiener and Yeh-Wiener integrals     

Evaluation of certain improper integrals arising in potential problems

The use of Schwarz‐Christoffel transform of the complex variable theory is shown to be well‐suited for the evaluation of certain improper integrals arising in potential problems, in closed form.     

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

A substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.     

Evaluation of singular integrals for anisotropic elastic boundary element analysis

To improve the numerical evaluation of weakly singular integrals appearing in the boundary element method, a logarithmic Gaussian quadrature formula is usually suggested in the literature. In this formula the singular function is expressed in terms of the distance between source point and field point, which is a real variable. When an anisotropic elastic solid is considered, most of the existing fundamental solutions are written in terms of complex variables. When the problems with holes, cracks, inclusions, or interfaces are considered, to suit for the shape of the boundaries usually a mapping function is introduced and then the solutions are expressed in terms of mapped complex variables. To deal with the trouble induced by the complex variables, in this study through proper change of variables we develop a simple way to improve the evaluation of weakly singular integrals, especially for the problems of anisotropic elastic solids containing holes, cracks, inclusions, or interfaces. By simple matrix expansion, the proposed method is extended to the problems with piezoelectric or magneto-electro-elastic solids. By using the dual reciprocity method, the proposed method employed for the elastostatic fundamental solution can also be applied to the elastodynamic analysis.     

Indefinite integrals involving complete elliptic integrals of the third kind

A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected.