Integral representations and summations of the modified Struve function

It is known that the Struve function H _ν and the modified Struve function L _ν are closely connected to the Bessel function of the first kind J _ν and to the modified Bessel function of the first kind I _ν and possess representations through higher transcendental functions like the generalized hypergeometric ₁ F ₂ and the Meijer G function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for L _ν (x). In this paper firstly, we obtain various another type integral representation formulae for L _ν (x) using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schlömilch series built by I _ν (x) and L _ν (x) which are connected by a Sonin–Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich–Wagner line integral expressions are derived for the Bessel function of first kind J _ν and for an associated generalized Schlömilch series.     

The evaluation of Bessel functions via exp-arc integrals

A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J.M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an “exp-arc” integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.     

On coefficients of Kapteyn-type series

Quite recently Jankov and Pogány [JANKOV, D.—POGÁNY, T. K.: Integral representation of Schlömilch series, J. Classical Anal. 1 (2012) 75–84] derived a double integral representation of the Kapteyn-type series of Bessel functions. Here we completely describe the class of functions Λ = {α}, which generate the mentioned integral representation in the sense that the restrictions $\alpha |_\mathbb{N} = (\alpha _n )_{n \in \mathbb{N}} $ is the sequence of coefficients of the input Kapteyn-type series.     

Estimates of the uniform modulus of continuity for Bessel potentials

In the theory of function spaces it is an important problem to describe the differential properties for the classical Bessel and Riesz potentials as well as for their generalizations. Bessel potentials are determined by the convolutions of functions with Bessel-MacDonald kernels G _α. In this paper we characterize the integral properties of functions by their decreasing rearrangements. The differential properties of potentials are characterized by their modulus of continuity of order k in the uniform norm. Estimates of such type were obtained by A. Gogatishvili, J. Neves, and B. Opic in the case k > α. Here, we remove this restriction and obtain the results for all values k ∈ N. We find order-sharp estimates from above for moduli of continuity and construct the examples confirming the sharpness. On the base of these results we obtain the order-sharp estimates for continuity envelope function in the space of potentials, and give estimates for the approximation numbers of the embedding operator.     

Bessel functions for GL 2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL ₂ over a finite field first to develop their formal properties and introduce the idea that the γ-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL ₂(?) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL ₂ over a p-adic field and the relation between γ-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of γ-factors under highly ramified twists.     

Fast integral equation solver for Maxwell’s equations in layered media with FMM for Bessel functions

The paper presents a new fast integral equation solver for Maxwell’s equations in 3-D layered media. First, the spectral domain dyadic Green’s function is derived, and the 0-th and the 1-st order Hankel transforms or Sommerfeld-type integrals are used to recover all components of the dyadic Green’s function in real space. The Hankel transforms are performed with the adaptive generalized Gaussian quadrature points and window functions to minimize the computational cost. Subsequently, a fast integral equation solver with O(N _z ² N _x N _y log(N _x N _y )) in layered media is developed by rewriting the layered media integral operator in terms of Hankel transforms and using the new fast multipole method for the n-th order Bessel function in 2-D. Computational cost and parallel efficiency of the new algorithm are presented.     

An Index Integral and Convolution Operator Related to the Kontorovich-Lebedev and Mehler-Fock Transforms

We deal with an index integral involving the product of the modified Bessel functions and associated Legendre functions. It was discovered by Ferrell (Nucl Instrum Methods Phys Res B 96:483?C485, 1995) while comparing solutions of the Laplace equation in different coordinate systems in his study of the so-called surface plasmons in various condensed matter samples. This integral is quite interesting from the pure mathematical point of view and it is absent in famous reference books for series and integrals. We give a rigorous proof of this formula and discuss its particular cases. We also construct a convolution operator associated with this integral, which is related to the classical Kontorovich-Lebedev and Mehler-Fock transforms. Mapping properties and the norm estimates in weighted L _p -spaces, 1 ?? p ?? 2 are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

The traces of Bessel potentials on regular subsets of Carnot groups

We prove the direct theorem on the traces of the Bessel potentials L _p ^α defined on a Carnot group, on the regular closed subsets called Ahlfors d-sets. The result is convertible for integer α, i.e., for the Sobolev spaces W _p ^α (the converse trace theorem was proven in [24]). This theorem generalizes A. Johnsson and H. Wallin’s results [13] for Sobolev functions and Bessel potentials on the Euclidean space.     

Application of pseudo‐trigonometry to Bessel series and integrals of Bessel functions

In this article, an extension of the Laplace transform of J_n (t) to pseudo‐trigonometric function is discussed. We are seeking elementary functions expressed by Bessel series. It is shown that the result is applicable to the solution of the first‐order differential equation. The expression of modified Bessel integral formulas in pseudo‐trigonometric function is also discussed.     

Approximate formulas for moderately small eikonal amplitudes

We consider the eikonal approximation for moderately small scattering amplitudes. To find numerical estimates of these approximations, we derive formulas that contain no Bessel functions and consequently no rapidly oscillating integrands. To obtain these formulas, we study improper integrals of the first kind containing products of the Bessel functions J₀(z). We generalize the expression with four functions J₀(z) and also find expressions for the integrals with the product of five and six Bessel functions. We generalize a known formula for the improper integral with two functions J_υ (az) to the case with noninteger υ and complex a.     

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

On the approximate solution of the abel integral equation with discontinuous solution

The classical Abel integral equation with discontinuous solution is solved numerically by replacing the inhomogeneous termg(x) (known on some finite subsetZ _N of the interval of integration) by a suitable linear familyψ(β, x) of continuous functions. The choice of these functions will be governed by two criteria: they are to reflect the discontinuous behavior of the exact solution of the integral equation, and they are to be such that the problem of finding a best (discrete) Chebyshev approximation tog onZ _N possesses a unique solution.     

Равносходимость разложений в кратный ряд и интеграл фурье, «прямоугольные частичные суммы» которых рассматриваются по некоторой подпоследовательности

In this paper we investigate the problem of the equiconvergence on T ^N = [-π, π) ^N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ L _p (T ^N ) and g ∈ L _p (? ^N ), where p > 1, N ≥ 3, g(x) = f(x) on T ^N , in the case when the “rectangular partial sums” of the indicated expansions, i.e.,– _n (x; f) and J _α(x; g), respectively, have indices n ∈ ? ^N and α ∈ ? ^N (n _j = [α _j ], j = 1,...,N, [t] is the integer part of t ∈ ?¹), in those certain components are the elements of “lacunary sequences”.     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

Separation of Eigenvalues of the Wave Equation for the Unit Ball in RN

It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of orders x ≥ ½ (as well as for x = 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. If W_l denotes the lth positive root for some fixed order x, the minimum gap property assures that {exp(±iw_lt|l = 1, 2,...} form a Riesz basis in L²(0, τ) for τ > 2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary values.     

On hyperfinite factors of type III0 and Krieger's factors

A factor M is the cross product of an abelian von Neumann algebra by a single automorphism iff there exists an increasing sequence of normal conditional expectations of M onto finite-dimensional subalgebras N_k with (U N_k)^? = M. Assuming the uniqueness of the hyperfinite factor of type II_∞, we prove then that any hyperfinite factor of type III₀ is the cross product of an abelian von Neumann algebra by a single automorphism.     

Commutation properties of automorphism groups and mappings of von Neumann algebras

We prove a commutation theorem for point ultraweakly continuous oneparameter groups of automorphisms of von Neumann algebras. If α_t, is such a group in Aut($\text{R}$) for a von Neumann algebra $\text{R}$, we show the equivalence of the following three conditions on an ultraweakly continuous linear transformation μ: $\text{R}$ → $\text{R}$: (a) μ commutes weakly with the infinitesimal generator for α_t; (b) μ ° α_t = α_t ° μ, t ∈ R; and (c) μ leaves invariant each of the spectral subspaces associated with α_t. A simple condition which is applicable when μ is an automorphism is pointed out.     

Application of Hankel transforms to boundary value problems of water flow due to a circular source

With the aid of Hankel transform technique, we obtain close-form solutions for discontinuous boundary-condition problems of water flow due to a circular source, which located on the upper surface of a confined aquifer. Owing to difficult evaluations of the original solutions that are in a form of an infinite range integral with a singular point and Bessel functions in integrands, we adopt two numerical algorisms to transform the original solutions as a series form for convenient practical applications. We apply the solutions in series form to numerical examples to analyze the characteristics of the flow in the confined aquifers subjected to pumping or recharge. By numerical examples, it indicates that: the drawdown will reduce with the increase of the layer thickness and the distance from the center of a circular source when pumping in a region with a finite thickness and a finite width; two algorisms for closed-form solutions of an infinite range integral have almost the same results, but the second algorism is superior for a faster convergence; in a semi-infinite confined aquifer, the drawdown due to a constant pumping rate Q and uplift due to recharge by a given hydraulic head s₀ will both decrease with the increase of K_r/K_v; however, the radius r₀ of the circular source has a reverse influence on the drawdown and the uplift, i.e., the drawdown decrease with the increase of r₀, while the uplift increase with r₀.