Riemann Function Research Articles

Uniform Convergence of Some Extremal Polynomials in Domain with Corners on the Boundary

The aim of this paper is to investigate approximation properties of some extremal polynomials in , space. We are interested in finding approximation rate of extremal polynomials to Riemann function in and -norms on domains bounded by piecewise analytic curve.

Asymptotic properties of Gabor frame operators as sampling density tends to infinity

We study the asymptotic properties of Gabor frame operators defined by the Riemannian sums of inverse windowed Fourier transforms. When the analysis and the synthesis window functions are the same, we give necessary and sufficient conditions for the Riemannian sums to be convergent as the sampling density tends to infinity. Moreover, we show that Gabor frame operators converge to the identity operator in operator norm whenever they are generated with locally Riemann integrable window functions in the Wiener space.

On Riemann sums and maximal functions in

We investigate problems on a.e. convergence of Riemann sums with the use of classical maximal functions in . A theorem on the equivalence of Riemann and ordinary maximal functions is proved, which allows us to use techniques and results of the theory of differentiation of integrals in in these problems. Using this method we prove that for a certain sequence the Riemann sums converge a.e. to , .Bibliography: 23 titles.

A certain boundary-value problem with shifts in a four-dimensional space

We consider a problem with shifts in boundary conditions for the Bianchi equation in a four-dimensional space. We establish sufficient conditions which allow one to evaluate its unique solution in terms of the Riemann function.

A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.

Construction of the Riemann function for the Bianchi equation in an n-dimensional space

In this paper we obtain a more general form of conditions which enable one to construct explicitly the Riemann function for the Bianchi equation.

On Goursat and Dirichlet problems for one equation of the third order

We study the problem of the unique solvability of Goursat and Dirichlet problems for one partial differential equation of the third order. We construct a Riemann function for a linear third-order equation with a hyperbolic operator in the principal part, study some properties of the Riemann function, and then use them to prove theorems on the existence and uniqueness of a solution of the problems indicated.

Functions not first-return integrable

Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [ 0 , 1 ] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C ∞ on ( 0 , 1 ] yet fails to be first-return integrable.

Holomorphic extensions in complex fiber bundles

The main result of this paper is the existence of solutions of the equation ∂ ¯ u = ω for closed ( 0 , 1 ) -forms with compact support in a complex fiber bundle from some class. This class of bundles contains, for example, vector bundles. What is important, the solution exists for any base manifold. The solvability property implies many results about extension and representation of complex analytic and Cauchy–Riemann functions: Riemann–Hilbert type representation, Hartogs and Hartogs–Bochner phenomena, vanishing of some cohomology groups, to mention few.

The holomorphic flow of Riemann's function ξ(z)

The holomorphic flow of Riemann's xi function is considered. Phase portraits are plotted and the following results, suggested by the portraits, proved: all separatrices tend to the positive and/or negative real axes. There are an infinite number of crossing separatrices. In the region between each pair of crossing separatrices—a band—there is at most one zero on the critical line. All zeros on the critical line are centres or have all elliptic sectors. The flows for ξ(z) and cosh(z) are linked with a differential equation. Simple zeros on the critical line and Gram points never coincide. The Riemann hypothesis is equivalent to all zeros being centres or multiple together with the non-existence of separatrices which enter and leave a band in the same half plane.

Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite–Pade approximation to the exponential function, defined by p(z)e-z + q(z) + r(z) ez = O(z3n+2) as z → 0. These polynomials are characterized by a Riemann–Hilbert problem for a 3 × 3 matrix valued function. We use the Deift–Zhou steepest descent method for Riemann–Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements the recent results of Herbert Stahl.

Application of the subdomain method to the approximate solution of singular integral equations on closed integration contours

In this section, we obtain theorems on the approximation of functions of the complex variable dened on closed contours bounding a simply connected domain in the complex plane; the approximations are constructed in the space of continuous functions and in Lebesgue spaces with the use of Lozinskii interpolation polynomials. These results are used to obtain a theoretical justication of the subdomain method for the approximate solution of singular integral equations dened on closed contours of various smoothness. The solvability of the computational scheme of this method is proved by the technique in [1]. Note that issues related to the approximate solution of singular integral equations have been considered in numerous papers (see [2{10] and the bibliography therein). However, the subdomain method for singular integral equations on closed contours other than the unit circle has not been studied. Let be a closed smooth contour bounding a connected domain F + containing the origin, and let F = CnfF + [ g ,w hereC is the entire complex plane. We say that belongs to the class (m) ,w herem is a positive integer, if the Riemann function t = (w), w2 0 =f : jj =1 g, (1 )= 1, 0 (1 )= a1 (a1 > 0), of this contour has continuous :

Symmetry operators for Riemann’s method

Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in 2 variables. Chaundy’s equation, with 4 parameters, is the most general self-adjoint equation for which the Riemann function is known. Here we show that Chaundy’s equation possesses a two-dimensional vector space of second-order symmetry operators. Hence a new equivalence class of Riemann functions, admitting no first-order symmetries and obtainable only via a higher order symmetry, is found. A new 5 parameter Riemann function is then subsequently derived.

Riemann functions and the group E(1,1)

Historically Lie algebras of first-order symmetry operators have proven to be a useful method for finding equivalence classes of Riemann functions. Here this idea is extended to higher order symmetries. The approach is to seek self-adjoint linear hyperbolic partial differential equations that separate variables in more than one coordinate system under the action of the group E(1,1). The equations derived admit no nontrivial first-order operators and can only be obtained from second-order symmetry operators. Using this symmetry structure, a new equivalence class of Riemann functions can then be found.

On the Cauchy Problem in the Four-Dimensional Space

in the Euclidean space R, where α = (α1, α2, α3, α4) = (2, 1, 1, 1), the multi-indices β have four components, and we write β < α if β is obtained from α by diminishing at least one component. This equation belongs to the class of equations of the form (D +M)u = f (x1, . . . , xn) , D ≡ D, α = (m1,m2, . . . ,mn) , (2) where M is a linear differential operator with variable coefficients that contains only derivatives obtained from D by omitting at least one differentiation. Formulas for the solution of the Cauchy problem were obtained in [1, 2] in terms of the Riemann function in the threeand four-dimensional Euclidean spaces for equations of the form (1) with α = (1, 1, . . . , 1). In [3], a similar result was obtained for equations of the form (2) in R with α = (2, 1) and in R with α = (2, 1, 1). Consider Eq. (1) with coefficients aβ ∈ C and right-hand side f ∈ C. The class C, β = (β1, β2, β3, β4), contains functions such that all of their derivatives D, γ = (γ1, γ2, γ3, γ4), γi = 0, . . . , βi, i = 1, 2, 3, 4, exist and are continuous. The Riemann function R (x1, x2, x3, x4; ξ1, ξ2, ξ3, ξ4) for Eq. (1) is defined as a solution of the integral equation

Wave front sets of the Riemann function of elastic interface problems

AbstractWe obtain an inner and an outer estimates of wave front sets and analytic wave front sets of the Riemann function of elastic interface problems by using the localization method due to Wakabayashi. In our problem the outer estimate of wave front sets and analytic wave front sets of the Riemann function coincides with the inner estimate of those. The strong point of our results is to catch the lateral wave as well as the incident, the reflected, and the refracted waves. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Perturbation method in the assessment of radiation reaction in the capture of stars by black holes

This work deals with the motion of a radially falling star in Schwarzschild geometry and correctly identifies radiation reaction terms by the perturbative method. The results are: (i) identification of all terms up to first order in perturbations, second in trajectory deviation, and mixed terms including lowest order radiation reaction terms; (ii) renormalization of all divergent terms by the ζ Riemann and Hurwitz functions. The work implements a method previously identified by one of the authors and corrects some current misconceptions and results.

Power spectrum of the difference between the prime-number counting function and Riemann's function: 1/ f2?

Gamba et al. [Phys. Lett. A 145 (1990) 106] numerically estimated the power spectrum of the difference between the prime-number counting function π( x) and its approximate given by Riemann's function R( x), but did not determine the functional dependence of the spectrum on the frequency. Our numerical estimates, based on samples of the difference function R( x)− π( x) at integer x up to x=10 9, strongly suggest that the difference function has a 1/ f 2 power spectrum.

The Riemann Function for Higher-Order Hyperbolic Equations and Systems with Dominated Lower-Order Terms

The Riemann Function for Higher-Order Hyperbolic Equations and Systems with Dominated Lower-Order Terms

On spherical harmonic representation of transient waves in dispersive media

Axisymmetric transient solutions to the inhomogeneous telegraph equation are constructed in terms of spherical harmonics. Explicit solutions of the initial-value problem are derived in the spacetime domain by means of the Smirnov method of incomplete separation of variables and the Riemann formula. The corresponding Riemann function is constructed with the help of the Olevsky theorem. Solutions for some source distributions on a sphere expanding with a velocity greater than the wavefront velocity are obtained. This allows an analogous solution in the case of a circle belonging to a sphere expanding with the wavefront velocity to be written at once. Application of the scalar solution to a description of electromagnetic waves is also discussed.