Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned. (Received 21 February 2000; in revised form 19 October 2000)     

On solutions of a second-order quasilinear differential system representable by Fourier series with slowly varying parameters

For a second-order quasilinear differential system whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove that, under certain conditions, there exists a particular solution with a similar structure in the case of purely imaginary roots of the characteristic equation for the matrix of coefficients of the linear part. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 654–664, May, 1998.     

Integral quadratic forms and Dirichlet series

A Dirichlet series with multiplicative coefficients has an Euler product representation. In this paper we consider the special case where these coefficients are derived from the numbers of representations of an integer by an integral quadratic form. At first we suppose this quadratic form to be positive definite. In general the representation numbers are not multiplicative. Instead we consider the average number of representations over all classes in the genus of the quadratic form. And we consider only representations of integers of the form tk ² with t square-free. If we divide the average representation number for these integers by a suitable factor, we do get a multiplicative function. Using results from Siegel (Ann. Math. 36:527–606, 1935), we derive a uniform expression for the Euler product expansion of the corresponding Dirichlet series. As a special case, we consider the standard quadratic form in n variables corresponding to the identity matrix. Here we use results from Shimura (Am. J. Math. 124:1059–1081, 2002). For 2≤n≤8, the genus of this particular quadratic form contains only one class, and this leads to a rather simple expression for the Dirichlet series, where the coefficients are just the number of representations of a square as the sum of n squares. Finally we consider the indefinite case, where we can get results similar to the definite case.     

A resonance case of the existence of solutions of a quasilinear second-order differential system, which are represented by Fourier series with slowly varying parameters

For a quasilinear second-order differential system, whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove, under certain conditions, the existence of a particular solution having a similar structure. This result is obtained in the case where the characteristic equation possesses purely imaginary roots, which satisfy a certain resonance relation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 285–288, February, 1999.     

Refinement equations with nonnegative coefficients

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σ_k∈ℤ ^Pk Ф(nx−k), where n≥2 is an integer, the coefficients {P_k} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.     

Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

On an Integral Equation in the Problem of Diffraction of a Plane Wave on a Transparent Circular Cone

The problem of diffraction on a transparent convex cone is studied. A uniqueness theorem is proved for the case where the cone is illuminated by a compact source. For a circular cone, the solution is obtained in the form of Kontorovich-Lebedev integrals and Fourier series expansions. A singular integral equation is deduced for the Fourier coefficients, and its regularization is carried out. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 101–123.     

Problem with nonlocal conditions for weakly nonlinear hyperbolic equations

For weakly nonlinear hyperbolic equations of order n, n≥3, with constant coefficients in the linear part of the operator, we study a problem with nonlocal two-point conditions in time and periodic conditions in the space variable. Generally speaking, the solvability of this problem is connected with the problem of small denominators whose estimation from below is based on the application of the metric approach. For almost all (with respect to the Lebesgue measure) coefficients of the equation and almost all parameters of the domain, we establish conditions for the existence of a unique classical solution of the problem. Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L’viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 186–195, February, 1997.     

A new explicit solution method for the diffraction through a slit – Part 2

In the paper [N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data, prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x|  ≥  1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x|  ≤  1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In the future it should be examined whether similar techniques as those which are presented in the present paper can be used to solve other canonical diffraction problems, inclusively vectorial diffraction problems.     

A new explicit solution method for the diffraction through a slit – Part 2

In the paper [N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data, prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x| ≥ 1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x| ≤ 1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In the future it should be examined whether similar techniques as those which are presented in the present paper can be used to solve other canonical diffraction problems, inclusively vectorial diffraction problems.     

Euler decompositions for theta-series of even quadratic forms

For a generating Dirichlet vector series with coefficients equal to the number of representations of a quadratic form by another one we abtain a decomposition into the product of a finite number of Dirichlet L-functions and an infinite number of matrix polynomials. The coefficients of the polynomials are the Eichler-Brandt matrices of the basis double cosets of the local orthogonal Hecke rings. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 97–113. Translated by N. Yu. Netsvetaev.     

A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

     

On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

On one class of solutions of a countable quasilinear system of differential equations with slowly varying parameters

For a countable quasilinear differential system whose coefficients are represented as Fourier series with slowly varying coefficients and frequency, we present conditions under which solutions of this system have analogous structure. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1121–1128, August, 1998.     

Elastic properties of two-dimensional two-phase composites with isotropic phases

Formal series of powers of Fourier coefficients for the effective elastic constants of a heterogeneous material (Herring’s series) are considered. It is demonstrated that, on their basis, all the known exact solutions of an elastic problem for a two-dimensional two-phase composite can be found. It is also shown how a full renormalization of the series for the inverse bulk modulus can be carried out. A general expression for Young’s modulus is deduced, leading to considerable simplifications in some special cases. All results have been obtained without any restrictions on the Fourier coefficients of local parameters of the composite.     

Application of the method of special series to the representation of solutions of the Lin-Reissner-Tsien equation

For the two-dimensional Lin-Reissner-Tsien equation, which describes nonstationary gas flows, we construct new classes of solutions with functional arbitrariness in the form of series in powers of specially chosen functions. Coefficients of such series are found successively as solutions of linear ordinary differential equations or as solutions of linear partial differential equations. The use of special series whose coefficients are determined by linear partial differential equations allowed us to satisfy two given additional boundary conditions exactly. For one class of flows, these coefficients were found in an explicit form from linear equations of the hyperbolic type; for another one, they were found from linear equations of the parabolic type. This circumstance was used to prove the convergence of such series and to study the asymptotics of the solutions constructed. We present results of numerical calculations on nonstationary transonic flow around a wedge.     

Lyapunov reducibility and stabilization of nonstationary systems with an observer

For a linear nonstationary control system with an observer, we assume that the coefficients are locally Lebesgue integrable and integrally bounded on ℝ and construct a linear feedback such that the closed-loop plant-controller system is Lyapunov reducible to the special triangular form corresponding to an independent shift of the diagonal coefficients in the original system and in the system of asymptotic estimation of the state by an arbitrary pregiven quantity. For a periodic system, we prove that the constructed controls and Lyapunov transformation are periodic. We obtain corollaries on the uniform stabilization and global controllability of the central and singular exponents of the system.     

Continuation of multiple power series in a sectorial domain

For multiple power series centered at the origin we consider the problem of its analytic continuability into a sectorial domain. The condition for continuability is formulated in terms of a holomorphic function that interpolates the series coefficients. For series in one variable this problem has been studied in the works of E. Lindelöf, N. Arakelian, and others.     

Profit maximization problem for Cobb–Douglas and CES production functions

Production functions are used to model the production activity of enterprises. In this article, we formulate the necessary and sufficient conditions of strict concavity for Cobb–Douglas and constant elasticity of substitution (CES) production functions. These conditions constitute the theoretical foundation for analyzing the profit maximization problem. An optimal solution is constructed in analytical form and some of its properties are described. Three approaches to solving the profit maximization problem are considered and their equivalence is established. For a Cobb–Douglas production function we investigate the dependence of the maximum profit on elasticity coefficients. A similar analysis is carried out also for the CES production function. The article presents a systematic and detailed discussion of the relevant topics. The topic is related to the investigation of innovation activity of enterprises. The theoretical results and the explicit analytical relationships provide a theoretical and algorithmic base for the “Planer” optimization software—a useful product for the analysis of the production activity of enterprises modeled by production function tools.