Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a differential operator

The question of the convergence of expansions in the eigenfunctions of a differential operator with discontinuous coefficients at a point x₀ of discontinuity of the coefficients is studied. Given an arbitrary function f(x) in the class L₂, a corresponding function \(\tilde f_{x_o } (x)\) is constructed which is such that at the point x₀ the eigenfunction expansion of f(x) diverges with the expansion of \(\tilde f_{x_o } (x)\) into a Fourier trigonometric series.     

A note on convergent polynomials of interpolation

In this note we define a sequence {L_n(f;x)} of interpolatory polynomials based on a system x_n={x_kn, k=1,2,…n} of nodes to be a sequence of QLIP if for every f(x)∈C[−1,1], L_n(f; x) tends uniformly to f(x) and ρ_n=1+o(1) as n→∞, where ρ_n is the ratio of the number of points in x_n, at which L_n(f;x) coincides with f(x), and the degree of L_n(f;x). Two sequences of QLIP are constructed, one of which is based on a Bernstein process and the other the Freud-Sharma's construction.     

A multi-step method for the numerical integration of ordinary differential equations

Summary In this paper we develop a multi-step method of order nine for obtaining an approximate solution of the initial value problemy'=f(x,y),y((x₀)=y ₀. The present method makes use of the second derivatives, namely, at the grid points. A sufficient criterion for the convergence of the iteration procedure is established. Analysis of the discretization error is performed. Various numerical examples are presented to demonstrate the practical usefulness of our integration method.

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Zusammenfassung In dieser Arbeit entwickeln wir eine mehrschrittige Methode der neunten Ordnung, um eine angenäherte Lösung des Anfangswertproblemsy'=f(x, y), y(x ₀)=y ₀. zu erhalten. Diese Methode bedient sich der Ableitungen zweiter Ordnung an den Schnittpunkten, d.h. . Ein hinreichendes Kriterium für die Konvergenz des Iterationsprozesses wird aufgestellt. Eine Analyse des Diskretionsfehlers ist durchgeführt. Verschiedene numerische Beispiele sollen den praktischen Nutzen unserer Integrationsmethode beweisen.

     

On Pal Type Weighted Lacunary (0 , 2 ; 0 )-Interpolation on Infinite Interval (?∞, +∞)

In this paper, we study the weighted (0, 2; 0)-interpolation on infinite interval, which means to determine a polynomial of degree 3n–1 when the function values are prescribed at two sets of points namely the zeros of H_n(x) and H_n(x) and the wieghted second derivative at the zeros of H_n(x).     

Approximation of function by Euler means

Let 蒖_n $$(q, f, x) = \frac{1}{{(1 + q)^n }}\sum\limits_{k = 0}^n {(_k^n )q^{n - k} s_k (f, x)} $$ denote the Euler means of the Fourier series of the 2π-perodic function f(x). For an integer q>0 and a function f(x)∈H^ω?C([0, 2π]), the main term of deviationf(x)-蒖_n(q, f, x) is calculated in this note. Asymptoteaally exact order 3 of decrease of the upper bound of such deviations over the class H^ω is also obtained.     

Bifurcation of periodic solutions for a semilinear wave equation

Bifurcation of time periodic solutions and their regularity are proved for a semilinear wave equation, u_tt?u_xx?λu=f(λ,x,u),x?(0,π), t?R, together with Dirichlet or Neumann boundary conditions at x = 0 and x = π. The set of values of the real parameter λ where bifurcation from the trivial solution u = 0 occurs is dense in $\text{R}$.     

Solvability of an integro-differential equation with partial derivatives on the half-axis

The paper considers the following integro-differential equation on the semi-axis: $ \frac{{\partial f(t,x)}} {{\partial t}} + \frac{{\partial f(t,x)}} {{\partial x}} + qf(t,x) = \int_0^\infty {k(x - x')f(t,x')dx'} , $ where 0 ≤ k(x) ? L ₁(?∞,∞), q = const and f(t, x) is the unknown function. This equation has significant applications in different fields of natural sciences, particularly in econometrics (see [1]), where the unknown function is considered as the density of the national income distribution function, q characterizes the mean savings etc., k(x) is the function of income rearrangement. A structural theorem of the solution is proved, whose asymptotic at infinity is found.     

Error bounds for derivative estimates based on spline smoothing of exact or noisy data

Estimates are found for the L₂ error in approximating the jth derivative of a given smooth function f by the corresponding derivative of the 2mth order smoothing spline based on an n-point sample from the function. The results cover both the case of an exact sample from f and the case when the sample is subject to some random noise. In the noisy case, the estimates are for the expected value of the approximation error. These bounds show that, even in the presence of noise, the derivatives of the smoothing splines of order less than m can be expected to converge to those of f as the number of (uniform) sample points increases, and the smoothing parameter approaches zero at a rate appropriately related to m, n, and the order of differentiability of f.     

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

ОптИМАльНыЕ кВАДРАт УРНыЕ ФОРМУлы Дль клА ссОВ ФУНкцИИ с ИНтЕгРИРУЕ МОИ ВL p r-ОИ пРОИжВОДНОИ

On the classW ^r L _p (1≦p≦∞;r=1, 2,…) of 1-periodic functions ?(x) having an absolutely continuous (r? l)st derivative such that $$\parallel f^{(r)} \parallel _{L_p } \leqq 1 (\parallel f^{(r)} \parallel _{L_\infty } = vrai \sup |f^{(r)} (x)|)$$ vrai sup ¦?(^r)(x)¦) an optimal quadrature formula of the form (0 ≦? ≦r?1, 0 ≦x ₀ < x₁ <…< x_m ≦ 1) is found in the cases ?=r?2 and ?=r? 3 (r=3, 5, …). An exact error bound is established for this formula. The statements proved forW ^r L _p allowed us also to obtain, under certain restrictions posed on the coefficientsp _kl, and the nodesx ₀ andx _m, optimal quadrature formulae for the classes $$W_0^r L_p = \{ f:f \in W^r L_p , f^{(i)} (0) = 0 (i = 0,1,...,r - 2)\} $$ and $$W_0^r L_p = \{ f:f \in \tilde W^r L_p , f^{(i)} (0) = f^{(i)} (1) = 0 (i = 0,1,...,r - 2)\} $$ for the same values ofp andr as above.     

Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on by-passing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable functionf(x + h) =E _α (h ^α D _x ^α )f(x).     

The Range of Possible Values of f(x)

The purpose of this paper is to explain how to compute the rangeof possible values of a function of one variable, f(x), givenvalues of the function at n distinct points x₁ < x₂ <... < x_M–1 < x_M, and given a finite bound on thekth derivative of f: ||f^(k)|| L, 1 k n.     

Some notes on Hermite-Fejer type interpolation

Let S_n(f,x) be the Hermite-Fejér type interpolation satisfying S_n(f,x_k)=f(x_k), S′_n(f,x_k)=0, k=1,2,…,n and S_n(f,y_i)=f(y_i), j=1,2,…,m. For m=0, let H_n(f,x)≔S_n(f,x). This paper investigates relationship between S_n(f,x) and H_n(f,x), as well as, the saturation of S_n(f,x).     

Second-order necessary and sufficient optimality conditions for minimizing a sup-type function

In this paper, we give second-order necessary and sufficient optimality conditions for a minimization problem of a sup-type functionS(x)=sup{f(x,t);t T}, whereT is a compact set in a metric space and f is a function defined on ⁿ ×T. Our conditions are stated in terms of the first and second derivatives of f(x, t) with respect tox, and involve an extra term besides the second derivative of the ordinary Lagrange function. The extra term is essential when {f(x,t)} _t forms an envelope. We study the relationship between our results, Wetterling [14], and Hettich and Jongen [6].     

Oscillation criteria for sublinear differential equations with damping

We present some criteria for the oscillation of the second order nonlinear differential equation [a(t)ψ(x(t))x'(t)]' + p(t)x'(t) + q(t)f (x(t)) =0, t≧t ₀> 0 with damping where a∈C ¹ ([t ₀,∞)) is a nonnegative function, p, q∈ C([t ₀,∞)) are allowed to change sign on [t ₀,∞), ψ, f∈C(R) with ψ(x) ≠ 0, xf(x)/ψ(x) > 0 for x≠ 0, and ψ, f have continuous derivatives on R{0} with [f(x) / ψ(x)]' ≧ 0 for x≠ 0. This criteria are obtained by using a general class of the parameter functions H(t,s) in the averaging techniques. An essential feature of the proved results is that the assumption of positivity of the function ψ(x) is not required. Consequently, the obtained criteria cover new classes of equations to which known results do not apply.     

Platitude d'une adherence schematique et lemme de Hironaka generalise

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied:

1. O _Y,y is reduced.
2. f is universally open at the generic points of the components of X_y which contain x.
3. For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to X_y, we have dim_x, (X_y')=dim_x(X_y)=d.
4. X_y is reduced at the generic points of the components of X_y which contain x and (X_y)_red is geometrically normal over K(y) in x.

Then there exist an open neighbourhood U of x in X and a subscheme U₀ of U which have the same underlying space as U such that f₀:U₀\arY is normal (i.e. f₀ is a flat morphism whose geometric fibers are normal).     

Behavior on the real line of entire functions represented by Dirichlet series

Conditions are found under which for an entire function f represented by a Dirichlet series with finite Ritt order on some sequence (x_k), 0 < x_k , as k one has ¦f(x_k)¦=M_t((1 + 0(1) x_k), M_f(x)=sup {¦ f (z) ¦:Re z x}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 265–269, February, 1991.     

Optimal accuracy approximation of functions and their derivatives

We consider the approximation of the function (x) and its derivative '(x) on [a, b] given that (x)C _2,N, i.e., belongs to the class of functions f(x) that satisfy the conditions f(x)L, f(x_i)=y_i, i=1,,N, where L and y_i are given real numbers and x_i are the nodes of an arbitrary grid, a=x₁<x₂<<X_N=b. A solution algorithm on the class of functions C_2,L,N is proposed which has optimal accuracy with a constant not exceeding 2. A bound on the approximation error of the function and its derivative is derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 57–61, 1985     

Implicit function theorems for generalized equations

We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 f(x, y) + F(x), wheref is a function andF is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution toz g(x) + F(x), whereg strongly approximatesf in the sense of Robinson. In particular, the inverse map (f + F)^–1 has a local selection which is Lipschitz continuous nearx ₀ and Fréchet (Gateaux, Bouligand, directionally) differentiable atx ₀ if and only if the linearization inverse (f (x ₀) + f (x₀) (× – x₀) + F(×))^–1 has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.This work was supported by National Science Foundation Grant Number DMS 9404431.