Trigonometric Approximation of SO(N) Loops

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a H?lder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order O(n^-a+e)\mathcal{O}(n^{-\alpha+\epsilon}) for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.     

Approximation of functions by (N,p) operator

The present paper deals with finding of constant occurring in the order of approximation of the function ofLipα(0<α<1) class by using (N, p) operator. Obviously the factor 1/(1−α) becomes large when α is close to 1. We have shown the role of this factor in the constant of approximation.     

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗_q on certain quasi-Banach spaces. We prove that the approximation spaces A^α_q(L₂) and A^α_q(H¹) equal tensor products of Besov spaces B^α_q(L_q), e.g., A^α_q(L₂([0,1]^d)) = B^α_q(L_q([0,1])) ⊗_q · ⊗_q B^α_{q · ·}(L_q([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.     

Convergence of Numerical Approximation for Jump Models Involving Delay and Mean-Reverting Square Root Process

The mean-reverting square root process with jump has been widely used as a model on the financial market. Since the diffusion coefficient in the model does not satisfy the linear growth condition and local Lipschitz condition, we can not examine its properties by traditional techniques. To overcome the difficulties, we develop several new techniques to examine the numerical method of jump models involving delay and mean-reverting square root. We show that the numerical approximate solutions converge to the true solutions. Finally, we apply the convergence to examine a path-dependent option price and a bond in the financial pricing.     

Approximation of the lauricella hypergeometric functionsF D (N) by branched continued fractions

Applying recursion relations for the Lauricella hypergeometric functions F _D ^Nl , we construct an expansion of a ratio of these functions in branched continued fractions. We study the convergence of the resulting expansion in the case of real parameters. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 70–74.     

关于模N的原根的逆及其整除性

本文研究了模ｎ的原根、逆及其整除性，并给出了一个关于ｍｉｎ（ａ－ａ＝１·ｋ１０、ａａＪ“＋ａ－ａ）的较强的渐近公式。     

Approximation of functions in the spaceL 2(ℝ N ;exp(−¦x¦2))

Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 3–19, January, 1995.     

关于模N的原根与它的逆的差的奇数幂及其推广

本文研究了模ｎ的原根ａ与它的逆ａ差的奇数幂分布性质，并给出了(?)的一个渐近公式．     

Approximation of the reproducing kernel

One discusses the asymptotic behavior of the reproducing kernel of the space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 195–198, 1980.     

Approximation of the Lerch Zeta-Function

We consider uniform in parameters approximations of the Lerch zeta-function by Dirichlet polynomials. It allows us to obtain uniform in parameters bounds in the critical strip.     

Approximation of the viability kernel

We study recursive inclusionsx ⁿ⁺¹ G(x ⁿ ). For instance, such systems appear for discrete finite-difference inclusionsx ⁿ⁺¹ G (x ⁿ) whereG :=1+F. The discrete viability kernel ofG , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx ⁿ⁺¹ (xⁿ) where (x) =x + F(x) + (ML/2) ². Secondly, we show that it can be approached by finite viability kernels associated withx _h ⁿ⁺¹ ( (x _h ⁿ⁺¹ ) +(h) X _h .     

Uniform Approximation of Functions with Discrete Approximation Functionals

Hoffman, Kouri, and collaborators have calculated nonrelativistic quantum scattering amplitudes by numerically evaluating Feynman path integrals. They observed that the errors introduced by their numerical scheme were uniform in coordinate space, implying that their scheme accurately reproduces both the shape and the phase of functions. Furthermore, they observed that the size and the uniform nature of the errors were preserved when the functions were allowed to evolve in time under the action of the kinetic energy operator. In this paper it is established that these observed properties of the errors are not numerical artifacts but follow from analytical properties of a general class of approximations that include those of Hoffman, Kouri, and collaborators as a special case.     

On the Relative Approximation Error of the Generalized Pareto Approximation for a High Quantile

Let F be a distribution function in the domain of attraction of an extreme value distribution . In case 0 and F has an infinite end-point, we study the asymptotic behaviour of the relative approximation error of a high quantile such that , where the order tends to 0. We use the approximation of the excesses over a high threshold u by a Generalized Pareto distribution. We give sufficient conditions under which tends to 0.AMS 2000 Subject Classification Primary—60G70, Secondary—62G20, 62G32     

多元周期函数的一类逼近及逼近阶估计

本文讨论了多元周期函数的一类逼近,得到一个稠密性的充要条件并构造出适当的逼近函数,给出了逼近阶的精确估计．     

论可微函数的共单调逼近和共凸逼近

对有限区间上可微函数借助于代数多项式的共单调逼近和共凸逼近的逼近度估计建立了更为精确的Jackson型不等式,扩充和改进了近期的一些结果。     

Probabilistic Approximation of the Evolution Operator

A method for approximation of the operator e^?itH, where \(H = - \frac{1}{2}\frac{{{d^2}}}{{d{x^2}}} + V(x)\), in the strong operator topology is proposed. The approximating operators have the form of expectations of functionals of a certain random point field.