Widths of classes of periodic differentiable functions in the space L 2[0, 2π]

We obtain exact values of different n-widths for classes of differentiable periodic functions in the space L ₂[0, 2π] satisfying the constraint $$ \left( {\int_0^h {\omega _m^p \left( {f^{\left( r \right)} ;t} \right)dt} } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \Phi \left( h \right) $$ , where 0 < h < ∞, 1/r < p ≤ 2, r ∈ ?, and ω _m (f ^(r); t) is the modulus of continuity of mth order of the derivative f ^(r)(x) ∈ L ₂[0, 2π].     

Widths of classes fromL 2[0, 2π] and minimization of exact constants in Jackson-type inequalities

The problem of minimizing the constant in Jackson-type inequalities with respect to all subspaces of dimensionN, i.e., with respect to the entire set of approximating subspaces of dimensionN, is solved. It is shown that the constant in question is equal to the value of different widths of classL ₂ (δ, τ, β, γ, θ). Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 816–820, June, 1999.     

L~2[-π,π]的Riesz-Fischer序列

主要研究了L~2[-π,π]的Riesz-Fischer序列,给出了复指数序列{e~(iλ_nt_}是L~2[-π,π]的Riesz-Fischer序列的两个充分条件以及Riesz-Fischer序列的一个性质.     

On the regularity of the growth of the modulus and argument of an entire function in the metric of L p [0, 2π]

Under sufficiently general assumptions, we describe sets of entire functions f, sets of growing functions λ, and sets of complex-valued functions H from L ^p [0, 2π], p ∈ [1, + ∞], for which

The Macaulay–Lex Ideals in k[x,y, z]

In this paper, the Macaulay–lex ideals of k[x, y, z] that do not contain any x^m(m ∈ N) are determined. Furthermore, using the results of ternary Macaulay–Lex ideals, some n’ary Macaulay–Lex ideals are determined.     

L[0,r_m]中人口发展渐近展开及人口系统的可控性

在文[3]中给出自然空间 L[0,r_m](‖(?)‖_(L(0,r_m))=integral from 0 to r_m |(?)(r)|dr) 人口发展的渐近展式,它是利用[4]中关于 sharpe-Lotka 人口模型所得结果给出的。本文给出人口发展渐近展开的表达式和人口系统的可控性。讨论 L[0,r_m]空间的原因是由于人口系统的解是非负函数,它是随时间变化的人口密度分布,其范数 integral from r_m to 0 |P(r,t)|dr=integral from r_m to 0 P(r,t)dr 表示在时刻 t 的人口总数。所以在 L[0,r_m]空间中,人口发展方程有特定的意义。     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

C（[—π,π]^n）上多项式逼近的一些结果

首先给出C（[－π，π]n）上连续函数算子序列的一个逼近定理及其在多元多项式逼近方面的一个推论．其次，在所获结果的基础上，再利用高维乘积核构造非正的n维Rogosinski型核及相应的逼近算子，进而又得到了以二阶连续模为逼近阶的高维Rogosinski型逼近定理，     

Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0,π]

Our aim in this paper is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the space ${\mathbb{L}}_2^{(\alpha ,\beta )}$.     

The Classification of Macaulay–Lex Ideals in k[x,y]

In this article, all Macaulay–Lex ideals of k[x, y] are determined. For k[x, y, z], many non-Macaulay–Lex ideals and several Macaulay–Lex ideals are determined.