Extremal Subspaces Research Articles

Estimates of Eigenvalues and Approximation Numbers for a Class of Degenerate Third-Order Partial Differential Operators

In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time.

Extremal subspaces in the problem about widths of $H_{\omega}$ classes in the space of continuous functions of two variables

In the paper, we have pointed out the conditions under which the subspaces of dimensionality $n^2$ ($n=2,3,\ldots$), extremal for $H_{\omega}$ classes of continuous functions of two variables, do not exist.

On extremal subspaces for widths of classes of convolutions

We obtained the exact values of the best $L_1$-approximations of the classes $K*F$ ($r\in \mathbb{N}$) of periodic functions $K*f$ such that $f$ belongs to a given rearrangement-invariant set $F$ and $K$ is $2\pi$-periodic, not increasing oscillation, kernel, by subspaces of generalized polynomial splines with nodes at points $2k\pi / n$ ($n\in \mathbb{N}$, $k\in \mathbb{Z}$). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

On n-widths of a Sobolev function class in Orlicz spaces

This paper considers the problem of n-widths of a Sobolev function class Ω∞r determined by Pr(D) = Dσ Πj=1l(D2 −tj2I) in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov’s widths, Gelfand’s widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given.

Extremal subspaces and their submanifolds

It was proved in the paper [KM1] that the properties of almost all points of $ \mathbb{R}^{n} $ being not very well (multiplicatively) approximable are inherited by nondegenerate in $ \mathbb{R}^{n} $ (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper a.ne subspaces, and prove that the aforementioned Diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces.

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces Eq(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes WrEp(Ω) and WrEp(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.

Some general local variational principles

Local variational min-sup characterizations are presented for the real spectrum of a selfadjoint operator pencil. Instead of minimizing over all subspaces of fixed codimension as in the classical result, the new characterizations minimize over subspaces that are close to extremal subspaces. In this way, the entire real spectrum, including continuous spectrum, can be characterized.

Some General Local Variational Principles

Local variational min-sup characterizations are presented for the real spectrum of a selfadjoint operator pencil. Instead of minimizing over all subspaces of fixed codimension as in the classical result, the new characterizations minimize over subspaces that are close to extremal subspaces. In this way, the entire real spectrum, including continuous spectrum, can be characterized.

A family of extremal subspaces

A family of extremal subspaces

ON LINEAR WIDTHS OF SOBOLEV CLASSES AND CHAINS OF EXTREMAL SUBSPACES

The linear and trigonometric -diameters of the class in are calculated in this paper. For the linear diameter it is proved that, when and , This formula, together with the known results for other , finishes the solution of the problem of asymptotic computation of the linear diameters for the Sobolev classes in the one-dimensional periodic case when .Bibliography: 28 titles.

On approximation of a class by another class and extremal subspaces inL 1

Получена оценка (в опр еделенном смысле неу лучшаемая) наилучшего приближе ния в метрикеL1=L1(0,2π) 2π-перио дических функций кла сса WrHω = {f:f∈Cr,ω(f(r),δ) ≦ ω(δ)}, r = 0, 1, ..., (ω(δ) — выпуклый вверх мо дуль непрерывности) ф ункциями класса W1r+vN = {ϕ: ϕr+v−1)(t) —локально абсолютн о непрерывна, ∥ϕ(r+v∥L1≦N}, v≧2.

On extremal subspaces and approximation of periodic functions by splines of minimal defect

Н. П. Корнейчук СССР, ДНЕПРОПЕТРОВСК 320 625 ПР. ГАГАРИНА 72 ДНЕПРОПЕТРОВСКИЙ ГО СУДАРСТВЕННЫЙ УНИВЕ РСИТЕТ

A Note on Optimal Approximating Manifolds of a Function Class

Concept of n-width and extremal subspaces, first introduced by Kolmogorov, plays an important part in mathematical problems of approximation of classes of functions and in engineering problems of signal representation and reconstruction. In this short paper, explicit expressions for n-width and extremal subspaces are obtained for a class which is of some engineering importance.

Reconstructions of signals of a known class from a given set of linear measurements

This paper investigates the error in reconstructions of a signal based on a given finite set of linear measurements, and presents two schemes that, if there is available a priori knowledge of the class of signals of which the measured signal is a member, can achieve a reduction of this error beyond the best that could be done without such knowledge. The error measure used is the supremum over the class of the \mathcal{L}_2 distance between a signal and its reconstruction. The essence of the proposed reconstruction techniques is a coordinate transformation from the sampling subspace to a new reconstruction subspace known to be efficient for representation of signals of the given class. This study applies the theory of extremal subspaces and n widths of signal classes originated by Kolmogorov. Results are applied to the much studied class of time-concentrated band-limited signals. The measurement process is here assumed to be the convenient one of Nyquist rate time sampling. For this problem, plots of the error bounds and of several test functions and their reconstructions are presented, both for the proposed reconstructions, and for conventional cardinal-sampling-theorem reconstructions.