Proceedings of the Steklov Institute of Mathematics最新文献

Abstract

It is known that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error measured in the square norm. In this paper we demonstrate how known results on universal sampling discretization of the uniform norm and recent results on universal sampling representation allow us to provide good universal methods of sampling recovery for anisotropic Sobolev and Nikol’skii classes of periodic functions of several variables. The sharpest results are obtained in the case of functions of two variables, where the Fibonacci point sets are used for recovery.

Abstract

We study the stability of solutions to nonlinear equations in finite-dimensional spaces. Namely, we consider an equation of the form (F(x)=overline{y}) in the neighborhood of a given solution (overline{x}). For this equation we present sufficient conditions under which the equation (F(x)+g(x)=y) has a solution close to (overline{x}) for all (y) close to (overline{y}) and for all continuous perturbations (g) with sufficiently small uniform norm. The results are formulated in terms of (lambda)-truncations and contain applications to necessary optimality conditions for a constrained optimization problem with equality-type constraints. We show that these results on (lambda)-truncations are also meaningful in the case of degeneracy of the linear operator (F'(overline{x})).

Using R. Wilson’s recent results, we prove the existence of triples ((mathfrak{X},G,H)) such that (mathfrak{X}) is a complete (i.e., closed under taking subgroups, homomorphic images, and extensions) class of finite groups, (G) is a finite simple group, and (H) is its (mathfrak{X})-maximal subgroup nonpronormal in (G). This disproves a conjecture stated earlier by the second author and W. Guo.

For a function (f(z)) analytic in a disk, a method of approximate reconstruction from known (or arbitrarily specified) boundary values of its real part (under the condition of its continuity) using interpolation wavelets is proposed; the method is easy to implement numerically. Despite the fact that there are known exact analytical formulas for solving this problem, the explicit formulas for approximating the function (f(z)) proposed here are much easier to apply in practice, since the previously known exact formulas lead to the necessity to apply numerical integration methods when calculating convolutions of functions with Poisson or Schwartz kernels. For the approximations used in this paper, effective upper bounds are obtained for the error of approximation of functions analytic in the disk by interpolation wavelets in the spaces (L_{p}(0,2pi)) for any (pgeq 2). These estimates can be used to find the parameters of the wavelets from a desired accuracy of recovering the function (f(z)). Note that if the real part of (f(z)) is continuous on the boundary of the disk, the continuity of (f(z)) in the closure of the disk cannot be guaranteed; that is why it is impossible to estimate the approximation error for (f(z)) in the uniform metric (for (p=infty)) in the general case.

We discuss the properties of the generalized translation operator generated by the system of functions (mathfrak{S}={{(sin kpi x)}/{(kpi x)}}_{k=1}^{infty}) in the spaces (L^{q}=L^{q}((0,1),{upsilon})), (qgeq 1), on the interval ((0,1)) with the weight (upsilon(x)=x^{2}). We find an integral representation of this operator and study its norm in the spaces (L^{q}), (1leq qleqinfty). The translation operator is applied to the study of Nikol’skii’s inequality between the uniform norm and the (L^{q})-norm of polynomials in the system (mathfrak{S}).

We consider a problem of convex parametric programming in which the objective function and the constraint functions are convex functions of an external parameter. Computational procedures are suggested for finding the maximum and minimum values of the optimal value function and for finding inner and outer approximations to the set of parameters for which the problem is consistent. All procedures are based on the application of support functions. Illustrative examples are provided.

Earlier, to confirm that one of the possibilities for the structure of vertex stabilizers of graphs with projective suborbits is realizable, we announced the existence of a connected graph (Gamma) admitting a group of automorphisms (G) which is isomorphic to Aut((Fi_{22})) and has the following properties. First, the group (G) acts transitively on the set of vertices of (Gamma), but intransitively on the set of (3)-arcs of (Gamma). Second, the stabilizer in (G) of a vertex of (Gamma) induces on the neighborhood of this vertex a group (PSL_{3}(3)) in its natural doubly transitive action. Third, the pointwise stabilizer in (G) of a ball of radius 2 in (Gamma) is nontrivial. In this paper, we construct such a graph (Gamma) with (G=mathrm{Aut}(Gamma)).

We consider the ill-posed problem of finding the position of the discontinuity lines of a function of two variables.It is assumed that the function is smooth outside the lines of discontinuity but has a discontinuity of the first kind on the line.At each node of a uniform grid with step (tau), the mean values of the perturbed function on a square with side (tau) are known.The perturbed function approximates the exact function in the space (L_{2}(mathbb{R}^{2})). The perturbation level (delta) is assumedto be known. Previously, the authors investigated (accuracy estimates were obtained) global discrete regularizing algorithmsfor approximating the set of lines of discontinuity of a noisy function provided that the line of discontinuity of the exact functionsatisfies the local Lipschitz condition. In this paper, we introduce a one-sided Lipschitz condition and formulate a new, widercorrectness class. New methods for localizing discontinuity lines are constructed that work on an extended class of functions.A convergence theorem is proved, and estimates of the approximation error and other important characteristics of the algorithmsare obtained. It is shown that the new methods determine the position of the discontinuity lines with guarantee in situationswhere the standard methods do not work.

The following results are proved. Let (d) be a natural number, and let (G) be a group of finite even exponent such that each of its finite subgroups is contained in a subgroup isomorphic to the direct product of (m) dihedral groups, where (mleq d). Then (G) is finite (and isomorphic to the direct product of at most (d) dihedral groups). Next, suppose that (G) is a periodic group and (p) is an odd prime. If every finite subgroup of (G) is contained in a subgroup isomorphic to the direct product (D_{1}times D_{2}), where (D_{i}) is a dihedral group of order (2p^{r_{i}}) with natural (r_{i}), (i=1,2), then (G=M_{1}times M_{2}), where (M_{i}=langle H_{i},trangle), (t_{i}) is an element of order (2), (H_{i}) is a locally cyclic (p)-group, and (h^{t_{i}}=h^{-1}) for every (hin H_{i}), (i=1,2). Now, suppose that (d) is a natural number and (G) is a solvable periodic group such that every of its finite subgroups is contained in a subgroup isomorphic to the direct product of at most (d) dihedral groups. Then (G) is locally finite and is an extension of an abelian normal subgroup by an elementary abelian (2)-subgroup of order at most (2^{2d}).

A smooth nonconvex optimization problem is considered, where the equality and inequality constraints and the objective function are given by DC functions. First, the original problem is reduced to an unconstrained problem with the help of I. I. Eremin’s exact penalty theory, and the objective function of the penalized problem also turns out to be a DC function. Necessary and sufficient conditions for minimizing sequences of the penalized problem are proved. On this basis, a “theoretical method” for constructing a minimizing sequence in the penalized problem with a fixed penalty parameter is proposed and the convergence of the method is proved. A well-known local search method and its properties are used for developing a new global search scheme based on global optimality conditions with a varying penalty parameter. The sequence constructed using the global search scheme turns out to be minimizing in the “limit” penalized problem, and each of its terms (z^{k+1}) turns out to be an approximately critical vector for the local search method and an approximate solution of the current penalized problem ((mathcal{P}_{k})triangleq(mathcal{P}_{sigma_{k}})). Finally, under an additional condition of “approximate feasibility,” the constructed sequence turns out to be minimizing for the original problem with DC constraints.