Problem of correctness of the best approximation in the space of continuous functions

Let W^rH _w be the subclass of those functions of C^r[a, b], for which (f ^(r),)(), where () is a given modulus of continuity, and P_n be the space of algebraic polynomials of degree at most n and _n(f) be the polynomial of best approximation for f(x) on [a, b]. Estimates for and moduli of continuity of the operators of best approximation on W^rH _w are established. For example, if ()=, then Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 351–360, March, 1978.The author thanks S. B. Stechkin for the formulation of the problem and assistance with the article.     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

Embeddings of Twin Trees

In this article we provide sufficient conditions for when a pair of trees having a semi-codistance function can be embedded in a twin tree with codistance function extending . We use these conditions to show that given a twin tree T of bidegree (d1,d2) there exists a twin tree of bidegree (e1,e2) containing T as a substructure as long as di ei.     

Windschiefe Kongruenzflächen konstanten Dralls

In this paper we study the curves of a surface ( _T -lines) which are base curves of ruled surfaces for which the parameter of distribution O has a constant value. Moreover we assume that the ruled surfaces belong to a given congruenceT of surface tangents of . Relations are established between the _T -lines and other curves of (asymptotic lines, lines of curvature, k _T -lines). The _T -lines are used to characterize the pseudospheres and the helicoids. The _T -lines of the surfaces of revolution are determined if the congruenceT consists of the tangents of the circles of latitude.

---

---

Herrn WERNER BURAU zum 70.Geburtstag     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

Homotopy classification of some four-dimensional manifolds

In this paper there is proved a generalization of the results of Whitehead and Pontryagin on the homotopy classification of closed, simply connected four-manifolds. Let W and M be compact four-dimensional simply connected oriented four-manifolds. By q_w is denoted the intersection index on the group H₂(W).Basic Result. THEOREM (Extension). Let the groups H₁(W)and H₁(M) be finite and suppose given a homotopy equivalence fWM. In order that f can be extended to a homotopy equivalence (W,W)(M,M), it is necessary and sufficient that there should exist an isomorphism , such that the diagram is commutative and *q_m=q_m.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 164–171, 1976.     

On the Absolute Convergence of Fourier Series

The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2-periodic functions which in [0, 2] have a finite number of intervals of convexity, and whose nth Fourier coefficients are O((l/n; f)/n), where ( f) is the continuity modulus of the function f.     

Generalization of Gδ*-Diagonals and ωΔ-Spaces

In this paper we introduce the concepts of a quasi-G*-diagonal and quasi--space as generalizations of the concepts of G*-diagonal and -space respectively. It is shown that a quasi-Moore space may be characterized in terms of these concepts. As a consequence we obtain the following metrization theorems: every paracompact -space with quasi-G-diagonal is metrizable and every collectionwise normal quasi--space is metrizable.     

Some applications of the nash-williams lemma to the edge-reconstruction conjecture

It is shown that a graph whose vertices have one of four possible degrees is edgereconstructible if either 8 ord 2log ₂ 68, where andd are the minimum and the average degree of a graph respectively.     

Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures

Let Z be a random variable with the distribution function G(x) and let s be a positive random variable independent of Z. The distribution function F(x) of the scale mixture X = sZ is expanded around G(x) and the difference between F(x) and its expansion is evaluated in terms of a quantity depending only on G and the moments of the powers of the variable of the form s/gr - 1, where (> 0) and (= ±1) are parameters indicating the types of expansion. For = -1, the bound is sharp under some extra conditions. Sharp bounds for errors of the approximations of the scale mixture of the standard normal and some gamma distributions are given either by analysis ( = -1) or by numerical computation ( = 1).     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness of is indestructible under ⁺ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.     

Morera type theorems on the unit disc

, . . : , , , , .     

An inequality linking packing and covering densities of plane convex bodies

For every convex body K in R ², let (K) denote the packing density of K, i.e. the density of the tightest packing of congruent copies of K in R ², and let (K) denote the covering density of K, i.e. the density of the thinnest covering of R ² with congruent copies of K. It is shown here that 4(K)3(K) for every convex body K in R ². This inequality is the strongest possible, since if E is an ellipse, then the equality 4(E)=3(E) holds. Two corollaries are presented, and a summary of known bounds for packing and covering densities is given.     

Some remarks on the existence of densities for sums with weights

One considers the weighted sums , where X_k, are independent random variables, having no densities, and _k() are nonnegative functions, defined on the interval (0, 1). One investigates the conditions under which S(()) has a density; one proves a local limit theorem; one studies the form of the distribution in the case when ()=^k and the variables X_k take only a finite number of values.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 83–94, 1986.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

Extremal problems in the class σ(Τ)

Let L_f(r)={w=f(x), ¦z¦=r}, 1 < r < , be a level line of the function f(z) . Sharp upper bounds are obtained for the diameter of the curve L_f(r) in the class () for functions f(z)=z+₀+₁z^–1+... for which there exists a domain _f that complements the exterior of the unit disk and has conformal radius at the point w=0 satisfying the condition R(_f,0) , 0 < < 1. Also, a set of values is found for the coefficient ₁ in the class ().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 138–153, 1991.     

On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

---

---

This paper was written during the author's scholarship at the State University of Odessa in the USSR.