Approximation of Classes B p,θ r by Linear Methods and Best Approximations

We investigate problems related to the approximation by linear methods and the best approximations of the classes , 1 p in the space L .     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Tensor products and Banach ideals of p-compact operators

A study is made of the norm w_p (1 p ) on the tensor product of two Banach spaces E and F. It is shown that w_p is a tensor norm, and a representation is deduced for the elements in the completion of E F equipped with w_p. Finally it is shown that the w_p-nuclear operators in the sense of Grothendieck [3] coincide with those operators factoring compactly through^p (if 1 p ) or Co (if p=), with related equalities concerning the idea1 norms.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

On the zetafunction of an arithmetical semigroup

The Zetafunction Z of an additive, multiplicative arithmetical semigroup is a power series with radius of convergence (01), a Dirichlet series with abscissa of convergence (0), respectively.Conditions are given which ensure that >0 and Z()=,< and Z()= hold true, respectively. Jürgen Neukirch zum Gedächtnis Mathematics Subject Classification (2000):11N45, 05A16.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

Cones in the space Lp (1 ≤ p ≤ ∞) and minimax problems of mathematical statistics

The structure of cones in the spaces L_p(l p ) and their application to minimax problems of statistical assumptions are considered.Translated from Staticheskie Metody, pp. 28–30, 1978.     

Some Problems for Cubic Nonlinear Equations on a Half-Line

The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q _t +q _xxx +6q ² q _x =0, 0x<, –<t<,q _t +q _xxx –6q ² q _x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L ₁(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q _t +q _xxx –6|q|² q _x =0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<.     

Sets of absolute convergence of double trigonometric series

We obtain a sufficient condition for a set of plane measure zero to be a set of absolute convergence (an A.C.-set) for a double trigonometric series. Specifically, let y=f(x) (0 x2) be a smooth curve and let inif (x) t <. Then, any set of positive linear measure lying on this curve is an A.C.-set.Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 473–480, May, 1972.In conclusion, I am indebted to E. M. Nikishin for suggesting the problem.     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

The Relations between Volume Ratios and New Concepts of GL Constants

In this paper we investigate a property named GL(p,q) which is closely related to the Gordon-Lewis property. Our results on GL(p,q) are then used to estimate volume ratios relative to _p, 1相似文献   

On the Approximation by Modified Interpolation Polynomials in Spaces L p

We consider certain modified interpolation polynomials for functions from the space L _p[0, 2], 1 p . An estimate for the rate of approximation of an original function f by these polynomials in terms of its modulus of continuity is obtained. We establish that these polynomials converge almost everywhere to f.     

On a Type of Mixed Norm Spaces

In this paper the mixed norm spaces L(B,p,q) and their duals are investigated. In the case p,q < it is proved that the dual of L(B,p,q) is L(B,p,q), where p-1 + p-1 = 1 and q-1 + q-1 = 1. For p = 2 and q = an isometric isomorphism is discussed between the mixed norm space L(B,2,) and L(B,2), the L-space of 2-valued functions. Here a measurability theorem is proved for 2-valued functions. The dual of an important subspace of L(B,2,) is characterized as a space of vector measures. Finally, as an application we show that if B is finitely generated then the dual of L(B,2,) is L(B,2,1).     

Method for symmetrization and estimation of solutions of the Neumann problem for the equation of a porous medium in domains with noncompact boundary for infinitely increasing time

We consider the initial boundary-value Neumann problem for the equation of a porous medium in a domain with noncompact boundary. By using a symmetrization method, we obtain exactL _p-estimates, 1p, for solutions as t.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 147–157, February, 1995.     

On pathwise behavior of multiserver queues

In this paper, by exploiting recent results on the pathwise behavior of the workload process in single server, work conserving queues of theG/G/1/ type, we show that the workload of multiserver, work conserving queues ofG/G/m/ (m<) (andG/G/) queues satisfies an o(t) growth condition, provided that the time average of the work brought into the system is less thanm form < (and finite form=).     

Diameters of certain classes of functions analytic in the unit disc. I

For specific classes of functions which are analytic on the unit disc, precise values are obtained for certain types of diameters in the Bergman space H q, q 1, and in the space B(p, q, ), 0 < p < q ; q, 1, studied by M. I. Gvaradze.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 873–881, July, 1990.     

Small into isomorphisms onL ∞ spaces

IfT is an isomorphism ofL (A, ) intoL (B, ) which satisfies the condition T T ^–11+, where (A, ) is a -finite measure space, thenT/T is close to an isometry with an error less than 4.     

Gaussian Version of a Theorem of Milman and Schechtman

Using Gordon's inequalities, we give a short proof of the existence of an embedding T:l2k to ln such that ||T||||T-1|| c k/ln(1+n/k). In the same way, we give a new proof of a theorem of Milman and Schechtman (1995).     

Green's theorem from the viewpoint of applications

Making use of a line integral defined without use of the partition of unity, Green's theorem is proved in the case of two-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces W^1,p () H^1,p () (1 p < ).     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.