A note on closed ideals in rings of smooth functions

We prove that if finitely many smooth functions on a manifoldM generate a closed ideal inC (M), then they generate a closed ideal inC (M×N).     

Manifolds of Maps in Riemannian Foliations

Let (M,F) and (M,F) be two (compact or not) foliated manifolds, C _F (M, M) the space of smooth maps which send leaves into leaves. In this paper we prove that C _F (M, M) admits a structure of an infinite-dimensional manifold modeled on LF-spaces, provided that F is a Riemannian foliation or, more generally, when it admits an adapted local addition.     

New Representation Formula for Convex Functions on Separable Normed Spaces

We derive a new representation formula for lower-semicontinuous convex functions on separable normed spaces. As a consequence of this formula, we obtain a C -approximation method for convex functions which are not necessarily differentiable.     

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<.     

A relation between spline approximation and the problem of the approximation of one class by another

An investigation of the approximation in L_q(–, ) of differentiable functions whose k-th derivatives belong to L_p(–, ), by splines S_m (x) with nonfixed nodes, under the extra assumption that the norms in L_s(–, ) of theirl-th derivatives have a common bound. A relation is established with the problem of approximating functions of one class by functions of another class.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 501–510, May, 1971.     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

Distributional control and generalized analyticity

Let S be a strongly continuous, separation-preserving representation of a locally compact abelian group G in L^p(), where 1p<, and is an arbitrary measure. We show that S is uniformly bounded with respect to the L^p-and L-norms if and only if it satisfies a certain boundedness condition for distribution functions. These equivalent conditions facilitate the transference from L^p(G) to L^p() of the a.e. convergence for a wide class of sequences of convolution operators. The result unifies and generalizes various aspects of ergodic theory--in particular, the ergodic singular integral operators and ergodic Hardy spaces.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

We investigate the relationship between the norm N _G() of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G() is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G() is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G() has finite index in G.     

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

Subspaces of Maximal Operator Spaces

We explore subspaces of maximal operator spaces ( submaximal spaces) and give a new characterization of such spaces. We show that the set of n-dimensional submaximal spaces is closed in the topology of c.b. distance, but not compact. We also investigate subspaces of MAX(L ) and prove that any homogeneous Hilbertian subspace of MAX(L ₁) is completely isomorphic to R + C.     

Stability of weighted difference schemes

Estimates of stability of weighted differences in norms of Banach spaces are constructed. On the basis of these, corresponding estimates of stability in the norms of the spaces C_h and L_ph, 1 P < , are obtained for difference schemes which approximate an initial-boundary value problem for the heat equation with boundary conditions of the first, second, and third kinds. In addition, the estimates of resolvents of difference elliptic operators in C_h and L_ph, 1 P < , obtained in this article are used in an essential way.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1254–1258, September, 1990.     

A r (Ω)-Weighted Imbedding Inequalities for A-Harmonic Tensors

We prove the basic A _r ()-weighted imbedding inequalities for A-harmonic tensors. These results can be used to estimate the integrals for A-harmonic tensors and to study the integrability of A-harmonic tensors and the properties of the homotopy operator T: C (D, ^l )C (D, ^l–1).     

Asymptotics for Sums of Random Variables with Local Subexponential Behaviour

We study distributions F on [0,) such that for some T , F ^*2(x, x+T] 2F(x, x+T]. The case T = corresponds to F being subexponential, and our analysis shows that the properties for T < are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.     

Saturation on locally compact abelian groups: An extended theorem

Let G be a locally compact abelian group. The concern of the present note is to extend (for exponents p>2) the saturation theorem on G stated as Theorem 4 in [5]. The extension will be established for approximation processes (I_t)_t>0 acting on the submodule C^P(G), p]1,+[, of the convolutionM ¹(G)-module L^P(G) which consists of all functions fL^P(G) admitting as their Fourier transformsF _Gf (in the sense of the theory of quasimeasures) complex Radon measures not necessarily absolutely continuous with respect to any Haar measure on the dual group . Moreover, the relationship of the complex vector spaces C^P(G) to some other function spaces, in particular to the vector spaces B^P(G) introduced in [5], will be investigated.     

Approximate solution of a class of singular control problems

A class of singular control problems involving amplitude constraints on the controls is examined. IfL is the space of control functionsU, the control constraint setS can be identified with the unit ball inL . Now, for anyn (1, ), an analogous problem may be set up withL ⁿ forU and the unit ball inL ⁿ forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn is investigated.This research was supported by the Science Research Council of Great Britain and the Commonwealth Fund (Harkness Fellowship).     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

Martingale transforms and Hardy spaces

Summary Burkholder's martingale transforms are especially useful in studying predictable martingale Hardy spaces. Characterizations of such spaces via martingale transforms are provided. In particular, it is shown that for 0<p<, a martingale inh ^p , defined by the conditioned square function, is the martingale transform of a bmo₂ martingale with a multiplier sequence whose maximal function is inL ^p .     

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.