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1.
Let [E(Ω)]p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in Rn with itself p-times. The finitely generated submodules of [E(Ω)]p are of the form im(F) where F: [E(Ω)]q → [E(Ω)]p is a p × q matrix of infinitely differentiable functions on Ω. Let r = max{rank(F(x)): x ? Ω}. The main results of the present paper are that for Ω ? Rn, if the finitely generated submodule im(F) is closed in [E(Ω)]p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? R1 the converse also holds.  相似文献   

2.
Let R denote the real line and L(R), the class of all Borel measurable L-functions of R. Let S ≠ {0} or φ, be a linear subspace of L(R) which is (i) translation invariant, (ii) weak1-closed, (iii) self-adjoint, i.e., f?S implies f?S, and (iv) an algebra. Then either (a) S = all constant functions in L; or (b) S = L; or (c) there is a unique c > 0 such that S consists of all L-functions which are periodic of period c.Extension of the above characterization of periodic subalgebras of L to LCA groups are presented. Also it is shown that the above characterization is in various ways best possible.  相似文献   

3.
Every Gaussian Radon measure (nonsymmetric) has the Riesz property. On R, a shift μ0 1 σb of the law μ0 of the Bernoulli sequence has the Riesz property iff b ? l2or b ? R0.  相似文献   

4.
Let H(Δ) denote the Banach algebra of bounded analytic functions on the open unit disc, let M denote its maximal ideal space, and let ? denote its Shilov boundary. D. J. Newman has shown that a homomorphism ? in M will be in ? if and only if ? is unimodular on all Blaschke products. We answer a question of K. Hoffman by showing that ? will be in ? if and only if ? is unimodular on every Blaschke product whose zero set is an interpolating sequence. Our method is based on a construction due to L. Carleson, originally developed for the proof of the Corona theorem.  相似文献   

5.
Let B be the open unit ball of Cn, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have ¦u¦ = 1 a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let ¦X¦ be the norm closure, in L = L(S), of X. Some results: set I is dense in the unit ball of H(B) in the compact-open topology. On S, Q?Q is weak1-dense in L, ¦Q? does not contain H, C(S) ?¦Q?H¦ ≠ ¦H?H¦ ≠ L. (When n = 1, ¦Q¦ = Hand ¦Q?Q¦ = L.) Every unimodular ? ? L is a pointwise limit a.e. of products uv?, u ? I, ν ? I. The zeros of every ? ? 0 in the ball algebra (but not of every H-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if ∥?∥ < 1. However, ?u cannot be bounded in B if u ? I is non-constant.  相似文献   

6.
Let Ol?L(S, μ) be a maximal abelian subalgebra of the factor F on separable Hilbert space with modular involution J. (OlJOlJ)″ is represented naturally as L(S × S, λ). If Takesaki's unitary equivalence relation R ? S × S is not λ-null, it is a measure groupoid. If it is conull, and (OlJOlJ)″ is maximal abelian, F and Ol are reconstructed by the σ-left regular representation procedure. Examples show that these hypotheses are not always satisfied. An application shows that the L spectrum of a properly infinite ergodic transformation is null with respect to the L2 spectrum.  相似文献   

7.
LetV ? 0, V?C0(Rv) with v ? 3 be such that H = ?12Δ + V ? 0 but for any ε > 0, ?12Δ + (1 + ε)V is not positive. We determine the exact rate of divergence of the norm of e?tH as a map from L to L. A number of related problems are discussed.  相似文献   

8.
We find the automorphisms and the spectra of several different topological convolution algebras of C-functions on the real line. Starting with the convolution algebra of compactly supported C-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C-functions of infinitly fast exponential decay at ∞, H J, on the one hand, and the algebra of C-functions of only a finite degree e?r¦x¦ decay at ∞, Jr0, on the other hand, have quite different automorphisms, although H J = ∩rJr0. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of Jr0, and that this representation does not extend to a representation on the Banach algebra L1(R).  相似文献   

9.
We give two characterizations of the ordering on Böhm trees induced by the D model, one of which formalizes a continuity property of infinite η-expansion: A?B if for any finite approximant A of A there exists a finite approximant B of B such that A is a sub-tree of B, modulo finitely many η-equalities and finitely many infinite η-expansions of variables. To cite this article: P.-L. Curien, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 77–82  相似文献   

10.
It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C0(Rm) of all positive powers of the Schrödinger operator T = ? Δ + q if q real and in C(Rm) and if T ? ?a ? b ¦ x ¦2on C0(Rm).  相似文献   

11.
The main result of this paper is that if F is a closed subset of the unit circle, then (H + LF)H is an M-ideal of LH. Consequently, if ? ∈ L then ? has a closest element in H + LF. Furthermore, if ¦F¦ >0 thenL(H + LF) is not the dual of any Banach space.  相似文献   

12.
We provide conditions on a finite measure μ on Rn which insure that the imbeddings Wk, p(Rndμ)?Lp(Rndμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by dμ = e?¦x¦αdx, where α > 1.  相似文献   

13.
Let D be the group of orientation-preserving diffeomorphisms of the circle S1. Then D is Fréchet Lie group with Lie algebra (δ)R the smooth real vector fields on S1. Let δR be the subalgebra of real vector fields with finite Fourier series. It is proved that every infinitesimally unitary projective positive-energy representation of δR integrates to a continuous projective unitary representation of D. This result was conjectured by V. Kac.  相似文献   

14.
A classical result of Kronecker, stated at the end of the Section 10 of Kronecker (J. Reine Angew. Math. 92 (1882) 1–123), is that any radical of a finitely generated ideal in a polynomial ring of n variables is the radical of an ideal generated by n+1 elements. We give a constructive and elementary proof of a generalisation presented in (Michigan Math. J. 31 (1984) (2) 167–180): in a ring of Krull dimension ?n a radical of a finitely generated ideal is the radical of an ideal generated by n+1 elements. To cite this article: T. Coquand, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  相似文献   

15.
The perturbed wave equation □u + q(x)u = 0 in R3 × R with C∞ (R3) compactly supported initial data at t = 0 is considered. It is proven that the Huygens' principle does not hold for this equation if the potential is (essentially) non-negative, well-behaved at infinity and small in a suitable sense. The treatment is elementary and based on energy estimates and the positivity of the Riemann function for the wave equation in three space dimensions. The result still holds if the solution u is “small” over some space-time propagation cone. In the ease in which q has compact support, stronger results of this type for the above equation are obtained.  相似文献   

16.
Let R be a ring with 1, Rop the opposite ring, and R-Mod the category of left unitary R-modules and R-linear maps. A characterization of well-powered abelian categories A such that there exists an exact embedding functor AR-Mod is given. Using this characterization and abelian category duality, the following duality principles can be established.Theorem. There exists an exact embedding functor AR-Mod if and only if there exists an exact embedding functor AopRop-Mod.Corollary. If R-Mod has a specified diagram-chasing property, then Rop-Mod has the dual property.A lattice L is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; L(R) denotes the quasivariety of all lattices representable by R-modules.Theorem. A lattice L is representable by R-modules if and only if its order dual L1 is representable by Rop-modules. That is, L(Rop)={L1:L?L(R)}.If R is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R-Mod, then the dual result is also satisfied in R-Mod. Furthermore, L(R) is self-dual: L(R)= {L1:L?L(R)}.  相似文献   

17.
Let Ω denote a connected and open subset of Rn. The existence of n commuting self-adjoint operators H1,…, Hn on L2(Ω) such that each Hj is an extension of i∂∂xj (acting on Cc(Ω)) is shown to be equivalent to the existence of a measure μ on Rn such that f → \̂tf (the Fourier transform of f) is unitary from L2(Ω) onto Ω. It is shown that the support of μ can be chosen as a subgroup of Rn iff H1,…, Hn can be chosen such that the unitary groups generated by H1,…, Hn act multiplicatively on L2(Ω). This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of Rn by some subgroup, i.e., iff Ω is essentially a fundamental domain.  相似文献   

18.
First we compute Brownian motion expectations of some Kac's functionals. This allows a complete study of the semigroups generated by the formal differential operator H = ?12Δ + V on the various Lebesgue's spaces Lq=LqRn, dx, whenever the negative part of V is in L + Lp for some p > max {1, n2}. Our approach is probabilistic and some of the proofs are surprisingly elementary. The negative infinitesimal generators of our semigroups are shown to be reasonable self-adjoint extensions of H. Under mild assumptions on V, H is unitary equivalent to the Dirichlet operator, say D, associated to its groundstate measure. We study regularity of the semigroups generated by D. We concentrate on hyper and supercontractivity and we give, using probabilistic techniques, new examples of potential functions V which give rise to hyper and supercontractive Dirichlet semigroups.  相似文献   

19.
We study, by means of flows in jet bundles, infinitesimal deformations of germs of b maps ?:Rn→Rm that depend on a finite number of derivatives of these maps. We show that for m = 1 such deformations cannot essentially depend on derivatives higher than the first and that the deformations generalize the notion of an infinitesimal contact transformation. Analogous results hold for m > 1. Some applications to differential equations are given.  相似文献   

20.
In this article we discuss the solution of boundary value problems which are described by the linear integrodifferential equation ?xu?t (t, x) + u(t, x) ? 1π12?∞exp(?y2) u(t, y) dy = 0, where tJ?R, xR. We interpret the equation in functional form as an ordinary differential equation for the mapping u:JL2(R,μ), where L2(R,μ) is a weighted L2-space. Emphasis is on the constructive aspects of the solution and on finding representations of the relevant isomorphisms.  相似文献   

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