共查询到20条相似文献,搜索用时 31 毫秒
1.
《Journal of Complexity》1996,12(1):58-79
LetBH∞(Ω) be the space of analytic functionsfin the region Ω for which |f(z)| ≤ 1,z∈ Ω, and letKbe a compact subset of Ω. How can we compute the values of any functionf∈BH∞(Ω) at an arbitrary pointz∈K? One of the approaches to this problem applies the results concerning then-widths and ϵ-entrophy of classBH∞(Ω) in the metricC(K). In the case whenKhas a simply connected complement inC and Ω is a canonical neighbourhood ofK, the classical tools for approximation off∈BH∞(Ω) inC(K) give the Faber series. This work is concerned with the following: the exact values of Kolmogorov and othern-widths of Hardy spacesHp, then-widths and ϵ-entrophy of classBH∞(Ω), the optimality of Faber approximations, and computing values of analytic functions with the help of Faber series. 相似文献
2.
Anne-Marie Charbonnel 《Israel Journal of Mathematics》1983,45(1):69-89
We study aC ∞ functional calculus with several variables forv pseudodifferential operatorsP 1, …,P v inR n . Whenf is a function belonging to the classS 1.0 r (R v ) of Hörmander, we prove that, under some conditions,f(P 1,…,P v) is a pseudodifferential operator, and we give an asymptotic formula for its symbol. 相似文献
3.
《Journal of Algebra》1999,211(2):562-577
LetRbe a Krull ring with quotient fieldKanda1,…,aninR. If and only if theaiare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb1,…,bninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(ai) = biandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringRvwith quotient fieldKanda1,…,aninSare pairwise incongruent mod allMkv ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of theaiamong residue classes of the idealsMkv ∩ S) such that for allb1,…,bn ∈ Rvthere exists a polynomialf ∈ K[x] of degree at mostdwithf(ai) = biandf(S) ⊆ Rv. 相似文献
4.
Denote by $\hat f$ the (complex) Fourier transform of a functionf which belongs toL 1(R 2). We shall assume thatf is odd inx andy, orf is even inx and odd iny, orf is odd inx and even iny. Among others, we prove that iff ∈L 1(R 2) and (x, y)=(0,0) is a strong Lebesgue point off, then $\left| t \right|\left| v \right|\hat f(t,v)$ tends to 0 as |t|, |v|→∞ in the sense (C;α,β) for allα,β>1. 相似文献
5.
Guido Cortesani 《Annali dell'Universita di Ferrara》1997,43(1):27-49
Let Ω be an open and bounded subset ofR
n
with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R
m
) whose jump setS
vis essentially closed and polyhedral and which are of classW
k, ∞ (S
v,R
m) for every integerk are strongly dense inGSBV
p(Ω,R
m
), in the sense that every functionu inGSBV
p(Ω,R
m
) is approximated inL
p(Ω,R
m
) by a sequence of functions {v
k{j∈N with the described regularity such that the approximate gradients ∇v
jconverge inL
p(Ω,R
nm
) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS
v
j converges to the (n−1)-dimensional measure ofS
u. The structure ofS
v can be further improved in casep≤2.
Sunto Sia Ω un aperto limitato diR n con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R m ) con insieme di saltoS v essenzialmente chiuso e poliedrale che sono di classeW k, ∞ (S v,R m ) per ogni interok sono fortemente dense inGSBV p(Ω,R m ), nel senso che ogni funzioneu∈GSBV p(Ω,R m ) è approssimata inL p(Ω,R m ) da una successione di funzioni {v j}j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v jconvergono inL p(Ω,R nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS v jconverge alla misura (n−1)-dimensionale diS u. La struttura diS vpuó essere migliorata nel caso in cuip≤2.相似文献
6.
Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <p < ∞ and μ > 0. ThenL has a bounded H∞ functional calculus in Lp(Ω), in the sense that ¦¦f (L +cI)u¦¦p ≤C sup¦arλ¦<μ ¦f¦ ¦‖u¦‖p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderón-Zygmund singular integral operators. 相似文献
7.
Let S be the unit sphere in Cn. We investigate the properties of Toeplitz operators on S, i.e., operators of the form Tφf = P(φf) where φ?L∞(S) and P denotes the projection of L2(S) onto H2(S). The aim of this paper is to determine how far the extensive one-variable theory remains valid in higher dimensions. We establish the spectral inclusion theorem, that the spectrum of Tφ contains the essential range of φ, and obtain a characterization of the Toeplitz operators among operators on H2(S) by an operator equation. Particular attention is paid to the case where φ ? H∞(S) + C(S) where C(S) denotes the algebra of continuous functions on S. Finally we describe a class of Toeplitz operators useful for providing counterexamples—in particular, Widom's theorem on the connectedness of the spectrum fails when n > 1. 相似文献
8.
If γ(x)=x+iA(x),tan ?1‖A′‖∞<ω<π/2,S ω 0 ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S ω 0 and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C 0(γ), ?z∈suppf, where Cc(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:
- T can be extended to be a bounded operator on L2(γ);
- there exists a function ?1 ∈H ∞(S ω 0 ) such that ?′1(z)=?(z)+?(-z), ?z∈S ω 0 ?z∈S w 0 .
9.
In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ?:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (f n ) inA, there exists a pointa inX such that ?(f n )=f n (a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC ∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceC m (E) (m=0,1,...,∞) ofC m -functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ? onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem. 相似文献
10.
The evaluation of matrix functions of the form f(A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type Km,mi+1(A)=span{A-m+1v,…,A-1v,v,Av,…,Amiv}, m=1,2,3,…, with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules. 相似文献
11.
Let denote the real line and L∞(R), the class of all Borel measurable L∞-functions of . 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
12.
《Advances in Mathematics》1985,56(3):238-282
Let n be the Lie algebra gln(C), let S(n) be the symmetric algebra of n, and let T(n) be the tensor algebra of n. In a recent paper, R. K. Gupta studied certain sequences of representations R∞ = (Rn)∞n = 1, where Rn is a representation of n. These sequences have the property that every irreducible representation occurring in S(n) is in exactly one of these sequences. Fixing f, she considers s(R∞, f) which is the limit on n of the multiplicity of Rn in Sf(n), the fth-graded piece of S(n). She and R. P. Stanley independently showed that the limit s(R∞, f) exists and is given by an amazingly elegant formula. They call s(R∞, f) the stable multiplicity of Rn in Sf(n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R∞ for all of the classical Lie algebras n are studied, and a simple formula for the stable multiplicity m(R)∞, ψ, f, ∞) of Rn in the ψ-isotypic component of Tf(n), where ψ is any irreducible character of the symmetric group tSf, is obtained. As in the work of Gupta and Stanley, the expressions for m(R)∞, ψ, f, ∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of n. As a byproduct of our proof, a “stable” decomposition of every isotypic component of T(n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S(n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f <n the combinatorial decomposition of Tf(n) projects onto Kostant's decomposition of Sf(n). 相似文献
13.
Ian N. Baker 《Aequationes Mathematicae》1997,54(1-2):87-101
Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr)
m+k,q=r
m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t
m) is a doubly-periodic function. 相似文献
14.
Given two functionsf(z),g(z) in the (usual) classS, we can form the new functions (arithmetric and geometric mean functions) F(itz)=∝(itf)(itz)+β(itg)(itz) and G(itz)=(itz)(f(itz)/(itz))(su∝)(g(itz)/(itz))(suβ),
whereα, β ∈ (0, 1) andα+β=1. This paper determines the maximum valence of the functionsF andG. 相似文献
15.
We consider functionsf(z),z∈D, of one complex variable that satisfy the following weakened asymptotic monogeny condition: for some positiveσ<1/2,f(z) is monogenic at each pointξ∈D with respect to some setG(ξ) such that the lower density ofG(ξ) atξ is greater than 1/2+σ. We show that if for somep σ ≥1 the function (log+|?(z)|) p σ is locally integrable inD with respect to the plane Lebesgue measure, thenf(z) is holomorphic inD. 相似文献
16.
Olof J Staffans 《Journal of Mathematical Analysis and Applications》1985,108(1):15-30
Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝0ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝0tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝0ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝0tr(t, s) f(s) ds maps a weighted L1 space continuously into another weighted L1 space, and a weighted L∞ space into another weighted L∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations. 相似文献
17.
Stephan Brandt 《Discrete Mathematics》2009,309(12):3786-237
As an edge variant of the well-known irregularity strength of a graph G=(V,E) we investigate edge irregular total labellings, i.e. functions f:V∪E→{1,2,…,k} such that f(u)+f(uv)+f(v)≠f(u′)+f(u′v′)+f(v′) for every pair of different edges uv,u′v′∈E. The smallest possible k is the total edge irregularity strength of G. Confirming a conjecture by Ivan?o and Jendrol’ for a large class of graphs we prove that the natural lower bound is tight for every graph of order n, size m and maximum degree Δ with m>111000Δ. This also implies that the probability that a random graph from G(n,p(n)) satisfies the Ivan?o-Jendrol’ Conjecture tends to 1 as n→∞ for all functions p∈[0,1]N. Furthermore, we prove that is an upper bound for every graph G of order n and size m≥3 whose edges are not all incident to a single vertex. 相似文献
18.
19.
Quanlei Fang 《Journal of Functional Analysis》2009,257(10):3082-673
Let H2(S) be the Hardy space on the unit sphere S in Cn, n?2. Consider the Hankel operator Hf=(1−P)Mf|H2(S), where the symbol function f is allowed to be arbitrary in L2(S,dσ). We show that for p>2n, Hf is in the Schatten class Cp if and only if f−Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership Hf∈C2n implies f−Pf=0, i.e., Hf=0. 相似文献
20.
Shelby J. Haberman 《Journal of multivariate analysis》1980,10(3):398-404
Let Y be an N(μ, Σ) random variable on Rm, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in Rm with closure . Let (·,-·) be the Eculidean inner product on Rm, and let μc be the conditional expected value of Y given Y ∈ C. For v ∈ Rm and s ≥ 0, let βs(v) be the expected value of |(v, Y) ? (v, μ)|s and let γs(v) be the conditional expected value of |(v, Y) ? (v, μc)|s given Y ∈ C. For s ≥ 1, γs(v) < βs(v) if and only if , and γs(v) < βs(v) for all v ≠ 0 if and only if for any v ∈ Rm such that v ≠ 0. 相似文献