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1.
Let the set of generalized polynomials having bounded coefficients beK={p= jgj. j j j,j=1, 2, ...,n}, whereg 1,g 2, ...,g n are linearly independent continuous functions defined on the interval [a, b], j, j are extended real numbers satisfying j<+, j>-, and j j. Assume thatf is a continuous function defined on a compact setX [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK.  相似文献   

2.
Résumé En généralisant un résultat de J. Aczél et M. Hosszú on donne des conditions nécessaires et suffisantes pour qu'une solution de l'équation de translationF(F(, x), y) = F(, xy), oùF: × G , est un ensemble arbitraire,G forme un groupe, soit de la formeF(, x) = f –1(f()·1(x)), oùf est une bijection de au groupeG 1 isomorphe avecG et 1 est un homomorphisme deG àG 1. On considère aussi le cas oùG forme un espace vectoriel sur le corps des nombres rationels.Si est un intervalle ayant plus qu'un point etG = R m avec l'addition comme l'opération on trouve des conditions pour que la fonction continueF soit de la formeF(, x 1,, x m ) =f –1(f() + c 1 x 1 + +c m x m ), oùf est une homéomorphie de àR et (c 1,,c m ) R m .
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3.
We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].  相似文献   

4.
The fundamental result: if and v are two finite Borel measures, defined in the spaceL p[0, 1] (1p<) or in C(K) (K is a metric compactum without isolated points), then from the equalities (B)=v(B) for all balls B of radius 1 there follows that =v. In addition, in the spaces C(K) and p (1p<) from the inequalities (B) v(B) there follows that v.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 122–128, 1989.  相似文献   

5.
We consider measurable subsets {ofR}n with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H k: 1kn} in L 2 () such that each H k is an extension of i(/x k) on C c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}n. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.  相似文献   

6.
In this paper we continue the study of structures of various types initiated by the author in the earlier paper Structures of extensions (Ref. Zh. Mat., 1974, 4A361). The present paper is devoted to the so-called structure of topological type. By a structure of topological type on the set X is meant a topological structure, defined on some set obtained from X, and possibly additional sets, by a totally ordered sequence of operations of unions of sets, products of sets, and passage to the set of subsets. We study certain structures of topological type: bitopological (Sec. 2) and settopological (Sec. 3). A bitopological structure on the set X is any topological structure on the set X×X, and a bitopological space is a pair (X,). This concept is a natural extension of the concept of a bitopological space as a set X on which there are given two topological structures 1 and 2-these structures define a structure =1×2 on the set X×X. A settopological structure on the set X is any topological structure on the set={A¦A. There are given representations of piecewise-linear structures (Sec. 4) and smooth structures (Sec. 5) as settopological structures.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 83, pp. 5–62, 1979.  相似文献   

7.
H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp n=, , ,n .. . , 117 234 . . -   相似文献   

8.
LetX be the solution of the SDE:dX t = (X t)dB t +b(X t)dt, with andb C b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of k , must be a constant function.  相似文献   

9.
Summary Let denote the class of infinite product probability measures = 1× 2× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and 0 n for some n 1, then 0× 1× 2×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556  相似文献   

10.
Summary We study a class of generalized gamma functions k (z) which relate to the generalized Euler constants k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for k is derived, and the constant term in the asymptotic expansion for log k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.  相似文献   

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