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1.
Weighted Composition Operators on Bergman and Dirichlet Spaces   总被引:3,自引:0,他引:3  
Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.  相似文献   

2.
Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes f and P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f f when prior P is fixed, and also the overall range over f f and P P . When multiple i.i.d. observations X 1,...,X k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.  相似文献   

3.
For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if fn(x) f(x) and f n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.  相似文献   

4.
Let and be two hyperbolic simply connected domains in the extended complex plane = {}. We derive sharp upper bounds for the modulus of the nth derivative of a holomorphic, resp. meromorphic function f: at a point z 0 . The bounds depend on the densities (z 0) and (f(z 0)) of the Poincaré metrics and on the hyperbolic distances of the points z 0 and f(z 0) to the point .  相似文献   

5.
For any functionf of L(0, 2), we prove that there is a function L(0, 2) such that ¦(x)¦ = ¦f(x)¦ almost everywhere and L(0, 2), where is the conjugate of.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 461–465, October, 1968.  相似文献   

6.
Summary The following Artin type characterization of : + + is proved: Assume thatf: + + satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim x 0 xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.  相似文献   

7.
Let M f(r) and f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.  相似文献   

8.
For a sectorial operator A with spectrum (A) that acts in a complex Banach space B, we prove that the condition (A) i R = Ø is sufficient for the differential equation where is a small positive parameter, to have a unique bounded solution x for an arbitrary bounded function f: R B that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as 0+ to the unique bounded solution of the differential equation x(t) = Ax(t) + f(t).  相似文献   

9.
We determined the maximum radius (r) of the disk r, possessing the property that every functionf (z), starlike of order, is starlike in ¦z¦r. The problem is reduced to that of finding the minimum of a certain functional for which extremal function is determined.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 287–293, September, 1971.  相似文献   

10.
It is shown that the precise result of G. Szegö on the asymptotic behavior of the Toeplitz determinants Dn(f), generated by the nonnegative summable function off() holds if Inf L1 (–, ).Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 693–702, June, 1968.  相似文献   

11.
Summary Relativistic and nonrelativistic data for the coaxial two-cylinder electrostatic electron-lens are compared. Values of the potential along the axis were measured on a resistance network analogue, whilst trajectories were calculated on a high speed digital computer.IndexR indicating relativistic values, andN non-relativistic ones, quotientsF R/FN of the focal point coordinates (image sided),f R/fN of the focal length (image sided) and differences of the principal plane coordinates (H RH N)/R (object sided) and (H RH N)/R (image sided) are represented graphically as functions of the imagespace potential andU. The gap between the cylinders was in all cases 0,8R (R radius of the cylinders).U being the potential on the object side, andU on the image side, =U/U was varied from 0,1 to 0,966 andU from 0 to 2,000 kV; in the relativistic case a maximal departure of about 20% from the nonrelativistic values was found for focal length. A ten-lens accelerator of 1,5 MeV overall tension has been calculated both ways.  相似文献   

12.
It is well known that for a large class of Markov process the associated semi-group T(t)f(x)=f(y)P(t,x;dy) satisfies the Kolmogorov backward differential equation, that is, if u(t,x)=T(t)f(x) then and .In this paper we are considering the opposite problem: given the diffusion and drift coefficients we study the differentiability preserving properties of the semigroup T(t) having as infinitesimal generator .More specifically, for a large class of functions a(x) and b(x), we will prove for k=0, ..., 3 the existence of T(t) such that T(t): C k (I) C k (I) and the existence of a constant k such that |T(t)f| k |f| k exp ( k t) for fC k (I). Moreover an explicit expression of k in terms of the coefficients a(x) and b(x) is obtained. As a side result we obtain the necessity of the boundary conditions imposed.This paper is a revised version of the author's Ph. D. dissertation at University of Massachusetts under W. Rosenkrantz  相似文献   

13.
(1–) + , R n =R j ×R k , ()=max{¦ 1¦, ¦ 1¦},=( 1, 2), 1R J , 2R k ,j,k1,n=j+k. n=3 , (1–) + [L 1(R n )]1, >1/2; j=4, (1–) + R L p (R n ). .

The author would like to thank Professor W. Trebels for encouragement and valuable advice.  相似文献   

14.
Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))2. LetH(x) be an arbitrary primitive of 1/h –1(x). Note that the functionH –1 x is positive and increasing on (–, +),H –1 0 asx– andH –1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.  相似文献   

15.
LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ * defined on the Borel sets of the extended complex plane * such thatQ *({})=0 andT=zdQ *(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ *-integrable functiong: * * which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.  相似文献   

16.
Reiterated homogenization is studied for divergence structure parabolic problems of the form u /t–div (a(x,x/,x/2,t,t/ k)u )=f. It is shown that under standard assumptions on the function a(x, y 1,y 2,t,) the sequence {u } of solutions converges weakly in L 2 (0,T; H 0 1 ()) to the solution u of the homogenized problem u/t– div(b(x,t)u)=f.This revised version was published online in April 2005 with a corrected missing date string.  相似文献   

17.
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   

18.
Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH xx-h, and the space variablex takes values on the unit circleS 1. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv 2 is aC k (S 1)-valued Markov process for each 0<1/2, whereC (S 1) is the Banach space of real-valued continuous functions onS 1 which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S 1) is a globally exponentially stable critical point of the unperturbed equation t 0 = 0 +f(x,0), that has a unique stationary distributionv K, on (C (S 1), (C K (S 1))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v K, tends tov K,0, the point mass centered on the zero element ofC (S 1). The main goal of this paper is to show that in factv K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S 1) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA  相似文献   

19.
The behavior of the poles zn(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |zn()–zn|=0((1/2)qn), where qn is the order of the pole of the scattering matrix for the operator 0u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.  相似文献   

20.
Let (S nn>-1) be a random walk on a hypergroup ( + , *), i.e., a Markov chain with transition kernelN(x, A) = x * (A), where is a fixed probability measure on + such that the second moment exists. Then depending on the growth of the hypergroup two situations can occur: when ( + , *) is of exponential growth then it is shown thatS n is asymptotically normal. In the case of polynomial growth {more precisely, if the densityA of the Haar measure of ( + , *) satisfies lim[A()/A()]=}, the normalized variablesS n/[n Var()/(+1)]1/2 converge to a Rayleigh distribution with parameter .  相似文献   

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