In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball Bn in , n?2, it is proved in this article that for any nonnegative, increasing, convex function ?(t) defined on , there exists satisfying such that there is no f∈Hp(Bn), 0<p<∞, with . Here Ng(ζ,1) denotes the integrated zero counting function associated with the slice function gζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces Hp(Bn), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner. 相似文献
In the open disk
of the complex plane, we consider the following spaces of functions: the Bloch space
; the Hardy--Sobolev space
; and the Hardy--Besov space
. It is shown that if all the poles of the rational function R of degree n,
, lie in the domain
, then
, where
and
depends only on
. The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions. 相似文献
Let be an analytic function mapping the unit disc to itself. We generalize a boundary version of Schwarz's lemma proven by D. Burns and S. Krantz and provide sufficient conditions on the local behavior of near a finite set of boundary points that requires to be a finite Blaschke product. Afterwards, we supply several counterexamples to illustrate that these conditions may also be necessary.
Let μ be a positive Borel measure having support supp μ ⊂ [1, ∞) and satisfying the conditionf(t−1)−1dμ(t)<∞. In this paper we study the order of the uniform approximation of the function
on the disk |z|≤1 and on the closed interval [−1, 1] by means of the orthogonal projection of
on the set of rational functions of degreen. Moreover, the poles of the rational functions are chosen depending on the measure μ. For example, it is shown that if supp
μ is compact and does not contain 1, then this approximation method is of best order. But if supp μ=[1,a],a>1, the measure μ is absolutely continuous with respect to the Lebesgue measure, and
fort∈[1,a] and some α>0, then the order of such an approximation differs from the best only by
.
Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 362–368, March, 1999. 相似文献
Let be a strip in the complex plane. For fixed integer let denote the class of -periodic functions , which are analytic in and satisfy in . Denote by the subset of functions from that are real-valued on the real axis. Given a function , we try to recover at a fixed point by an algorithm on the basis of the information
where , are the Fourier coefficients of . We find the intrinsic error of recovery
Furthermore the -dimensional optimal information error, optimal sampling error and -widths of in , the space of continuous functions on , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class , consisting of all -periodic functions, which are analytic in with -integrable boundary values. In the case sampling fails to yield optimal information as well in odd as in even dimensions.
Rigidity of nondegenerate Blaschke surfaces in is studied. The rigidity criteria are given in terms of , where is the curvature of the Blaschke connection . If the rank of is 2, then the surface is rigid. If , it is nonrigid. In the case where the rank of is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.
This paper is concerned with Fredholm operator valued Hp – functions on the unit disc, where the Fredholm operators action a Banach space. Sufficient conditions are presented which guarantee that Fatou's theorem is valid. Using the theory of traces and determinants on quasi – Banach operator ideals, we develop conditions that guarantee that the zeros of Fredholm operator valued Hp – functions satisfy the Blaschke condition. 相似文献