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64.
《Mathematische Nachrichten》2017,290(8-9):1317-1327
The volume of the unit ball of the Lebesgue sequence space is very well known since the times of Dirichlet. We calculate the volume of the unit ball of the mixed norm , whose special cases are nowadays popular in machine learning under the name of group Lasso. We give two proofs of the main results, one in the spirit of Dirichlet, the other one using polarization identities. The result is given by a closed formula involving the gamma function, only slightly more complicated than the one of Dirichlet. We consider the real as well as the complex case. We also consider the anisotropic unit balls. We close by an overview of open problems. 相似文献
65.
《Mathematische Nachrichten》2017,290(10):1544-1552
Let K and S be locally compact Hausdorff spaces and X be an abstract space. Suppose that T is a positive Banach lattice isomorphism from into . Then for each ordinal α the cardinalities of the αth derivatives and satisfy the following inequality Moreover, if then is a continuous image of a subset of which can be taken closed when K is compact. The first statement of this result for is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number is sharp in these vector‐valued theorems. 相似文献
66.
《Mathematische Nachrichten》2017,290(14-15):2388-2400
In this paper, we study the high‐dimensional Hausdorff type operators and establish their boundedness on the power weighted Hardy spaces for . As a consequence, we obtain that the Hausdorff type operator is bounded on if Φ is the Gauss function, or the Poisson function. 相似文献
67.
《Mathematische Nachrichten》2017,290(17-18):2725-2739
In this paper we introduce and study the fractional Hardy spaces of the half space and of the unit ball in the quaternionic setting. In particular, we discuss their properties of invariance and of factorization in terms of functions in the Hardy space of the half space in the first case, and in terms of a suitable reproducing kernel Hilbert space in the case of the unit ball. 相似文献
68.
《Mathematische Nachrichten》2017,290(4):507-519
We show a picture of the relations among different types of summability of series in the space of integrable functions with respect to a vector measure m of relatively norm compact range. In order to do that, we study the class of the so‐called m‐1‐summing operators. We give several applications regarding the existence of copies of c 0 in , as well as on m‐1‐summing operators which are weakly compact, Asplund or weakly precompact. 相似文献
69.
《Mathematische Nachrichten》2017,290(17-18):3020-3028
Let X be a measurable space, let be a family of measurable subsets of it, and let be a subspace of complex measures on X that is also closed under restrictions of measures. In this paper we introduce the ‐convergence topology and the ‐strict topology on . Among other results, we find necessary and sufficient conditions for Hausdorff‐ness and coincide‐ness of these topologies. Applications to Lebesgue spaces, and also examples in Hausdorff topological spaces and locally compact groups are given. 相似文献
70.
In this paper, the parakähler Hom-Lie algebras or phase space of Hom-Lie algebras in terms of Hom-left-symmetric algebras are studied. A structure theory of parakähler Hom-Lie algebras in terms of matched pairs of Hom-Lie algebras is also presented. Besides, the bimodules of Hom-left-symmetric algebras are investigated. Furthermore, Hom-left-symmetric bialgebra which is precisely equivalent to parakähler Hom-Lie algebra is introduced. Especially, the coboundary Hom-left-symmetric bialgebra is considered. 相似文献