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On L p Space Formed by Real-Valued Partial Functions

Yasushige Watase, Noboru Endou, Yasunari Shidama (2010)

Formalized Mathematics

This article is the continuation of [31]. We define the set of Lp integrable functions - the set of all partial functions whose absolute value raised to the p-th power is integrable. We show that Lp integrable functions form the Lp space. We also prove Minkowski's inequality, Hölder's inequality and that Lp space is Banach space ([15], [27]).

On L₁-subspaces of holomorphic functions

Anahit Harutyunyan, Wolfgang Lusky (2010)

Studia Mathematica

We study the spaces H μ ( Ω ) = f : Ω h o l o m o r p h i c : 0 R 0 2 π | f ( r e i φ ) | d φ d μ ( r ) < where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, H μ ( Ω ) is either isomorphic to l₁ or to ( A ) ( 1 ) . Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

On limiting embeddings of Besov spaces

V. I. Kolyada, A. K. Lerner (2005)

Studia Mathematica

We investigate the classical embedding B p , θ s B q , θ s - n ( 1 / p - 1 / q ) . The sharp asymptotic behaviour as s → 1 of the operator norm of this embedding is found. In particular, our result yields a refinement of the Bourgain, Brezis and Mironescu theorem concerning an analogous problem for the Sobolev-type embedding. We also give a different, elementary proof of the latter theorem.

On locally convex extension of H in the unit ball and continuity of the Bergman projection

M. Jasiczak (2003)

Studia Mathematica

We define locally convex spaces LW and HW consisting of measurable and holomorphic functions in the unit ball, respectively, with the topology given by a family of weighted-sup seminorms. We prove that the Bergman projection is a continuous map from LW onto HW. These are the smallest spaces having this property. We investigate the topological and algebraic properties of HW.

On measure-preserving transformations and doubly stationary symmetric stable processes

A. Gross, A. Weron (1995)

Studia Mathematica

In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral representation which is itself stationary, and they gave an example of a stationary symmetric stable process which they claimed was not doubly stationary. Here we show that their process actually had a moving average representation, and hence was doubly stationary. We also characterize doubly stationary processes in terms of measure-preserving regular set isomorphisms...

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