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Mixed-norm spaces and interpolation

Joaquín Ortega, Joan Fàbrega (1994)

Studia Mathematica

Let D be a bounded strictly pseudoconvex domain of ℂ n with smooth boundary. We consider the weighted mixed-norm spaces A δ , k p , q ( D ) of holomorphic functions with norm ∥ f ∥ p , q , δ , k = ( ∑ | α | ≤ k ʃ 0 r 0 ( ʃ ∂ D r | D α f | p d σ r ) q / p r δ q / p - 1 d r ) 1 / q . We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces A δ , k p ( D ) and we give results about real and complex interpolation between them. We apply these results to prove that A δ , k p , q ( D ) is the intersection of a Besov space B s p , q ( D ) with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm...

Model selection for regression on a random design

Yannick Baraud (2002)

ESAIM: Probability and Statistics

We consider the problem of estimating an unknown regression function when the design is random with values in ℝ k . Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small probability....

Model selection for regression on a random design

Yannick Baraud (2010)

ESAIM: Probability and Statistics

We consider the problem of estimating an unknown regression function when the design is random with values in ℝ k . Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small...

Modifications of the double arrow space and related Banach spaces C(K)

Witold Marciszewski (2008)

Studia Mathematica

We consider the class of compact spaces K A which are modifications of the well known double arrow space. The space K A is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of K A spaces and on the isomorphic classification of the Banach spaces C ( K A ) .

Modular inequalities for the Hardy averaging operator

Hans P. Heinig (1999)

Mathematica Bohemica

If P is the Hardy averaging operator - or some of its generalizations, then weighted modular inequalities of the form u (Pf) Cv (f) are established for a general class of functions φ . Modular inequalities for the two- and higher dimensional Hardy averaging operator are also given.

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