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A remark on functions continuous on all lines

Luděk Zajíček (2019)

Commentationes Mathematicae Universitatis Carolinae

We prove that each linearly continuous function f on ℝ n (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016). The same result holds also for f on an arbitrary Banach space X , if f has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such f on a separable X is continuous at all points outside a first category set which is also null in any usual sense.

A remark on p-convolution

Rafał Sałapata (2011)

Banach Center Publications

We introduce a p-product of algebraic probability spaces, which is the definition of independence that is natural for the model of noncommutative Brownian motions, described in [10] (for q = 1). Using methods of the conditionally free probability (cf. [4, 5]), we define a related p-convolution of probability measures on ℝ and study its relations with the notion of subordination (cf. [1, 8, 9, 13]).

A remark on supra-additive and supra-multiplicative operators on C ( X )

Zafer Ercan (2007)

Mathematica Bohemica

M. Radulescu proved the following result: Let X be a compact Hausdorff topological space and π C ( X ) → C ( X ) a supra-additive and supra-multiplicative operator. Then π is linear and multiplicative. We generalize this result to arbitrary topological spaces.

A remark on the approximation theorems of Whitney and Carleman-Scheinberg

Michal Johanis (2015)

Commentationes Mathematicae Universitatis Carolinae

We show that a C k -smooth mapping on an open subset of ℝ n , k ∈ ℕ ∪ { 0 , ∞ } , can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.

A remark on the asymmetry of convolution operators

Saverio Giulini (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A convolution operator, bounded on L q ⁢ ( ℝ n ) , is bounded on L p ⁢ ( ℝ n ) , with the same operator norm, if p and q are conjugate exponents. It is well known that this fact is false if we replace ℝ n with a general non-commutative locally compact group G . In this paper we give a simple construction of a convolution operator on a suitable compact group G , wich is bounded on L q ⁢ ( G ) for every q ∈ [ 2 , ∞ ) and is unbounded on L p ⁢ ( G ) if p ∈ [ 1 , 2 ) .

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