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Derivative of the Donsker delta functionals

Herry Pribawanto Suryawan (2019)

Mathematica Bohemica

We prove that derivatives of any finite order of Donsker's delta functionals are well-defined elements in the space of Hida distributions. We also show the convergence to the derivative of Donsker's delta functionals of two different approximations. Finally, we present an existence result of finite product and infinite series of the derivative of the Donsker delta functionals.

Descriptive properties of mappings between nonseparable Luzin spaces

Petr Holický, Václav Komínek (2007)

Czechoslovak Mathematical Journal

We relate some subsets G of the product X × Y of nonseparable Luzin (e.g., completely metrizable) spaces to subsets H of ℕ ℕ × Y in a way which allows to deduce descriptive properties of G from corresponding theorems on H . As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points y in Y with particular properties of fibres f - 1 ( y ) of a mapping f X → Y . Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.

Descriptive set-theoretical properties of an abstract density operator

Szymon Gła̧b (2009)

Open Mathematics

Let 𝒦 (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that { K ∈ 𝒦 ( ℝ ) : ∀ x ∈ K ( d + ( x , K ) = 1 o r d - ( x , K ) = 1 ) } is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.

Dichotomies for 𝐂 0 ( X ) and 𝐂 b ( X ) spaces

Szymon Głąb, Filip Strobin (2013)

Czechoslovak Mathematical Journal

Jachymski showed that the set ( x , y ) ∈ 𝐜 0 × 𝐜 0 : ∑ i = 1 n α ( i ) x ( i ) y ( i ) n = 1 ∞ is bounded is either a meager subset of 𝐜 0 × 𝐜 0 or is equal to 𝐜 0 × 𝐜 0 . In the paper we generalize this result by considering more general spaces than 𝐜 0 , namely 𝐂 0 ( X ) , the space of all continuous functions which vanish at infinity, and 𝐂 b ( X ) , the space of all continuous bounded functions. Moreover, we replace the meagerness by σ -porosity.

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