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Equicontinuity and Convergent Sequences in the Spaces $𝓞 '_C$ and $𝓞_M$

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Kisyński J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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tom 59

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nr 3

223-235

EN

Characterizations of equicontinuity and convergent sequences are given for the space $𝓞 '_C(ℝ ⁿ)$ of rapidly decreasing distributions and the space $𝓞_M(ℝ ⁿ)$ of slowly increasing infinitely differentiable functions.

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On Cohen's proof of the Factorization Theorem

100%

Kisyński J.

Annales Polonici Mathematici

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2000

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tom 75

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nr 2

177-192

EN

Various proofs of the Factorization Theorem for representations of Banach algebras are compared with its original proof due to P. Cohen.

3

Around Widder's characterization of the Laplace transform of an element of $L^{∞}(ℝ^{+})$

100%

Kisyński J.

Annales Polonici Mathematici

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2000

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tom 74

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nr 1

161-200

EN

Let ϰ be a positive, continuous, submultiplicative function on $ℝ^{+}$ such that $lim_{t→∞} e^{-ωt}t^{-α}ϰ(t) = a$ for some ω ∈ ℝ, α ∈ $\overline{ℝ^{+}}$ and $a ∈ ℝ^{+}$. For every λ ∈ (ω,∞) let $ϕ_{λ}(t) =e^{-λt}$ for $t ∈ ℝ^{+}$. Let $L^{1}_{ϰ}(ℝ^{+})$ be the space of functions Lebesgue integrable on $ℝ^{+}$ with weight $ϰ$, and let E be a Banach space. Consider the map $ϕ_{•}: (ω,∞) ∋ λ → ϕ_{λ} ∈ L_{ϰ}^{1}(ℝ^{+})$. Theorem 5.1 of the present paper characterizes the range of the linear map $T → Tϕ_{•}$ defined on $L(L_{ϰ}^{1}(ℝ^{+});E)$, generalizing a result established by B. Hennig and F. Neubrander for $ϰ(t)=e^{ωt}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder's characterization of the Laplace transform of a function in $L^{∞}(ℝ^{+})$. Some applications of Theorem 5.1 to the theory of one-parameter semigroups of operators are discussed. In particular a version of the Hille-Yosida generation theorem is deduced for $C_0$ semigroups $(S_t)_{t ∈ \overline{ℝ^{+}}}$ such that $sup_{t ∈ \overline{ℝ^{+}}} (ϰ(t))^{-1}∥ S_t∥ < ∞$.

4

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Metrization of D_E[0,1] by Hausdorff distance between graphs

44%

Kisyński J.

Annales Polonici Mathematici

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1990

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tom 51

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nr 1

195-203

Ograniczanie wyników

3 Annales Polonici Mathematici

1 Bulletin of the Polish Academy of Sciences. Mathematics

4 Kisyński J.

1 2011

2 2000

1 1990

Equicontinuity and Convergent Sequences in the Spaces $𝓞 '_C$ and $𝓞_M$

On Cohen's proof of the Factorization Theorem

Around Widder's characterization of the Laplace transform of an element of $L^{∞}(ℝ^{+})$

Metrization of D_E[0,1] by Hausdorff distance between graphs