The irregular Cantor sets Ce ([0, 1]) and Cπ ([0, 1]), and the Cantor- Lebesgue irregular functions Ge and Gπ
Abstract: In this article, we introduce and study a new class of perfect nowhere-dense sets, which are not selfsimilar in any subset, also, we constructed the correspondent singular functions. We construct a twodimensional irregular Cantor set Ce,π ([0, 1]) on the real plane.
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References 8 publications
An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
The generalization of Hardy uncertainty principle for the windowed Bochner–Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function Ψ : G → X {\Psi:G\to X} , where X is a completely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. We establish that if ϕ ∈ C C ( G ) {\phi\in C_{C}(G)} is a non-trivial window function and Ψ ∈ L 2 ( G ) {\Psi\in L^{2}(G)} satisfies ∥ V ϕ ( Ψ ) ( g , χ ) ∥ HB ≤ c 2 ( g ) exp ( - β ∥ χ ∥ 2 ) \|V_{\phi}(\Psi)(g,\chi)\|_{\rm HB}\leq c_{2}(g)\exp(-\beta\|\chi\|^{2}) , β > 0 {\beta>0} , then Ψ = 0 {\Psi=0} almost everywhere.
An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
The generalization of Hardy uncertainty principle for the windowed Bochner–Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function Ψ : G → X {\Psi:G\to X} , where X is a completely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. We establish that if ϕ ∈ C C ( G ) {\phi\in C_{C}(G)} is a non-trivial window function and Ψ ∈ L 2 ( G ) {\Psi\in L^{2}(G)} satisfies ∥ V ϕ ( Ψ ) ( g , χ ) ∥ HB ≤ c 2 ( g ) exp ( - β ∥ χ ∥ 2 ) \|V_{\phi}(\Psi)(g,\chi)\|_{\rm HB}\leq c_{2}(g)\exp(-\beta\|\chi\|^{2}) , β > 0 {\beta>0} , then Ψ = 0 {\Psi=0} almost everywhere.
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