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Bourgain’s discretization theorem asserts that there exists a universal constant with the following property. Let be Banach spaces with . Fix and set . Assume that is a -net in the unit ball of and that admits a bi-Lipschitz embedding into with distortion at most . Then the entire space admits a bi-Lipschitz embedding into with distortion at most . This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem.We also obtain an improvement...
A typical wavelet system constitutes an unconditional basis for various function spaces -Lebesgue, Besov, Triebel-Lizorkin, Hardy, BMO. One of the main reasons is the frequency localization of an element from such a basis. In this paper we study a wavelet-type system, called a brushlet system. In [3] it was noticed that brushlets constitute unconditional bases for classical function spaces such as the Triebel-Lizorkin and Besov spaces. In this paper we study brushlet expansions of functions in the...
Let H be a separable complex Hilbert space, 𝓐 a von Neumann algebra in 𝓛(H), ϕ a faithful, normal state on 𝓐, and 𝓑 a commutative von Neumann subalgebra of 𝓐. Given a sequence (Xₙ: n ≥ 1) of operators in 𝓑, we examine the relations between bundle convergence in 𝓑 and bundle convergence in 𝓐.
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