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Register a new userOn the extension of Whitney ultrajets, II
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We characterize the validity of the Whitney extension theorem in the ultradifferentiable Roumieu setting with controlled loss of regularity. Specifically, we show that in the main Theorem 1.3 of [15] condition (1.3) can be dropped. Moreover, we clarify some questions that remained open in [15].
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In this memoir, we develop a general framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in $mathbb R^D$ and the Whitney near isometry extension problem for discrete and non-discrete subsets of $mathbb R^D$ with certain geometries. In addition, we survey related work of ours on clustering, dimension reduction, manifold learning, vision as well as minimal energy partitions, discrepancy and min-max optimization. Numerous open problems in harmonic analysis, computer vision, manifold learning and signal processing connected to our work are given. A significant portion of the work in this memoir is based on joint research with Charles Fefferman in the papers [48], [49], [50], [51].
Let $ f $ be a real-valued function on a compact subset in $ mathbb{R}^n $. We show how to decide if $ f $ extends to a nonnegative and $ C^1 $ function on $ mathbb{R}^n $. There has been no known result for nonnegative $ C^m $ extension from a general compact set $ E $ when $ m > 0 $. The nonnegative extension problem for $ m geq 2 $ remains open.
A classical theorem of Kuratowski says that every Baire one function on a G_delta subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index beta(f). We prove a refinement of Kuratowskis theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that beta_{Y}(f)<omega^{alpha}, alpha < omega_1, then f has an extension F onto X so that beta_X(F)is not more than omega^{alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.
We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, BV and maximal functions.
We prove an analogue of Chernoffs theorem for the Laplacian $ Delta_{mathbb{H}} $ on the Heisenberg group $ mathbb{H}^n.$ As an application, we prove Ingham type theorems for the group Fourier transform on $ mathbb{H}^n $ and also for the spectral projections associated to the sublaplacian.
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