Orthonormal bases of regular wavelets in spaces of homogeneous type
By P. Auscher, Tuomas Hytönen. 2011
105 citations recorded by the source corpus.
Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have H\"older regularity. This is used to build an orthonormal basis of H\"older-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calder\'on reproducing formula to study and develop function space theory and singular integrals. We discuss the examples of $L^p$ spaces, BMO and apply this to a proof of the T(1) theorem. As no extra condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the space of homogeneous type is required, our results extend a long line of works on the subject.
Subfields: Applied Mathematics, Computer Vision and Pattern Recognition
View Paper: https://www.semanticscholar.org/paper/e1fd7dc9c61694183d255317479830939acd0fcd
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In Applied Mathematics
In Computer Vision and Pattern Recognition
Citations in the corpus
- A class of Laplacian multiwavelets bases for high-dimensional data (2015, background): “Another construction for non-Euclidean setting is presented in [3] for general spaces of homogeneous type.”
- Multilinear analysis on metric spaces (2014, background): “[1, 67, 68] constructed a randomized dyadic structure by only assuming that the underlying metric space is geometrically doubling.”
- Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases (2015, methodology): “ original quasi-metric d and a doubling measure µ? Recently, Auscher and Hyto¨nen constructed an orthonormal wavelet basis with H¨older regularity and exponential decay for spaces of homogeneous type [AH]. This result is remarkable since there are no additional assumptions other than those defining s”
- Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces (2021, methodology): “Recently, a breakthrough on the analysis over a space of homogeneous type without any additional geometrical assumptions was made by Auscher and Hytönen [4] who constructed an orthonormal wavelet basis, with exponential decay, of L2() by using the system of random dyadic cubes.”