Abstract
In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel–Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.
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Acknowledgments
The authors would like to thank the organizers of the 2014 trimester program “Harmonic Analysis and Partial Differential Equations” at the Hausdorff Research Institute for Mathematics in Bonn, where this work has been initiated, for providing a pleasant and fruitful research atmosphere. Both authors would like to thank Peter Oswald, Winfried Sickel, Vladimir N. Temlyakov, and Hans Triebel for several valuable remarks.
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Communicated by Wolfgang Dahmen.
Research supported in part by the National Science Foundation and the DFG Emmy-Noether Programme UL403/1-1.
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Seeger, A., Ullrich, T. Lower Bounds for Haar Projections: Deterministic Examples. Constr Approx 46, 227–242 (2017). https://doi.org/10.1007/s00365-016-9353-3
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DOI: https://doi.org/10.1007/s00365-016-9353-3