Almost-orthogonality of restricted Haar functions

Weigt, Julian

doi:10.1090/proc/14752

Almost-orthogonality of restricted Haar functions
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by Julian Weigt PDF

Proc. Amer. Math. Soc. 148 (2020), 601-609 Request permission

Abstract:

We consider the Haar functions $h_I$ on dyadic intervals. We show that if $p>\frac 23$ and $E\subset [0,1]$, then the set of all functions $\|h_I1_E\|_2^{-1}h_I1_E$ with $|I\cap E|\geq p|I|$ is a Riesz sequence. For $p\leq \frac 23$ we provide a counterexample.

References

Marcin Bownik, Pete Casazza, Adam W. Marcus, and Darrin Speegle, Improved bounds in Weaver and Feichtinger conjectures, J. Reine Angew. Math. 749 (2019), 267–293. MR 3935905, DOI 10.1515/crelle-2016-0032
Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1946982, DOI 10.1007/978-0-8176-8224-8
Vjekoslav Kovač, Christoph Thiele, and Pavel Zorin-Kranich, Dyadic triangular Hilbert transform of two general functions and one not too general function, Forum Math. Sigma 3 (2015), Paper No. e25, 27. MR 3482272, DOI 10.1017/fms.2015.25
Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. (2) 182 (2015), no. 1, 327–350. MR 3374963, DOI 10.4007/annals.2015.182.1.8
F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909–928. MR 1685781, DOI 10.1090/S0894-0347-99-00310-0
Julian Weigt, Almost-orthogonality of restricted haar functions, Master’s thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 2018. http://math.aalto.fi/~weigtj1/Thesis_updated.pdf.