Recorded Zoom presentations


  • Discretizing Lp norms, (joint work with Dorsa Ghoreishi), March 16 2021, 1 hour talk in Codex Seminar. We discuss the problem of discretizing the Lp-norm on subspaces and its connection with frame theory. It is known that if X is an n-dimensional subspace of L2[0,1] then the L2-norm may be discretized on X using m on the order of n sampling points as long as X satisfies a necessary boundedness condition. In contrast to this, we construct an n-dimensional subspace X of L1[0,1] which satisfies the necessary boundedness condition but the L1-norm cannot be discretized on X using m on the order of n sample points. We conclude the talk by sketching a proof that discretizing a continuous frame to do stable phase retrieval requires simultaneously discretizing both the L2-norm and the L1-norm on the range of the analysis operator.
  • A positive Schauder basis for L_2, (joint work with Alex Powell and Mitchell Taylor), May 1 2020, 1 hour talk in Banach spaces webinar. Johnson and Schechtman recently constructed a Schauder basis for L1 using only non-negative functions. We present this construction and explain why it does not work in Lp for p>1. We then discuss our construction of a Schauder basis for L2 using only non-negative functions. Furthermore, for the case p not equal to 2 our construction allows us to build a basic sequence of positive functions in Lp whose closed span contains Lp isomorphically as a subspace.

Research papers


  1. W. Alharbi, D. Freeman, D. Ghoreishi, C. Lois, and S. Sebastian Stable phase retrieval and perturbations of frames, submitted, 13 pages.
  2. P. Balazs, D. Freeman, R. Popescu, and M. Speckbacher, Quantitative bounds for unconditional pairs of frames, submitted, 19 pages.
  3. D. Freeman, T. Oikhberg, B. Pineau, and M.A. Taylor, Stable phase retrieval in function spaces, submitted, 57 pages.
  4. W. Alharbi, S. Alshabhi, D. Freeman, and D. Ghoreishi, Locality and stability for phase retrieval, submitted, 14 pages.
  5. K. Beanland and D. Freeman, Shrinking Schauder frames and their associated spaces, submitted 18 pages.
  6. R. Calderbank, I. Daubechies, D. Freeman, and N. Freeman, Stable phase retrieval for infinite dimensional subspaces of L2(ℝ), submitted, 27 pages.
  7. D. Freeman and D. Ghoreishi, Discretizing Lp norms and frame theory, J. Math. Anal. and Applications, (2022) 17 pages. doi.org/10.1016/j.jmaa.2022.126846
  8. D. Freeman, Th. Schlumprecht, and A. Zsák, Banach spaces for which the space of operators has 2𝔠 closed ideals, Forum of Mathematics, Sigma, 9, E27, (2021) 20 pages. doi:10.1017/fms.2021.23
  9. D. Freeman, Th. Schlumprecht, and A. Zsák, Addendum Closed ideals of operators between the classical sequence spaces. Bull. Lond. Math. Soc, 53, no. 2, (2021), 593-–595.
  10. D. Freeman, A.M. Powell, and M. Taylor, A Schauder basis for L2 consisting of non-negative functions, Mathematische Annalen, (2021) 28 pages, https://doi.org/10.1007/s00208-021-02143-4
  11. J. Eisner and D. Freeman, Continuous Schauder frames for Banach spaces, J. Fourier Anal. and Apps., 26 , no. 4, (2020), 30 pages, https://doi.org/10.1007/s00041-020-09776-0
  12. J.A. Chavez-Dominguez, D. Freeman, and K. Kornelson Frame potential for finite-dimensional Banach spaces. , Linear Algebra and Applications, 578 (2019), 1--26.
  13. D. Freeman and D. Speegle, The discretization problem for continuous frames, Advances in Math., 345 (2019), 784--813.
  14. D. Freeman, E. Odell, B. Sari, and B. Zheng, On spreading sequences and asymptotic structures, (with , Trans. AMS, 370, no. 10, (2018) 6933--6953.
  15. D. Freeman, Th. Schlumprecht, and A. Zsak, Closed ideals of operators between the classical sequence spaces, Bulletin of the London Math. Soc., 49 , no. 5 (2017), 859--876.
  16. K. Beanland, D. Freeman, R. Causey, and B. Wallis Classes of operators determined by ordinal indices, J. Functional Analysis, 271, no. 1, (2016) 1691--1746.
  17. P. G. Casazza, D. Freeman, and R. Lynch, Weaving Schauder frames, J. Approximation Theory, 211 (2016) 42--60.
  18. F. Baudier, D. Freeman, Th. Schlumprecht, and A. Zsak, The metric geometry of the Hamming Cube and applications, Geometry and Topology, 20 (2016), 1427--1444.
  19. K. Beanland, D. Freeman, and P. Motakis The stabilized set of p's in Krivine's Theorem can be disconnected, Advances in Math., 281 (2015), 553--577.
  20. K. Beanland, D. Freeman, and R. Liu, Upper and lower estimates for Schauder frames and atomic decompositions, Fund. Math. 231 (2015), 161--188.
  21. D. Freeman, R. Hotovy, and E. Martin, Moving finite unit norm tight frames for Sn, Illinois J. Math., 58 (2014), no. 2, 311--322
  22. D. Freeman, E. Odell, Th. Schlumprecht, and A. Zsak, Unconditional structures of translates for Lp(Rd). Israel J. Math., 203 (2014), no. 1, 189--209.
  23. P.N. Dowling, D. Freeman, C.J. Lennard, E. Odell, B. Randrianantoanina, and B. Turett A weak Grothendiek compactness principle for Banach spaces with a symmetric basis. Positivity, 18 (2014), no. 1, 147--159.
  24. K. Beanland and D. Freeman, Uniformly factoring weakly compact operators. J. Functional Anal, 266, (2014), no. 5, 2921--2943.
  25. D. Freeman, E. Odell, B. Sari, and Th. Schlumprecht, Equilateral sets in uniformly smooth Banach spaces. Mathematika, 60 (2014), no. 01, 219--231.
  26. D. Freeman, D. Poore, A. R. Wei, and M. Wyse, Moving Parseval frames for vector bundles. Houston J. of Math., 40, (2014), no. 3, 817--832.
  27. P.N. Dowling, D. Freeman, C.J. Lennard, E. Odell, B. Randrianantoanina, and B. Turett, A weak Grothendiek compactness principle. J. Functional Analysis 263 (2012), no. 5, 1378--1381.
  28. S.A. Argyros, D. Freeman, R. Haydon, E. Odell, Th. Raikoftsalis, Th. Schlumprecht, and D.Z. Zisimopoulou, Embedding Banach spaces into spaces with very few operators. J. Functional Anal. 262 (2012), no. 3, 825--849.
  29. K. Beanland and D. Freeman, Ordinal ranks on weakly compact and Rosenthal operators. Extracta Mathematicae, 26 (2) (2011), 173--194.
  30. S. Dilworth, D. Freeman, E. Odell, and Th. Schlumprecht, Greedy bases for Besov spaces. Constructive Approx., 34 (2011), no. 2, 281--296.
  31. D. Freeman E. Odell, and Th. Schlumprecht, The universality of l_1 as a dual space. Math. Annalen. 351 (2011), no. 1, 149--186.
  32. D. Freeman, E. Odell, Th. Schlumprecht, and A. Zsak. Banach spaces of bounded Szlenk index II , Fund. Math. 205 (2009) 161--177.
  33. D. Freeman, Weakly null sequences with upper estimates. Studia Math. 184 (2008), no. 1, 79--102.
  34. K. Dykema, D. Freeman, K. Kornelson, D. Larson, M. Ordower, and E. Weber, Ellipsoidal tight frames and projection decompositions of operators. Illinois J. Math. 48 (2004), no. 2, 477--489.

Theses


  • Doctoral Dissertation: Upper estimates for Banach spaces, 2009. (advised by Thomas Schlumprecht)
  • B.S. thesis: Ellipsoidal and convex tight frames, 2003. (advised by Ruddy Gordh and Elwood Parker)