- Discretizing L
_{p}norms, (joint work with Dorsa Ghoreishi), March 16 2021, 1 hour talk in Codex Seminar. We discuss the problem of discretizing the L_{p}-norm on subspaces and its connection with frame theory. It is known that if X is an n-dimensional subspace of L_{2}[0,1] then the L_{2}-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 L_{1}[0,1] which satisfies the necessary boundedness condition but the L_{1}-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 L_{2}-norm and the L_{1}-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 L
_{1}using only non-negative functions. We present this construction and explain why it does not work in L_{p}for p>1. We then discuss our construction of a Schauder basis for L_{2}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 L_{p}whose closed span contains L_{p}isomorphically as a subspace.

- W. Alharbi, D. Freeman, D. Ghoreishi, C. Lois, and S. Sebastian Stable phase retrieval and perturbations of frames, submitted, 13 pages.
- P. Balazs, D. Freeman, R. Popescu, and M. Speckbacher, Quantitative bounds for unconditional pairs of frames, submitted, 19 pages.
- D. Freeman, T. Oikhberg, B. Pineau, and M.A. Taylor, Stable phase retrieval in function spaces, submitted, 57 pages.
- W. Alharbi, S. Alshabhi, D. Freeman, and D. Ghoreishi, Locality and stability for phase retrieval, submitted, 14 pages.
- K. Beanland and D. Freeman, Shrinking Schauder frames and their associated spaces, submitted 18 pages.
- R. Calderbank, I. Daubechies, D. Freeman, and N. Freeman, Stable phase retrieval for infinite dimensional subspaces of L2(ℝ), submitted, 27 pages.
- 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
- D. Freeman, Th. Schlumprecht, and A. Zsák, Banach spaces for which the space of operators has 2
^{𝔠}closed ideals, Forum of Mathematics, Sigma,**9**, E27, (2021) 20 pages. doi:10.1017/fms.2021.23 - 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.
- D. Freeman, A.M. Powell, and M. Taylor, A Schauder basis for L
_{2}consisting of non-negative functions, Mathematische Annalen, (2021) 28 pages, https://doi.org/10.1007/s00208-021-02143-4 - 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 - J.A. Chavez-Dominguez, D. Freeman, and K. Kornelson Frame potential for finite-dimensional Banach spaces. , Linear Algebra and Applications,
**578**(2019), 1--26. - D. Freeman and D. Speegle, The discretization problem for continuous frames, Advances in Math.,
**345**(2019), 784--813. - D. Freeman, E. Odell, B. Sari, and B. Zheng, On spreading sequences and asymptotic structures, (with , Trans. AMS,
**370**, no. 10, (2018) 6933--6953. - 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. - 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. - P. G. Casazza, D. Freeman, and R. Lynch, Weaving Schauder frames, J. Approximation Theory,
**211**(2016) 42--60. - 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. - 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. - K. Beanland, D. Freeman, and R. Liu, Upper and lower estimates for Schauder frames and atomic decompositions,
Fund. Math.
**231**(2015), 161--188. - D. Freeman, R.
Hotovy, and E. Martin, Moving finite unit norm tight frames for S
^{n}, Illinois J. Math.,**58**(2014), no. 2, 311--322 - D. Freeman, E. Odell, Th. Schlumprecht, and A. Zsak, Unconditional structures of translates for
L
_{p}(R^{d}). Israel J. Math.,**203**(2014), no. 1, 189--209. - 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. - K. Beanland and D. Freeman, Uniformly factoring weakly compact operators. J. Functional Anal,
**266**, (2014), no. 5, 2921--2943. - D. Freeman, E. Odell, B. Sari, and Th. Schlumprecht, Equilateral sets in uniformly smooth Banach spaces.
Mathematika,
**60**(2014), no. 01, 219--231. - 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. - 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. - 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. - K. Beanland and D. Freeman, Ordinal ranks on weakly compact and Rosenthal operators. Extracta Mathematicae,
**26**(2) (2011), 173--194. - S. Dilworth, D. Freeman, E. Odell, and Th. Schlumprecht, Greedy bases for Besov spaces. Constructive Approx.,
**34**(2011), no. 2, 281--296. - D. Freeman E. Odell, and Th. Schlumprecht, The universality of l_1 as a dual space. Math. Annalen.
**351**(2011), no. 1, 149--186. - D. Freeman, E. Odell, Th. Schlumprecht, and A. Zsak. Banach spaces of bounded Szlenk index II , Fund. Math.
**205**(2009) 161--177. - D. Freeman, Weakly null sequences with upper estimates.
Studia Math.
**184**(2008), no. 1, 79--102. - 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.

- 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)