Research

My research is in the area of harmonic analysis, specifically in the study of frames. A frame is a generalization of a basis for a Hilbert space. Frame vectors are building blocks that can be combined to construct the elements in the space. The use of frames has contributed to many fields including wavelets, time-frequency analysis, and digital signal/image processing. The study of frames has connections to some very deep mathematical theory as well. I am currently interested working on emerging problems in dynamical sampling, i.e. a structured version of sampling in space and time.

I also study the harmonic properties of a collection of measures called Bernoulli convolutions. The supports of these measures are fractals contained in the real line, and the properties we can determine arise out of a self-similarity property (like a fractal - the object can be built by combining smaller versions of the whole).

Recently, I have participated in some mathematics education (RUME) projects with my colleagues at OU. One involves the development of two educational video games, one for use in Calculus class and the other to assist College Algebra students with the covariational aspect of functions. On another team, we take a qualitative look at how a mathematician (the expert) brings Calculus concepts to the students (novices) and trains the students to think more like experts. A newer project involves an analysis of the mathematical strengths and weaknesses of students as they arrive in their first mathematics course.

Please contact me if you are an undergraduate or graduate student interested in a reading course or research project in the areas of frames and dynamical sampling.

List of publications

  1. The Iteration and Design and Assessment for a Digital Game to Support Reasoning in a College Algebra Course, with X. Ge, S. Wilson, J. Mania Singer, W. Thompson, J. Lajos, B. Roper, J. Elizondo, S. Reeder, L. Williams, and M. Kleiser. In: Aprea C., Ifenthaler D. (eds) Game-based Learning Across the Disciplines. Advances in Game-Based Learning, Springer, Cham., 273-295.
    DOI: https://doi.org/10.1007/978-3-030-75142-5_12
  2. Considering the evolution of the STEM mathematical pathway at the University of Oklahoma using organizational development and change theory with Moore-Russo, D., Savic, M., and Andrews, C., PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, vol. 31 (3-5), 2021, 343-357.
    DOI: https://www.tandfonline.com/doi/full/10.1080/10511970.2020.1793852
  3. Frame potential for finite-dimensional Banach spaces with Chavez-Dominguez, J. A. and Freeman, D., Linear Algebra and its Applications, vol. 578, 2019, 1-26.
    arXiv Listing: arXiv:1804.03677 [math.FA] .
  4. Digital game based learning for undergraduate calculus education: Immersion, calculation, and conceptual understanding with Lee, Y.-H., Dunbar, N., Wilson, S., Ralston, R., Savic, M., Stewart, S., Lennox, E., Thompson, W., and Elizondo, J., International Journal of Gaming and Computer-Mediated Simulations, vol. 8 (1), 2016, 13-27.
  5. Additive spectra of the 1/4 Cantor measure, with P. Jorgensen and K. Shuman, "Operator Methods in Wavelets, Tilings, and Frames", Contemporary Mathematics, vol. 626, American Mathematical Society, Providence, RI, 2014, 121-128.
    arXiv Listing: arXiv: 1310:7274 [math.SP] .
  6. Scaling by 5 on the 1/4-Cantor measure, with P. Jorgensen and K. Shuman, Rocky Mountain Journal of Mathematics, vol. 44, no. 6, 2014, 1881-1901.
    arXiv Listing: arXiv:1111.4487v2 [math.FA] .
  7. Scalar spectral measures of an operator-fractal, with P. Jorgensen and K. Shuman. Journal of Mathematical Physics, vol. 55, 022103 (2014).
    arXiv Listing: arXiv:1204.5116v1 [math.SP] .
  8. Necessary and sufficient conditions to perform Spectral Tetris, with P. Casazza, A. Heinecke, Y. Wang, and Z. Zhou, Linear Algebra and its Applications, vol. 438 (2013), no. 5, 2239-55.
    arXiv Listing: arXiv:1204.3306v1 [math.FA] .
  9. An Operator-Fractal, with P. Jorgensen and K. Shuman, Numerical Functional Analysis and Optimization, vol. 33 (2012), no. 7-9, 1070-1094.
    arXiv Listing: arXiv:1109.3168v1 [math.OA] .
  10. Iterated function systems, moments, and transformations of infinite matrices , with P. Jorgensen and K. Shuman, Memoirs of the American Mathematical Society, vol. 213 (2011), no. 1003.
    arXiv Listing: arXiv:0809.2124v1 [math.CA].
  11. Families of spectral sets for Bernoulli convolutions, with P. Jorgensen and K. Shuman. Journal of Fourier Analysis and Applications, vol. 17 (2011), no. 3, 431-456.
    arXiv Listing: arXiv:0911.2435v1 [math.OA].
  12. Invariance of a shift-invariant space, with A. Aldroubi, C. Cabrelli, C. Heil, and U. Molter,
    Journal of Fourier Analysis and Applications, vol. 16 (2010), no. 1, 60-75.
    arXiv Listing: arXiv:0804.1597 [math.FA].
  13. Orthogonal Exponentials for Bernoulli Iterated Function Systems, with P. Jorgensen and K. Shuman,
    Current Trends in Harmonic Analysis and Its Applications: Wavelets and Frames Workshop in Honor of Larry Baggett, Birkhauser, 2008, 217-237.
    arXiv Listing: math.OA/0703385.
  14. Frames for Undergraduates, with D. Han, D. Larson, and E. Weber. Student Mathematical Library, vol. 40. American Mathematical Society, Providence, RI, 2007.
  15. Affine Systems: Asymptotics at Infinity for Fractal Measures , with P. Jorgensen and K. Shuman. Acta Applicandae Mathematica, vol. 98 (2007), no. 3, 181-222.
    arXiv Listing: math-DS/0707.1263.
  16. Harmonic Analysis of Iterated Function Systems with Overlap, with P. Jorgensen and K. Shuman. Journal of Mathematical Physics, vol. 48 (2007), no. 8, 083511, 35 pp.
    arXiv Listing: math-ph/0701066.
  17. Convolutional Frames and the Frame Potential , with M. Fickus, B. Johnson, and K. Okoudjou, Appl. Comput. Harmon. Anal., vol. 19 (1),2005, 77-91.
  18. Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group, Canadian J. Math. vol.57 (3), 2005, 598-615.
  19. Rank-One Decomposition of Operators and Construction of Frames, with D. Larson, "Wavelets, Frames, and Operator Theory", Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, 203-214.
    arXiv Listing: arXiv:math/0309236.
  20. Ellipsoidal Tight Frames and Projection Decomposition of Operators, with K. Dykema, D. Freeman, D. Larson, M. Ordower, and E. Weber, Illinois J. Math, Vol. 48, Number 2, Summer 2004, 477 - 489.
    arXiv Listing: arXiv:math/0304387.

Refereed Conference Proceedings

  1. Minding the gaps: Algebra skills of university calculus students, with D. Moore-Russo and S. Reeder, 2020. Paper presented at the Conference on Research in Undergraduate Mathematics Education 2020, Boston, MA and appears in the Proceedings from that conference.
  2. Dynamical sampling with a burst-like forcing term, with A. Aldroubi, L. Huang, and I. Kryshtal, Proceedings from Sampling Theory and Applications 2019; journal article in progress.
  3. Dynamical sampling with an additive forcing term, with A. Aldroubi, Proceedings from Sampling Theory and Applications 2015; journal article in progress.
  4. Investigating the Effectiveness of an Instructional Video Game for Calculus: Mission Prime, with S. Wilson, N. Dunbar, Y.-H. Lee, W. Thompson, R. Ralston, S. Stewart, M. Savic, and E. Lennox (2014). Paper presented at Conference on Research in Undergraduate Mathematics Education 2015, Pittsburgh, PA and appeared in the Proceedings of that conference.
  5. Balancing Formal, Symbolic, and Embodied World Thinking in First Year Calculus Lectures, with S. Stewart, C. Thompson, N. Brady, and L. Lifschitz (2014). Paper presented at Conference on Research in Undergraduate Mathematics Education 2015, Pittsburgh, PA and appeared in the Proceedings of that conference; journal article in progress.
  6. Digital game based learning for undergraduate calculus education: Immersion, calculation, and conceptual understanding with Lee, Y.-H., Dunbar, N., Wilson, S., Ralston, R., Savic, M., Stewart, S., Lennox, E., Thompson, W., and Elizondo, J. (2014). Paper was one of five Top Paper Award recipients and was presented at Meaningful Play 2014, East Lansing, MI. USA.
  7. A comparison of Fisher vectors and Gaussian super vectors for document versus non-document image classification, with D. Smith. Proceedings from SPIE Optics and Photonics, Applications of Image Processing Conference, San Diego, CA. August, 2013.

Frames for Undergraduates:

book logo

This book was written to be an accessible introduction to the intriguing and rapidly evolving subject of frame theory. It can be used in teaching either a special-topics course about frames or a second linear algebra course, where the frames serve as examples to the theory. This book could also serve as a reference for students doing research projects about frames.

Conferences and Events