Research

My research interests include wavelets, frames, and iterated function systems. A frame can be thought of as a generalization of an orthonormal basis for a Hilbert space. The use of frames has greatly broadened a variety of fields including wavelets, time-frequency analysis, and digital signal processing. The study of frames has connections to some very deep mathematical theory as well, including the 1959 Kadison-Singer problem.

I am currently looking at harmonic properties of the equilibrium measures which arise from iterated function systems. The supports of these measures are fractals, and the properties we can determine arise out of a self-similarity which results from the iterative scheme used to create the measures. One area of interest is whether or not the associated L² Hilbert space contains Fourier bases or frames, i.e. orthonormal bases or frames consisting of exponential functions.

Publications

Non-Integer Translation Invariant Systems, with A. Aldroubi, C. Cabrelli, C. Heil, and U. Molter, accepted for publication in Journal of Fourier Analysis and Applications.
The online publication is now available.

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.

Frames for Undergraduates, with D. Han, D. Larson, and E. Weber. Student Mathematical Library, vol. 40. American Mathematical Society, Providence, RI, 2007.

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.

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.

Convolutional Frames and the Frame Potential , with M. Fickus, B. Johnson, and K. Okoudjou, Appl. Comput. Harmon. Anal., vol. 19 (1),2005, 77-91.

Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group, Canadian J. Math. vol.57 (3), 2005, 598-615.

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.

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.

Work in Preparation

Iterated function systems, moments, and transformations of infinite matrices , with P. Jorgensen and K. Shuman, submitted.
arXiv Listing: arXiv:0809.2124v1 [math.CA].

Families of spectral sets for Bernoulli convolutions, with P. Jorgensen and K. Shuman, submitted.
arXiv Listing: arXiv:0911.2435v1 [math.OA].

Work in Progress

Ergodic properties of isometries on fractals, with P. Jorgensen and K. Shuman.
Fourier duality properties in IFS measures, with P. Jorgensen and K. Shuman.

Upcoming Conferences

January 16, 2010 Optimal Frames and Operator Algebras, Special Session at the Joint Mathematics Meetings, San Francisco, CA.

April 10-11, 2010 Fractals, Convolution Measures, and Frames, Special Session at the Spring Central Section meeting of the AMS.

Recent Conferences

July 6-10, 2009 Workshop on Fractals and Tilings , Strobl, Austria.

February 5-8, 2009 FL-IA-CO-OK Workshop on Fractal Measures, University of Iowa, Iowa City, IA.

Keri's Home Page	Curriculum Vitae	Teaching Information	OU Mathematics

My email address: kkornelson "at" math "dot" ou "dot" edu