My research interests lie in the areas of applied harmonic analysis, applied linear algebra, and approximation theory, with focus on frame theory and signal processing. Publications

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A key idea in harmonic analysis is to decompose a given function into "simpler" components referred to as a representative set. This makes it easier to study the properties of the function through simpler functions and can give an efficient representation of the function. In this sense, choosing an appropriate representative set is important.

• Finite frames (Applied Linear Algebra): In finite dimensions, most things can be done using linear algebra. A basic idea in linear algebra is to express a given object in some space in terms of elements in a representative set like a basis. When dealing with different kinds of data sets, the structure of this representative set becomes crucial for efficient storage and transmission. Frames are representative sets like bases but are redundant. The redundancy allows more flexibility and freedom of choice. Frames have now become standard tools in signal processing due to their resilience to noise and transmission losses. This topic is appropriate for students who have taken some undergraduate linear algebra, and are familiar with the concept of a basis and its fundamental properties. A nominal knowledge of complex numbers is needed.
  
  Here is a book chapter on undergraduate research in Finite Frame Theory that appears in A Primer for Undergraduate Research: From Groups and Tiles to Frames and Vaccines of the series Foundations for Undergraduate Research in Mathematics, Springer, 2017.
  
  
• General frames: In infinite dimensions, things are more subtle, and one needs advanced tools from analysis. For instance, one might have to take into account the various notions of basis and/or deal with issues arising from convergence of infinite sums.
  
  
• Frame design: A lot of my researcch involves frame design issues. In some situations, one might seek a "sparse" representation of a signal. In other situations, one might have to use a subset of a frame to deal with transmission loss. Given the goal, properties like "equiangularity", "equal-norm", or "tightness" might be desirable. Consequently, one wishes to design frames having some "desirable" properties. When working with infinite dimemsional function spaces, generalizing some concepts such as equiangularity becomes tricky, but one might seek frames with a specific structure such as a Gabor structure or a wavelet structure. Such studies involve concepts from real, complex, and functional analysis.

Frame theory is a very active area of research that is interdisciplinary. If you are a Ph.D. student interested in working in this area feel free to contact me.