Research

Research Interests

Moment Problem, Applied Analysis, Control Systems, Finance Mathematics, Continued Fractions, Mathematics Education

Current Research Projects

1. Sparse Moment Problem

    Moments of a geometric object conveys a lot of its topological information such as how round it is, where it is located, what direction it is tapered, where its mass is centered, etc. For simple objects like an ellipsoid, just a few terms in the moment sequence are enough to identify it. However, only infinitely many terms can uniquely identify more complicated shapes like a polygon or a quadrature domain. While numerous applications of moment problems with a complete set of moments have been identified, they are mostly limited to theoretical observations. For practical implementation of moment problems, it is vital to be able to deal with missing moment data since data obtained from physical experiments and phenomena are often corrupt or incomplete. We call a positive subsequence of a moment sequence a sub-moment sequence and the moment problem of sub-moment sequence a sub-moment problem. The main goal of this research is to give a complete characterization of the sub-moment problem and to develop a deep connection between the measures arising from the sub-moment problem and the original moment problem. Recent advances in image reconstruction from sparse MRI data, partial data transmission and sparse signals, make this problem timely and of broad interest.

    $\int +C+xy$ The Hamburger Theorem is stated as: a sequence {s_k}_k=0^∞ {sk}k=0 can be represented as
    moments sk= xk dσ(u), k=0,1,2,....
    , where σ(u) is positive measure, if and only if for any finite set of real numbers xi,
    m i,j=0 xixjsi+j≥ 0, for every m=0,1,2,3,....

    Later, mathematicians including R. Nevanlinna, M. Riesz, T. Carleman, F. Hausdorff, M. Stone, and C. Carathéodory further studied moment problems in further details. There are several variations of this moment problem that have their own specialties of applications. Hausdorff's moment problem deals with finding a function σ(u) with support on the closed unit interval [0,1], while Stieltjes moment problem is concerned with the measures on the positive real numbers.

    The problem of moments stands as a very important problem in analysis up to the present day. Results on moment problems have numerous applications in the areas of extremal problems, optimization theory and limit theorems in probability theory. Rather than the usual infinite moment sequence problem, the truncated moment problem is more applicable in mathematical and physical sciences. Further, multivariable moment problem is more applicable than a single variable one.

    Various generalizations and extensions of the problems of moments have been studied to deal with several applications. Some generalizations include replacing the sequence of functions {un} with a more general sequence fn(u) such as {einu}, and replacing integrals with more general functionals in abstract spaces. An important variation of the moment problem I am interested in is the moment problem with some of the moments missing. Moment data obtained from various experiments are often corrupt and incomplete. Such reconstruction of moments can be then applied in problems associated with realization of a signal or an image. Reconstruction of a function from moment sequences with missing terms is an interesting problem leading to several advances in image and/or signal reconstruction. I am mostly interested in finding techniques for reconstructing the missing moment data.

    Several authors have studied this problem by replacing the missing moments with an appropriate value, leading to a perturbation of the moment sequences followed by the question of sensitivity of the corresponding orthogonal polynomials. Because of necessary requirements for a sequence to be moments, this approach turns out to be very restrictive in choosing and replacing the missing moments. Thus, we need to look for ideas to extract a positive subsequence of a positive sequence and look for its moment solution.

    There are primarily two questions to answer:

      1. How do we characterize positive subsequences of a positive sequence?
      2. How do we characterize the corresponding non-decreasing measures arising from the sub-moment data and what, if any, is the relationship of these measures to the original measures?

    We have established some results that explicitly gives some methods for extracting positive subsequences. We have established necessary and sufficient conditions for a subsequence of a moment sequence to be positive. These results give a complete answer to the problem of Hankel matrix completion. We have also carved out a relation between a moment solution and one of its sub-moment solution in terms of their Hilbert transforms.

    One of the current projects we are working on is multidimensional case of the sparse moment problem. Data obtained from physical experiments and natural processes are often moment multisequences. We want to establish conditions for a subsequence of a moment multisequence to be positive. Positivity is not enough to guarantee that a multisequence be a moment sequence. Generalization of the Hamburger moment problem to higher dimensions is non-trivial. While the integral representation of a multisequence in itself is a very interesting question, a deeper question in fact is the case when some of the terms in a multisequence are missing. We want to establish a connection between a moment solution of a subsequence of a multisequence and its original moment solution.

    Another work in progress deals with methods of continued fractions for sparse moment sequences. The connection between continued fractions and moment sequences date back to the seminal work of Stieltjes. In the case of missing moment data, as it may be in the case of sparse moment sequences, the continued fractions will be significantly different from those associated with the original moment data. We have been studying the relationship between the asymptotic behavior of these continued fractions and the original continued fractions.

    For future work, the top agenda of my research interests is finding a precise relation between solution of sparse moment problem and the original moment problem remains. To achieve this, I plan to identify various approaches and techniques. An approach I am interested in is applying numerical and computational methods in approximating the sub-moment measures. Then the approximate sub-moment measure can be used in applications concerning reconstruction of data.




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