Mathematics Colloquia and Seminars

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Let f be a real valued function defined on a subset E of the real line. How do we know if there is a nonnegative, twice continuously differentiable F defined on the entire real line such that F agrees with f on E? What if we require the graph of F to lie between two horizontal lines? How can we answer the analogous questions in R^n instead of the real line? If an F satisfying the constraints exists, can we efficiently construct one? These types of problems have significant applications in engineering, computer graphics, statistics, and physics. For example, in statistics, when fitting a probability density function to a collection of points, the graph is naturally constrained to be nonnegative everywhere; in physics, when we are fitting energy function or (absolute) temperature function to data, the underlying function is intrinsically nonnegative. In this talk, I will explain the progress on these problems.