Mathematics Colloquia and Seminars

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Suppose at each point in a closed subset of R^n, we are given a degree m polynomial. How can we tell there exists an M-times continuously differentiable function F whose m-th degree Taylor polynomial agrees with the given polynomials and whose derivatives are controlled by the coefficients of the polynomials? Here M is an integer bigger than or equal to m. I will survey some known results and pose some related questions.