Interpolation and Approximation

When you can't represent a function exactly, you settle for a polynomial (or spline) that's close enough, this course is about making "close enough" precise, asking when a best approximation exists, when it's unique, and how fast the error shrinks as you raise the degree. You'll work through written problem sets and exams proving classical results: Chebyshev's equioscillation, Jackson and Bernstein's links between smoothness and convergence rate, Haar conditions for uniqueness, and the behavior of Lebesgue constants that decide whether interpolation actually converges. It sits on top of real analysis and linear algebra, and feeds directly into numerical analysis, spectral methods, and any computational work where you need to bound how badly a discrete model can lie about a continuous object.