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Simultaneous approximation by polynomials

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ASC - Approximation, sampling and compression in data science

Least square polynomials in an $L2$ space are partial sums of the Fourier orthogonal expansions. If we were to approximate functions and their derivatives simultaneously on a domain in $Rd$ (as desired in spectral method), we would need to consider orthogonal expansions in a Sobolev space, for which the orthogonality is defined with respect to an inner product that contains derivatives. Since multiplication operators are no longer self-adjoint under such an inner product, the orthogonality is hard to understand and analyze. In the talk we will explain what is known.



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