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Pseudo-differential equations on the sphere

PDE and Applied Math Seminar

Speaker: Hrushikesh Mhaskar, Cal State, LA
Location: 1147 MSB
Start time: Thu, Jan 17 2008, 11:00AM

We study the solutions of an equation of the form Lu = f , where L is a pseudo-differential operator defined for functions on the unit sphere embed- ded in a Euclidean space, f is a given function, and u is the desired solution. We assume that the eigenvalues of L are known in advance, and demon- strate a method to solve such an equation, starting with the values of f at arbitrary sites. One has no control on the choice of these sites. By using carefully constructed quadrature formulas and approximation operators, we obtain a spectral rate of convergence. Our solution is a single spherical poly- nomial, but the accuracy of approximation on different parts of the sphere is commensurate with the smoothness of f on these parts. Once the quadrature formula is constructed, we no longer need to solve any system of equations, unlike the collocation method. This is joint work with Q. T. Le Gia.