It is well known that an algebraic (or trigonometric) polynomial is completely determined by its values on a small interval. Therefore, such traditional methods of function approximation as least squares fit, Fourier projection, and interpolation are unsatisfactory for approximating functions which are generally ``good'' but have a few ``bad points''. We discuss the construction of polynomials whose degree of approximation on different parts of the domain is commensurate with the local smoothness of the target function on a neighborhood of this domain. Our constructions are based on either Fourier information or values of the target function, where one has no control on the location of the points at which the function is sampled. The approximating polynomial is the same on the whole domain; only the approximation behavior changes on different parts in an almost spline-like manner. The constructions are not adaptive, and do not require an a priori knowledge about the behavior of the target function on different parts of the domain. Applications include direction finding in linear phased array antennas, and solutions of pseudo-differential equations on the sphere.
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