On Certain Approximations of Vector-Valued Functions
For best approximations of vector-valued functions in certain norms, specialications of the Kolmogorov criterion are given. These specializations depend on the norm used in the range ℝn of the vector-valued functions. The case where the norm in ℝn is expressible in terms of a scalar product in ℝn and the case where the norm is the maximum norm are treated explicitly. Hints are given for other norms. There are examples of complex approximations given, which may be regarded as simultaneous approximations of two real functions defined on a two dimensional region.
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