Uniqueness and non-uniqueness in bivariate \(L^ 1\)-approximation. (English) Zbl 0551.41035
Approximation theory IV, Proc. int. Conf., Tex. A&M Univ. 1983, 509-514 (1983).
[For the entire collection see Zbl 0533.00014.]
We approximate continuous functions h over a fundamental square in the sense of the \(L^ 1\)-norm. Thereby we ask for such functions which have more than one proximum in a given set of approximators. As such we employ trigonometric and algebraic blending functions. Our central results state that in both cases non-uniqueness appears. On the other hand uniqueness is assured under appropriate restrictions for the approximated functions h. For comparison and contrast we report results concerning approximation by bivariate polynomials.
We approximate continuous functions h over a fundamental square in the sense of the \(L^ 1\)-norm. Thereby we ask for such functions which have more than one proximum in a given set of approximators. As such we employ trigonometric and algebraic blending functions. Our central results state that in both cases non-uniqueness appears. On the other hand uniqueness is assured under appropriate restrictions for the approximated functions h. For comparison and contrast we report results concerning approximation by bivariate polynomials.
