Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation |
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Authors: | Martin Bartelt John Swetits |
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Institution: | aDepartment of Mathematics, Christopher Newport University, Newport News, VA 23606, USA;bDepartment of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA |
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Abstract: | When G is a finite-dimensional Haar subspace of C(X,Rk), the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,Rk) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C(X,Rk). |
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Keywords: | Best Chebyshev approximation Vector-valued best approximation |
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