Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

  • Authors:

  • Martin Bartelt;John Swetits

  • Affiliations:

  • Department of Mathematics, Christopher Newport University, Newport News, VA 23606, USA;Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2007

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Abstract

When G is a finite-dimensional Haar subspace of CX,R^k, 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 CX,R^k the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Holder) condition of order 12. 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 CX,R^k.