On the Role of ℓ∞ in Approximation Theory

B. L. Chalmers and B. Shekhtman
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B. L. Chalmers: University of California, Department of Mathematics
B. Shekhtman: University of South Florida, Department of Mathematics

A chapter in Approximation, Probability, and Related Fields, 1994, pp 151-160 from Springer

Abstract: Abstract Given a family of finite-dimensional subspaces V n in C[0,1], it is important to develop an algorithm that to a given function f ∈ C[0,1] assigns an approximation P n f ∈ V n . This algorithm should be “ simple ” and “ good ”. That translates into P n being a continuous linear map from C[0,1] in V n such that ∥f − P n f∥ ≤ C dist(f, V n ) where the constant C does not depend on n. Using the principle of uniform boundedness it is easy to show that the above-mentioned conditions are equivalent to the existence of projections (linear, idempotent operators) P n from C[0,1] onto V n such that ∥P n∥ ≤ C.

Keywords: Unit Ball; Minimal Projection; Isometric Copy; Projection Constant; Idempotent Operator (search for similar items in EconPapers)
Date: 1994
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DOI: 10.1007/978-1-4615-2494-6_10

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