An Introduction to the Approximation of Functions

By T. J. Rivlin. 2003

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Citations in the corpus

Reinforcement learning in robotics: A survey (2013, methodology): “Function approximation (Rivlin, 1969) is a family of mathematical and statistical techniques used to represent a function of interest when it is computationally or informationtheoretically intractable to represent the function exactly or explicitly (e.”
Akaike-type criteria and the reliability of inference: Model selection versus statistical model specification (2010, background): “This is commonly justified on the basis that the net G(x) is dense in the interval [ ] in the sense that as →∞ the discrete metric converges to its continuous analogue (see Rivlin, 1981): lim →∞ ( P =1() |()−(;)”
Best uniform rational approximation ofxα on [0, 1] (2003, background): “=:£(«) m—»oo exists for each a > 0; however, an explicit expression for the constant ß(a), a > 0, still is not known.”
Sparse Recovery for Orthogonal Polynomial Transforms (2019, background): “Diﬀerent families of orthogonal polynomials give rise to diﬀerent quadrature rules for numerical integration [12,33].”
Linear barycentric rational quadrature (2012, background): “But it is well known that polynomial interpolation from equidistant samples is unstable and that the corresponding Lebesgue constant grows very fast with n (see [24] or [8] and the references therein).”
A ‘best points’ interpolation method for efficient approximation of parametrized functions (2008, background): “A Priori Framework In interpolation theory, the Lebesgue constants [21, 22, 9] measure how good the approximation of a function is in comparison with the best approximation.”
Bernstein series solutions of pantograph equations using polynomial interpolation (2012, background): “[18]Using the above notation, if f [ C1⁄2a; b , we get from (2.”
Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials (2014, background): “f ; x/ in terms of the modulus of continuity Theorem 5 ([39]) If f is bounded on Œ0, 1 , then”
A Bessel collocation method for numerical solution of generalized pantograph equations (2012, neutral): “[28] Using the above notation, if f ∈ C[a, b], we get from (16) and (17) ‖f − pnf ‖∞ ≤ 6(1 + n(T ))w ( b − a 2n )”
Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions (2013, background): “The Chebyshev polynomial on x 2 Œ 1, 1 of degree n is denoted by Cn, which can be defined as follows [45] Cn.”
Interpolation and Approximation by Polynomials (2003, methodology): “The proof given here is based on that given by Rivlin [48].”
The Bernstein polynomial basis: A centennial retrospective (2012, background): “Complete details on these proofs can be found in standard introductory texts [42, 46, 156, 169] on approximation theory.”