An Introduction to the Approximation of Functions
This graduate-level text offers a concise but wide-ranging introduction to methods of approximating continuous functions by functions depending only on a finite number of parameters. It places particular emphasis on approximation by polynomials and not only discusses the theoretical underpinnings of many common algorithms but also demonstrates their practical applications. 1969 edition.
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adjoined to Pn Ak(X alternating set consisting approximation problem approximation to/on array of nodes Bernstein polynomials best approximation to/out best uniform approximation bound called changes sign chapter Chebyshev nodes Chebyshev polynomials conclude constant continuous function converges uniformly Corollary defined denote distinct points distinct zeros En(f equations example Exercise exists extreme point finite number finite point set follows hence Hint hypernormal implies inequality integral interpolating polynomials interval Jackson's Theorem Jacobi polynomials leading coefficient least-first-power approximation least-squares approximation Legendre polynomials Lemma linear space mathematical max f(x modulus of continuity nomials nonnegative obtain orthogonal polynomials p e Pn pePn points of Xm polynomial interpolation polynomial of degree rational function real numbers result Rolle's Theorem S(Xn satisfies Show spline strictly convex subintervals subset Suppose Theorem 1.1 theorem is proved Tn(x trigonometric polynomial uniform norm unique weight function xeXm