## An Introduction to the Approximation of Functions |

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adjoined to Pn alternating set consisting approximation problem approximation to/on Bernstein polynomials best approximation to/out best uniform approximation called changes sign 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 f(xj finite number finite point set 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 Ln(f 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 sn(g spline strictly convex subintervals subset Suppose Theorem 1.1 Tn(x trigonometric polynomial uniform norm unique weight function