Prove that given a continuous function f on [a, b], there exists a sequence (Pn) of polynomials such that pn →f uniformly on [a, b], and for each n, pn(a) = f(a) and pn(b) = f(b). Hint: Select a sequence (gn) of polynomials such that qnf uniformly on [a, b]. For each n, let sn be the function on R whose graph is the straight line passing through the points (a, f(a)- qn(a)) and (b, f(b) — qn (b)). Set pn = qn + Sn.
Prove that given a continuous function f on [a, b], there exists a sequence (Pn) of polynomials such that pn →f uniformly on [a, b], and for each n, pn(a) = f(a) and pn(b) = f(b). Hint: Select a sequence (gn) of polynomials such that qnf uniformly on [a, b]. For each n, let sn be the function on R whose graph is the straight line passing through the points (a, f(a)- qn(a)) and (b, f(b) — qn (b)). Set pn = qn + Sn.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 74E
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