1. Let (a, bj be an interval in R and let be a continuous function on (a, b). (i) Check that the function z+ (b – a)r+ a is a bijection from [0,1] to [a, b) and its inverse is z+ (ii) For z € [0, 1], let g(r) = »((b-a)r+a) (which is equivalent to say that (z) = 9(=). Check that g is continuous on [0, 1]. (iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges uniformly to g on [0, 1]. Construct a sequence of polynomials (p)n such that | – Pa|l.ja.bj = ||g -
1. Let (a, bj be an interval in R and let be a continuous function on (a, b). (i) Check that the function z+ (b – a)r+ a is a bijection from [0,1] to [a, b) and its inverse is z+ (ii) For z € [0, 1], let g(r) = »((b-a)r+a) (which is equivalent to say that (z) = 9(=). Check that g is continuous on [0, 1]. (iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges uniformly to g on [0, 1]. Construct a sequence of polynomials (p)n such that | – Pa|l.ja.bj = ||g -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![4. Let [a, b] be an interval in R and let be a contimuous function on [a, b).
(i) Check that the function r+ (b- a)x + a is a bijection from [0, 1] to [a, b] and its inverse is I+E.
(ii) For r € [0, 1], let g(x) = »((b-a)x+a) (which is equivalent to say that (r) = g(). Check that
g is continuous on [0, 1].
(iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges
uniformly to g on [0, 1]. Construct a sequence of polynomials (pn)n such that || – Pn|.ja.bj = ||g –
(iv) Deduce the Weiestrass approximation theorem for the continuous function on [a, b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae257d6e-8d1d-4070-b937-ad737dc001ea%2F5b751ccb-3daa-40ca-a30a-50d06687179b%2F67nddkz_processed.gif&w=3840&q=75)
Transcribed Image Text:4. Let [a, b] be an interval in R and let be a contimuous function on [a, b).
(i) Check that the function r+ (b- a)x + a is a bijection from [0, 1] to [a, b] and its inverse is I+E.
(ii) For r € [0, 1], let g(x) = »((b-a)x+a) (which is equivalent to say that (r) = g(). Check that
g is continuous on [0, 1].
(iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges
uniformly to g on [0, 1]. Construct a sequence of polynomials (pn)n such that || – Pn|.ja.bj = ||g –
(iv) Deduce the Weiestrass approximation theorem for the continuous function on [a, b].
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