3. Let d 2 2 and let P(z) = ao + ajz + ... + a4z“, ad # 0 be a polynomial with real coefficients such that la;| < 1 for j = 0,1, ...,d. The purpose of this exercise is to show the existence of a polynomial f(z) = co + c1z + . ..+ caz, where c; = ±1 for j = 0,1, ...,d and f well approximates P, i.e., max |P(z) – f(z)| = 0( vd log d), by employing the probabilistic method based on the next steps. (a) Show that P satisfies |P(z) – P(w)| < d²]z – w[ for all z, w e [-1, 1). Hint; you may show that |P'(z)| < d² for all |z] < 1.
3. Let d 2 2 and let P(z) = ao + ajz + ... + a4z“, ad # 0 be a polynomial with real coefficients such that la;| < 1 for j = 0,1, ...,d. The purpose of this exercise is to show the existence of a polynomial f(z) = co + c1z + . ..+ caz, where c; = ±1 for j = 0,1, ...,d and f well approximates P, i.e., max |P(z) – f(z)| = 0( vd log d), by employing the probabilistic method based on the next steps. (a) Show that P satisfies |P(z) – P(w)| < d²]z – w[ for all z, w e [-1, 1). Hint; you may show that |P'(z)| < d² for all |z] < 1.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.3: Change Of Basis
Problem 17EQ
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![+ aaz", as + 0 be a polynomial with real coefficients
3. Let d 2 2 and let P(z) = ao + aịz +
such that Ja;| < 1 for j = 0,1,...,d. The purpose of this exercise is to show the existence of a
polynomial f(z) = co + c1z + ...+ caz“, where c; = ±1 for j = 0,1,...,d and f well approximates
...
P, i.e.,
max |P(z) – f(z)| = 0( Vd log d),
by employing the probabilistic method based on the next steps.
(a) Show that P satisfies |P(z) – P(w)| < d²]z – w[ for all z, w e [-1, 1].
Hint; you may show that |P(z) < d² for all |z| < 1.
(b) Show that there exist independent random signs fo. E1....Ea such that EF;] = a; for
j = 0, 1, ..., d.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6326f55b-710c-40e0-9762-37e302c84bca%2F019c95b4-5fb4-491c-bf8f-4b9622a294e1%2Fy3dr4j_processed.png&w=3840&q=75)
Transcribed Image Text:+ aaz", as + 0 be a polynomial with real coefficients
3. Let d 2 2 and let P(z) = ao + aịz +
such that Ja;| < 1 for j = 0,1,...,d. The purpose of this exercise is to show the existence of a
polynomial f(z) = co + c1z + ...+ caz“, where c; = ±1 for j = 0,1,...,d and f well approximates
...
P, i.e.,
max |P(z) – f(z)| = 0( Vd log d),
by employing the probabilistic method based on the next steps.
(a) Show that P satisfies |P(z) – P(w)| < d²]z – w[ for all z, w e [-1, 1].
Hint; you may show that |P(z) < d² for all |z| < 1.
(b) Show that there exist independent random signs fo. E1....Ea such that EF;] = a; for
j = 0, 1, ..., d.
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