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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![4. 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 (an) 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 = 9n + Sn.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe591e3e-c1e9-4180-ad99-642a1f427277%2F2220ee5b-8c3e-4701-aab9-d0a93776ea21%2Fnydb9r_processed.png&w=3840&q=75)
Transcribed Image Text:4. 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 (an) 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 = 9n + Sn.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
