3. This question concerns the L metric d and the L¹ metric e on C[0, 1] (Examples 1.1 i)). a) Calculate d(f.g) and e(f.g), where f(x) = 1+z g(x) = x². b) * Explain why we have e(f.g) ≤ d(f.g) for all functions f and g. Under what conditions on f and g is e(f.g) = d(f.g)? and
3. This question concerns the L metric d and the L¹ metric e on C[0, 1] (Examples 1.1 i)). a) Calculate d(f.g) and e(f.g), where f(x) = 1+z g(x) = x². b) * Explain why we have e(f.g) ≤ d(f.g) for all functions f and g. Under what conditions on f and g is e(f.g) = d(f.g)? and
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
![3. This question concerns the L metric d and the L' metric e on C[0, 1] (Examples
1.1 1)).
a) Calculate d(S,9) and e(f,g), where
f(z) = 1+ z
and
9(z) =z.
b) Explain why we have e(f.g) s d(f.g) for all functions f and g. Under what
conditions on f and g is e(f.g) = d(f,g)?
%3D
In the remainder of the question, we treat an explicit example of the "function with
a very narrow tall bump" of Figure 1.14. For each integer n 2 2, define a function
InE C10,1] by
Tz+1-
Sn(z) -nz +1 + if ss+4
otherwise.
e)
Sketch the graph of fs, and describe the graph of fn for any n.
d) Let g(x) 0 be the zero function. Calculate d(fn.g) and e(fn.g).
e) ** Deduce that there is no number A such that
elf.9) Sd.g) S Ae(f.g)
for all f.ge C0,1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbebff674-1050-4cc1-9cfe-3ef2e154e613%2Ff0003ced-7796-47ff-8a81-a9d32c407828%2F8rfe9lw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. This question concerns the L metric d and the L' metric e on C[0, 1] (Examples
1.1 1)).
a) Calculate d(S,9) and e(f,g), where
f(z) = 1+ z
and
9(z) =z.
b) Explain why we have e(f.g) s d(f.g) for all functions f and g. Under what
conditions on f and g is e(f.g) = d(f,g)?
%3D
In the remainder of the question, we treat an explicit example of the "function with
a very narrow tall bump" of Figure 1.14. For each integer n 2 2, define a function
InE C10,1] by
Tz+1-
Sn(z) -nz +1 + if ss+4
otherwise.
e)
Sketch the graph of fs, and describe the graph of fn for any n.
d) Let g(x) 0 be the zero function. Calculate d(fn.g) and e(fn.g).
e) ** Deduce that there is no number A such that
elf.9) Sd.g) S Ae(f.g)
for all f.ge C0,1.
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