4 Using V; to approximate functions in L²([0, 1)) A function f e V; has the form f (x) = a,21/26(2'x) + a,2'/²¢(2'x – 1) +· … +azi-12'/² p(2'x – (2' – 1)). (7) Since the functions on the right side form an orthonormal set, the coefficients ak are given by the formula ar =< f(x), 2i/² (2°x – k) >= [ f(x) · 2i/²6(2'x – k) dæ (8) Exercise 2 Take the scalar product with 2/2$(2'x – k) on both sides of (7) to verify Formula (8).
4 Using V; to approximate functions in L²([0, 1)) A function f e V; has the form f (x) = a,21/26(2'x) + a,2'/²¢(2'x – 1) +· … +azi-12'/² p(2'x – (2' – 1)). (7) Since the functions on the right side form an orthonormal set, the coefficients ak are given by the formula ar =< f(x), 2i/² (2°x – k) >= [ f(x) · 2i/²6(2'x – k) dæ (8) Exercise 2 Take the scalar product with 2/2$(2'x – k) on both sides of (7) to verify Formula (8).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![# 4 Using \( V_j \) to Approximate Functions in \( L^2([0,1]) \)
A function \( f \in V_j \) has the form
\[
f(x) = a_0 2^{j/2} \phi(2^j x) + a_1 2^{j/2} \phi(2^j x - 1) + \cdots + a_{2^j - 1} 2^{j/2} \phi(2^j x - (2^j - 1)).
\]
(Since the functions on the right side form an orthonormal set, the coefficients \( a_k \) are given by the formula)
\[
a_k = \langle f(x), 2^{j/2} \phi(2^j x - k) \rangle = \int_0^1 f(x) \cdot 2^{j/2} \phi(2^j x - k) \, dx.
\]
**Exercise 2**
Take the scalar product with \( 2^{j/2} \phi(2^j x - k) \) on both sides of (7) to verify Formula (8).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc11d54a1-f970-40d5-8a22-4b336286a5a0%2Fb2277f34-8648-48b6-8589-a956ac902e0a%2F7mkvs4k_processed.png&w=3840&q=75)
Transcribed Image Text:# 4 Using \( V_j \) to Approximate Functions in \( L^2([0,1]) \)
A function \( f \in V_j \) has the form
\[
f(x) = a_0 2^{j/2} \phi(2^j x) + a_1 2^{j/2} \phi(2^j x - 1) + \cdots + a_{2^j - 1} 2^{j/2} \phi(2^j x - (2^j - 1)).
\]
(Since the functions on the right side form an orthonormal set, the coefficients \( a_k \) are given by the formula)
\[
a_k = \langle f(x), 2^{j/2} \phi(2^j x - k) \rangle = \int_0^1 f(x) \cdot 2^{j/2} \phi(2^j x - k) \, dx.
\]
**Exercise 2**
Take the scalar product with \( 2^{j/2} \phi(2^j x - k) \) on both sides of (7) to verify Formula (8).
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