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showed that certain sine and cosine function formed an orthogonal system. 1. How would you construct an orthonormal system from that orthogonal system? 2. What would the Fourier decomposition of a function ƒ € C([-, π]) look like with respect to the orthonormal system you have constructed?

showed that certain sine and cosine function formed an orthogonal system. 1. How would you construct an orthonormal system from that orthogonal system? 2. What would the Fourier decomposition of a function ƒ € C([-, π]) look like with respect to the orthonormal system you have constructed?

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.3: Orthonormal Bases:gram-schmidt Process
Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...
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On page 125, our textbook refers to a representation v=1 (V, Vk)V of a vector V with respect to an orthonormal basis {V1, V2,..., Vn} as (a baby version of) the abstract orthogonal Fourier decomposition, and then claims that "the classical Fourier decomposition deals with a concrete orthonormal system (sines and cosines)". Refer back to last week's discussion, where you showed that certain sine and cosine function formed an orthogonal system. 1. How would you construct an orthonormal system from that orthogonal system? 2. What would the Fourier decomposition of a function f € C([-, π]) look like with respect to the orthonormal system you have constructed?
Transcribed Image Text:On page 125, our textbook refers to a representation v=1 (V, Vk)V of a vector V with respect to an orthonormal basis {V1, V2,..., Vn} as (a baby version of) the abstract orthogonal Fourier decomposition, and then claims that "the classical Fourier decomposition deals with a concrete orthonormal system (sines and cosines)". Refer back to last week's discussion, where you showed that certain sine and cosine function formed an orthogonal system. 1. How would you construct an orthonormal system from that orthogonal system? 2. What would the Fourier decomposition of a function f € C([-, π]) look like with respect to the orthonormal system you have constructed?
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