2. In this exercise we show how the symmetries of a function imply certain properties of its Fourier coefficients. Let ƒ be a 27-periodic Riemann integrable function defined on R. (a) Show that the Fourier series of the function f can be written as f(0) ~ ƒ(0) + Ll(n) + ƒ(-n)] cos no + i[f(n) – ƒ(-n)] sin nô. n21 (b) Prove that if ƒ is even, then f (n) = ƒ(-n), and we get a cosine series.

2. In this exercise we show how the symmetries of a function imply certain properties of its Fourier coefficients. Let ƒ be a 27-periodic Riemann integrable function defined on R. (a) Show that the Fourier series of the function f can be written as f(0) ~ ƒ(0) + Ll(n) + ƒ(-n)] cos no + i[f(n) – ƒ(-n)] sin nô. n21 (b) Prove that if ƒ is even, then f (n) = ƒ(-n), and we get a cosine series.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.5: Applications

Problem 27EQ

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