Application: Fourier seriesReview. Given an orthogonal basis v1, v2,, we express a vector x asx = c1v1+ c2v2+, ci=A Fourier serie s of a function f(x) is an infinite expansion:f(x) = a0+ a1cos(x) + b1sin(x) + a2cos(2x ) + b2sin(2x) +• We are working in the infinite dimens ional vector space of functions.More precisely, we are working with (say, continuous) functions that are periodic with period 2π.• The functions1, cos (x), sin (x), cos (2x), sin (2x),are a basis of this space. In fact, an orthogonal basis!That’s the reason for the success of Fourier series.What is the inner product on the space of functions?• Vectors: hv , w i =• Functions: hf , gi =Why these limits?Example 1. Show that cos (x) and sin (x) are orthogonal.Solution.More generally, 1, cos (x), sin (x), cos (2x), sin (2x),are all orthogonal to eac h other.Armin [email protected] 2. What is the norm of cos (x)?Solution.Example 3. How do we find a1?Or: how much cosine is in a functionf (x)?Solution.f(x) has the Fourier seriesf (x) = a0+ a1cos(x) + b1sin(x) + a2cos(2x) + b2sin(2x) +whereak=hf (x), cos (kx)ihcos (kx), cos (kx)i=bk=hf (x), sin (kx)ihsin (kx), sin (kx)i=a0=hf (x), 1ih1, 1i=Armin [email protected] 4. Find the Fourier series of the 2π-periodic function f(x) defined byf (x) =−1, for x ∈ (−π, 0),+1, for x ∈ (0, π).−π π 2π 3π 4πSolution.Note. We just observed the following gene r a l principle: an odd function is orthogon a lto ...f(x) is odd and the cosines are even functions, so ...Armin [email protected] 5. Consider the spa c e of 1-periodic functions.• What does a Fourier series for a 1-periodic f(x) look like?• What should be our inner prod uct for Fourier series?• How are the Fourier coefficients computed?Solution.Armin
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