We dene an inner product on v by means of the formula

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Unformatted text preview: e1 (t) + f (t), e2 (t) e2 (t) + · · · . ˆ In other words, 11 f (t) = f , √ √ + f , cos t cos t + f , cos 2t cos 2t + . . . 22 + f , sin t sin t + f , sin 2t sin 2t + . . . , a0 = + a1 cos t + a2 cos 2t + . . . + b1 sin t + b2 sin 2t + . . . , 2 where a0 = f , 1 , a1 = f , cos t , a2 = f , cos 2t , b1 = f , sin t , ... , b2 = f , sin 2t , ... . Use of the inner product makes the formulae for Fourier coefficients almost impossible to forget. We say that a function f (t) is odd if f (−t) = −f (t), even if f (−t) = f (t). Thus sin t, sin 2t, sin 3t, ... are odd functions, while 1, cos t, cos 2t, cos 3t, ... are even functions. Let Wodd = {f ∈ V : f is odd}, Weven = {f ∈ V : f is odd}. Then f, g ∈ Wodd ⇒ f + g ∈ Wodd and cf ∈ Wodd , for every choice of real number c. Thus we can say that Wodd is a linear subspace of V . Similarly, Weven is a linear subspace of V . It is not difficult to show that f ∈ Wodd , g ∈ Weven ⇒ f , g = 0; in other words, the linear subspaces Wodd and Weven are orthogonal to ea...
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This document was uploaded on 01/12/2014.

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