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Functional analysis - Parseval

I'm trying to find the infimum of

∫−ππ∣t+t2−∑i=1najsin(jt)∣2 dt \int^\pi_{-\pi} |t+t^2-\sum_{i=1}^n a_j sin(jt)|^2\ dt

as n ranges through N\mathbb{N} and a_j through R\mathbb{R}

This obviously seems to use parseval in some manner, but I'm stuck for how to proceed. A hint with the question mentions very few calculations are necessary, so it's obviously something quite simple. I'm guessing that the infimum is when the a_j are fourier coefficients of some relevant function, but the lack of a cosine term seems to be an issue. Any advice much appreciated.
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DFranklin
18
Somewhat making this up as I go along, but t+t2=∑j=1∞(bjsin⁡(jt)+cjcos⁡(jt)) t+t^2 = \sum_{j=1}^\infty (b_j \sin(jt) + c_j \cos(jt)) for suitable choices of b_j and c_j (i.e. the Fourier series).

You can then use Parseval to find what the a_j have to be.

I'm fairly sure that a bit of thought will allow you to avoid having to actually find the Fourier series involved.
(edited 12 years ago)

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