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    Say I have this, where Q_k is a polynomial.

    P_N(x)=\sum_{k=0}^N{\frac{2}{N+1  }\left(\sum_{i=0}^N{f(x_i)\cos( \frac {k\pi (i+0.5)}{N+1})}\right)Q_k(x)}

    I want to take the limit of P_N as N tends to infinity. Obviously I can't replace the Ns on the RHS with the symbol infinity. Is there any way to um, make it simpler? I want a nice expression and I don't want to leave it as just lim_{N -> infinity} P_N.

    Any tips in the right direction would be helpful, thanks.
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    Depending on what you mean by f(x_i), you could possibly look at the sum of cosines being the real part of a geometric complex series.
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    I just tried looking at the geometric series... unfortunately I can't seem to find a f(x) that doesn't involve N such that it gets rid of the 1/N+1 in the outer sum (and some other nasty bits).
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    Without some restrictions on f and Q, I don't think you can even assume the limit function exists.
 
 
 
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Updated: April 9, 2011
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