I want to show that if $f(x)\in L^{p}(p>1)$ and $\phi(x)\in L^{q}$, where $\displaystyle \frac{1}{p}+\frac{1}{q}=1$ then the trigonometric series $\displaystyle \frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx)$ is a Fourier series of function $f(x)$,where for every function $\phi(x)\in L^{q}$ with Fourier coefficients $\alpha_{n}, \beta_{n}$, the series $\displaystyle \frac{a_{0}\alpha_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}\alpha_{n}+b_{n}\beta_{n})$ is convergent.
Trigonometric series as a Fourier series.
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fourier-series
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0Do you mean that the trigonometric series converges a.e. to $f\in L^p$ ? – 2012-05-26
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0what you need to know is that $\frac {a_0} 2 + \sum^N a_n cos(nx) + b_n sin(nx) \rightarrow f$ in $\mathcal L^p$, which is true and can be found in Katznelson. The rest is straightforward $\mathcal L^p \mathcal L^q$ duality stuff. – 2012-05-26