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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.

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Do you mean that the trigonometric series converges a.e. to $f\in L^p$ ? – AD. May 26 '12 at 6:36
what 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. – mike May 26 '12 at 14:48

Let $f_n$ and $\phi_n$ be the partial sums of Fourier series of $f$ and $\phi$, respectively. Then (see below) $\|f-f_n\|_{L^p}\to 0$ and $\|\phi-\phi_n\|_{L^q}\to 0$. It follows that $$ \sum_{k\le n} (a_k\alpha_k+b_k\beta_k) = \int_{\mathbb T} f_n \phi_n \to \int_{\mathbb T} f \phi $$ which proves the convergence of the series on the left.

The convergence of Fourier series in $L^p$ is a standard fact, found (as mike suggested) in An Introduction to Harmonic Analysis by Katznelson. The ingredients of the proof are:

  • Since $f_n\to f$ uniformly for smooth functions, it suffices to prove that the operators $S_nf = f_n$ are uniformly bounded on $L^p$.
  • $S_n$ can be written in terms of the Hilbert transform $H$. Indeed, $(1+iH)/2$ truncates the Fourier series to a half-infinite interval of indices. From there it is routine to obtain truncation to a finite interval.
  • The fact that $H$ is bounded on $L^p(\mathbb T)$ for $1<p<\infty$ is a theorem of M.Riesz, which can also be expressed in terms of conjugate harmonic functions: if a harmonic function has boundary values in $L^p$, then so does its conjugate.
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