For $f\in L_1+L_2$ we can define its Fourier transform $\hat f$ and the partial sums as $$S_Rf(x)=\int_{|\xi|<R}\hat f(\xi)e^{2\pi i\xi.x}d\xi$$
My question is, for $p>2$, how do we define the partial sums, if we cannot define the Fourier transform $\hat f$? I'd like an answer that doesn't involve distributions.
For $p\leq 2$ we have the inequality $||\hat{\phi}||_{p'}\leq ||\phi||_p$ for any Schwartz function $\phi$. Because of the density of Schwartz functions in $L_p$ we can pick a sequence $(\phi_n)$ converging to $f\in L_p$ and due to the above inequality, $(\hat \phi_n)$ converges to some $g\in L_{p'}$. We define this $g$ to be $\hat f$. Since I cannot do this for $p>2$, I am stuck while trying to define $S_R$ on each $L_p$ space.