One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$ \sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(p,q,\lambda) \int_{-\pi}^{\pi}|F(\theta)|^p|\theta|^{p\alpha}d\theta, $$ where, $a_n$'s are the Fourier series coefficients, K is independent of $F$, $1<p\leq q <\infty$, ${1}/{p} + {1}/{p'}=1$, $0\leq \alpha < 1/p'$ and $\lambda = 1/p + 1/q + \alpha -1 \geq 0$.

Another theorem for Fourier transforms (from "Pitts Inequality with sharp convolution estimates" by W. Beckner, 2007) states that for a function $f$ in $L^2(R)$, $$ \int_{R} \phi\left\{1/|x|\right\}|f(x)|^2 dx \leq C_{\phi} \int_{R} \phi\left\{|x|\right\}|\hat{f}(x)|^2 dx, $$ where $\hat{f}$ is the FT of $f$, and $\phi$ is an increasing function.

Is there a way to obtain the first theorem from the second?



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