# $L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$

It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in terms of the coefficients?

• isn't any operation on the Fourier series expressible "in principle" in terms of an operation on the coefficients? it won't be a simple expression (except for p=2), but why would it fail "in principle"? – Carlo Beenakker Sep 27 '13 at 16:47

Consider a function $f \in L^2$ and $f \notin L^{p}$ (for $p>2$). Now multiple the Fourier coefficients by random signs. Almost surely, the new function, $g$, will be in $L^{p}$ (by Khinchin's inequality and Fubini's theorem). Thus we have two functions, $f$ and $g$, both of whose Fourier coefficients have the same absolute values, one of which has finite $L^{p}$ norm and the other of which has infinite $L^{p}$ norm. Thus, no expression involving only the absolute values of the Fourier coefficients can compute (or even bound!) the $L^p$ norm of a function.