Let $f \in L^2[0, 1]$, Plancherel's Equality says that $$ \|f\|^2 = \sum_{n = -\infty}^\infty |\hat{f}(n)|^2 = \sum_{n = -\infty}^\infty |\langle f, e^{2 \pi i n t}\rangle_{[0, 1]}|^2 .$$ Do a similar equality/inequality hold for a different Hilbert spaces? In particular, is there a $E \subseteq \mathbb{R}$ unbounded with $0 < |E| < \infty$ such that no equality/inequality does not hold in $L^2(E)$? I believe it is true, but cannot come up with a way to construct $E$. For example, take $$E = \bigcup_{k = 1}^\infty \left[k, k+\frac{1}{k^2}\right].$$ Taking indicator functions over each of these then normalizing was my first intuition to show that there is no $K > 0$ that would satisfy $$ K\|f\|^2 \geq \sum_{n = -\infty}^\infty |\langle f, e^{2 \pi i n t}\rangle_{E}|^2 .$$
Is this even possible?