Let $ X\_N = \text{span} \{\cos(2\pi lx): l=0, \cdots, N-1 \} $ with $ x \in [0, 1] $ and $ Y\_N = \{v =(v\_0, \cdots, v\_{N-1}): v\_j \in \mathbb{C}\} = \mathbb{C}^N $. Then $ X\_N $ is the space of trigonometric polynomials. We equip $ X\_N $ with the usual $ L^p (1 \leq p \leq \infty ) $ norm and equip $ Y\_N $ with the discrete $ l^p (1 \leq p \leq \infty)$:


$$ \| v \|\_{l^p} = \left( \frac{1}{N}\sum\_{j=0}^{N-1} |v\_j|^p \right)^\frac{1}{p}, \quad 1 \leq p < \infty, $$


$$ \| v \|\_{l^\infty} = \max\_{0 \leq j \leq N-1} |v\_j|, \quad v \in Y\_N. $$


Let $ R\_N:X\_N \rightarrow Y\_N $ be the restriction of the trigonometric polynomials to some discrete points: for $ u \in X\_N $,


$$ v = R\_N u, \quad v\_j = u(x\_{j+1/2}), \quad j=0, \cdots, N-1, $$
where $ x\_{j+1/2} = (j+1/2)/N $.


**Question**:
can we find a constant $ C $ independent of $ N $ such that


$$ \| R\_N u \|\_{l^4} \leq C\| u \|\_{L^4}, \quad u \in X\_N. $$


Moreover, does it hold for higher dimensions though currently it is only stated in 1D?


Note that by Parseval's identity, one has,


$$ \| R\_N u \|\_{l^2} = \| u \|\_{L^2}. $$



Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials
Yes, this goes back to the work of


*Plancherel, M.; Pólya, George*, [**Fonctieres entières et intégrales de Fourier multiples**](http://dx.doi.org/10.1007/BF01258191), Comment. Math. Helv. 9, 224-248 (1937). [ZBL0016.36004](https://zbmath.org/?q=an:0016.36004).


(see for instance Theoreme III). Nowadays one would usually prove such a result using Littlewood-Paley projections or similar devices to express $R\_N u$ as a suitable integral expression of $u$ to which Schur's test (for instance) can be applied.


