Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials

Question

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,

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

Answer 1

Yes, this goes back to the work of

Plancherel, M.; Pólya, George, Fonctieres entières et intégrales de Fourier multiples, Comment. Math. Helv. 9, 224-248 (1937). ZBL0016.36004.

(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.