Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j},~ 1\leq j \leq 2^{n} \} \right\} \right)_{n \in \mathbb{N}_{0}} $$ be a sequence of partitions of $ [0,1) $ such that $$\underset{1\leq j\leq 2^{n}}{\max}|I^{n}_{j}|\leq \lambda ^{n} $$ for some $\lambda \in (0,1).$
Next, define a sequence $ (F_{n}: [0,1) \to \mathbb{C})_{n \in \mathbb{N}_{0}} $ of functions by $$ \forall n \in \mathbb{N}_{0}, ~ \forall x \in [0,1), ~ \forall j \in \{ 1,\ldots,2^{n} \}: \\ {F_{n}}(x) \stackrel{\text{df}}{=} \frac{1}{\left| I^{n}_{j} \right|} \int_{I^{n}_{j}} f(t) ~ \mathrm{d}{t}, \quad \text{if $ x \in I^{n}_{j} \in D_{n} $}. $$
Question. Is it true that $ \displaystyle \| f - F_{n} \|_{{L^{p}}([0,1))} \leq \sup_{|t| \leq \lambda^{ n}} \| f(\bullet + t) - f(\bullet) \|_{{L^{p}}([0,1))} $?
Or what kind of estimates can be obtained for $ \displaystyle \| f - F_{n} \|_{{L^{p}}([0,1))}$ depending on $\lambda^{n}$?
Note that if $(D_{n})_{n \in \mathbb{N}_{0}}$ is the sequence of dyadic partitions then the inequality is true, in the proof of this inequality I used the equality of the lengths of atoms of dyadic partition. But in case of different lengths of atoms I have got some problem.
