Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x_{0})$ is the integer part of $2^{j} x_{0}$. Show that for all $j \in \mathbf{N}$, $|c_{j,\widetilde{k}_{j}(x_{0})}(f)| \leq 2^{s(1-j)} c_{1}$. Deduce, by setting $c_{3} = (1 - 2^{-s})^{-1} 2^{s} c_{1}$, $$\sum_{j=n_{0}+1}^{+\infty} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x_{0})| \leq c_{3} |x - x_{0}|^{s}$$
Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$,
$$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$
We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x_{0})$ is the integer part of $2^{j} x_{0}$.
Show that for all $j \in \mathbf{N}$, $|c_{j,\widetilde{k}_{j}(x_{0})}(f)| \leq 2^{s(1-j)} c_{1}$. Deduce, by setting $c_{3} = (1 - 2^{-s})^{-1} 2^{s} c_{1}$,
$$\sum_{j=n_{0}+1}^{+\infty} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x_{0})| \leq c_{3} |x - x_{0}|^{s}$$