We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$ The purpose of Part III is to construct a function of class $C ^ { \infty }$ on $\mathbb { R }$, non-zero, whose all moments of order $p$ ($p \in \mathbb { N }$) are zero. Using the results of questions 36 and 37, conclude.
We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by
$$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
The purpose of Part III is to construct a function of class $C ^ { \infty }$ on $\mathbb { R }$, non-zero, whose all moments of order $p$ ($p \in \mathbb { N }$) are zero. Using the results of questions 36 and 37, conclude.