We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$ Let $k \in \mathbb { N } ^ { * }$. Prove that there exist $( 2 k + 1 ) ^ { 2 }$ complex numbers $\left( a _ { j , l } \right) _ { - k \leq j , l \leq k }$ such that for every $( u , v ) \in \mathbb { R } ^ { 2 } : f _ { k } ( u , v ) = \sum _ { - k \leq j , l \leq k } a _ { j , l } e ^ { i u j } e ^ { i v l }$. Justify that the function $f _ { k }$ satisfies the Ergodic Theorem.
We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set:
$$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $k \in \mathbb { N } ^ { * }$. Prove that there exist $( 2 k + 1 ) ^ { 2 }$ complex numbers $\left( a _ { j , l } \right) _ { - k \leq j , l \leq k }$ such that for every $( u , v ) \in \mathbb { R } ^ { 2 } : f _ { k } ( u , v ) = \sum _ { - k \leq j , l \leq k } a _ { j , l } e ^ { i u j } e ^ { i v l }$. Justify that the function $f _ { k }$ satisfies the Ergodic Theorem.