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2018-10-23 Nonlinear Seminar

Adam Nawrocki
Some remarks concerning the convolution of a certain almost periodic function

For $$ \alpha \in \mathbb R \setminus \mathbb Q $$ we define the family of the functions

f_\alpha(x)=\frac{1}{2+\cos x +\cos(\alpha x)} $$ dla $$ x\in \mathbb R $$.

Moreover for $$ \lambda<0 $$ let $$ g(x)=e^{\lambda x} $$ if $$ x\geq 0 $$ and $$ g(x)=0 $$ if $$ x<0 $$. Then the sets $$ 
S_{\lambda}=\{\alpha \in \mathbb R \setminus \mathbb Q\colon \text {the convolution $f_\alpha *g_\lambda$ exists}\} $$ and $$ S_{\lambda}^{\prime}=\{\alpha \in \mathbb R \setminus \mathbb Q\colon \text {the convolution $f_\alpha *g_\lambda$ does not exist}\} $$ are dense subsets of $$ \mathbb R $$. During the seminar we give properties of these sets from the point of view of set theory, measure theory and topology.

The seminar starts at 10:45.

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