grandes-ecoles 2021 Q25

grandes-ecoles · France · centrale-maths2__official Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences

We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Let $k \in \mathbb{N}$. Show that, if $g$ is of class $\mathcal{C}^k$ and if the functions $g^{(j)}$ are bounded for $j \in \llbracket 0, k \rrbracket$, then $f * g$ is of class $\mathcal{C}^k$ and $(f * g)^{(k)} = f * (g^{(k)})$.

We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Let $k \in \mathbb{N}$. Show that, if $g$ is of class $\mathcal{C}^k$ and if the functions $g^{(j)}$ are bounded for $j \in \llbracket 0, k \rrbracket$, then $f * g$ is of class $\mathcal{C}^k$ and $(f * g)^{(k)} = f * (g^{(k)})$.

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