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LSpice

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Analycity Analyticity of $fg$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^$

Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\backslash\{0\}$$\mathbb{R}\setminus\{0\}$ but non-analytic at $x=0$. Is the convolution (assumed to be well defined) defined as:

$$(f*g)(x) = \int_\mathbb{R}f(x-z)g(z)dz$$

an analytic function ?

Otherwise, under which conditions the convolution is analytic ?

Any references or solutions will be highly appreciated.

Thank you!

Simplifying conditions

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edited Jul 2 at 20:11

NancyBoy

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When is the convolution of two elements Analycity of $\mathcal{C}^\infty(\mathbb$fg$ with $f$ and $g$ smooth on $\mathbb{R},\mathbb$ and analytic on $\mathbb{R})$ an analytic function?^$

Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ without compact supportanalytic on $\mathbb{R}\backslash\{0\}$ but non-analytic at $x=0$. Is the convolution (assumed to be well defined) defined as:

$$(f*g)(x) = \int_\mathbb{R}f(x-z)g(z)dz$$

an analytic function ?

Otherwise, under which conditions the convolution is analytic ?

Any references or solutions will be highly appreciated.

Thank you!

Notice added Draw attention by NancyBoy

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Bounty Started worth 50 reputation by NancyBoy

occurred Jun 29 at 17:46

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edited Jun 14 at 17:37

LSpice

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When is the convolution of two elements of $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ an analytic function?

fixed latex in title

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edited Jun 14 at 8:27

Carlo Beenakker

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asked Jun 14 at 7:56

NancyBoy

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