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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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Can we find an operator that satisfies this equation? [closed]

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$ Is there a function $f$ from $R^{\infty} …