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User:PMajer/sb

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1. First one proves half of the claim: that is, with only . Given , an entire function $$ is easily found of the form $$ with a convenient increasing sequence of even positive integers $n_k$. 2. Then one treats the case with $f=0 < g$, that is settled with a $$ of the form $$ (warning: $$ wouldn't work, for it may have poles.) This in particular gives positive real entire convolution kernels with any prescribed decay. 3. Case of $$ A correseponding $\phi$ is a mollification of $h$ with an entire convolution kernel $$ : $$, which is still an entire function. 4. If $$ is any continuous function, one writes $$ with $$, a series totally convergent on compacta: $$ for all $$.