Conider two power series is definded as follows :

$$f(x) = \sum_{n=0}^\infty f_n x^n$$ $$g(x) =\sum_{n=0}^\infty g_n x^n$$

Then , Hadamard product of $$f$$ and $$g$$:

$$(f\odot g)(x) =\sum_{n=0}^\infty f_n g_n x^n$$ $$= (1/2π) \int_0^{2π}f(\sqrt{x}e^{it})g(\sqrt{x}e^{-it})dt$$

Questions :

1. consider the situation where we know everything about $$(f\odot g)$$ and $$g$$ then how to find (constructively) $$f$$ ? (Equivalent "Hadamard Division")
2. In general can we construct inter-relative "Hadamard Algebra" ?
3. Can we get upper bound on $$f$$ if we know the upper bounds on $$(f\odot g)$$ and we know everything ( poles and zeroes ) about $$g$$

Note : I'm seeing in comment section there is an induction of trivial confusion due to my insufficiency of stating . By " we know everything about .." I mean we know the compact and series representation of them ( $$(f\odot g)$$ and $$g$$ ) . The problem is we only know infinite series representation of $$f$$ and due to nature of it's mysterious coefficients we can't get certain properties of $$f$$ . so I'm trying to get alternative representation (likely to be compact ) of $$f$$. Also all $$f_n$$ & $$g_n$$ are non zero .

• If we "know everything" just divide $(f\odot g)_n$ by $g_n$. What is " inter-relative algebra"? There is a nice theory of Hadamard products for rational functions and algebraic functions. Jan 13, 2020 at 7:05
• If there was such a formula you'd get an integral formula for "Hadamard reciprocal". But series with even a single zero coefficient have no reciprocals, so there is no regular representation of this sort. You have to recover the coefficients and divide them, if possible. Jan 13, 2020 at 10:55