= Solution
For each <coefficient> index $k\geq0$, let $H_k$ have constant diagonal blocks $f_kJ$ and cross blocks $g_kJ$. The preceding block-constant argument gives $H_k\succeq0$. Define the <Hadamard powers> $X^{\circ k}$, with
$$
X^{\circ0}=J_{n+m,n+m}.
$$
This is the constant entrywise power, including at zero entries; it is not the <identity matrix>. Repeated use of the <Schur product theorem> shows $X^{\circ k}\succeq0$ for every $k\geq0$.
Applying the <Schur product theorem> again makes every summand $H_k\circ X^{\circ k}$ <positive semidefinite>. The partial sums are <positive semidefinite>, and their entrywise limit is exactly $Y$. In finite <dimension> this is a matrix-norm limit; the <positive semidefinite cone> is closed. Consequently
$$
\boxed{Y=\sum_{k=0}^\infty H_k\circ X^{\circ k}\succeq0}.
$$
Endpoint convergence follows from the stated expansions on the full interval: $f_k\geq0$ and $\sum f_k=f(1)<\infty$, while $\sum|g_k|\leq\sum f_k$. Thus the expansions converge absolutely at every <matrix> entry. This is <coefficient-dominated entrywise positivity>.
\b[The <coefficient> condition must include index zero]. We interpret the PDF's $\mathbb N$ accordingly. If it means only positive integers and no condition is imposed on $f_0,g_0$, the assertion is false: take $n=m=1$, $X=0$, $f=-1$ and $g=0$. All positive-index inequalities hold, but $Y=-I_2$.
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