Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 339 / 2 / d / Solution

For each coefficient index $k\ge 0$, let $H_k$ have constant diagonal blocks $f_{k}J$ and cross blocks $g_{k}J$. The preceding block-constant argument gives $H_k⪰0$. Define the Hadamard powers $X^{\circ k}$, withThis is the constant entrywise power, including at zero entries; it is not the identity matrix. Repeated use of the Schur product theorem shows $X^{\circ k}⪰0$ for every $k\ge 0$.

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. ConsequentlyEndpoint convergence follows from the stated expansions on the full interval: $f_k\ge 0$ and $\sum f_k=f(1)<\infty$, while $\sum |g_k|\le \sum f_k$. Thus the expansions converge absolutely at every matrix entry. This is coefficient-dominated entrywise positivity.

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 $Y=-I_2$.