  We initiate the analytical functional bootstrap study of conformal field
theories with large $N$ limits. In this first paper we particularly focus on
the 1D $O(N)$ vector bootstrap. We obtain a remarkably simple bootstrap
equation from the $O(N)$ vector crossing equations in the large $N$ limit. The
bootstrap bound is saturated by the generalized free field theory. We study the
analytical extremal functionals of this crossing equation, for which the total
positivity of the $SL(2,\mathbb{R})$ conformal block plays a critical role. We
prove the $SL(2,\mathbb{R})$ conformal block is totally positive for large
scaling dimension $\Delta$ and show that the total positivity is violated below
a critical value $\Delta_{\textrm{TP}}^*\approx 0.32315626$. The conformal
block forms a surprisingly sophisticated mathematical structure, which for
instance can violate total positivity at the order $10^{-5654}$ for a normal
value $\Delta=0.1627$! We construct a series of analytical functionals
$\{\alpha_M\}$ which satisfy the bootstrap positive conditions up to a range
$\Delta\leqslant \Lambda_M$. The functionals $\{\alpha_M\}$ have a trivial
large $M$ limit. Surprisingly, due to total positivity, they can approach the
large $M$ limit in a way consistent with the bootstrap positive conditions for
arbitrarily high $\Lambda_M$, therefore proving the bootstrap bound
analytically. Our result provides a concrete example to illustrate how the
analytical properties of the conformal block lead to nontrivial bootstrap
bounds. We expect this work paves the way for large $N$ analytical functional
bootstrap in higher dimensions.
