The all-α extension of the super-4 theorem

Complete proof for every real $\alpha>2$, including the Lorentz condition and strong convergence. Let $c_{n,k}$ be the source's Boţ–Nguyen coefficient array, extended by $c_{n,k}=0$ outside $0\le k\le n$. Source equation (43e) states

$$x_{n+1}=\sum_{k=0}^{n}c_{n,k}T^kx_0.$$

The source's Theorem 4.4 already proves, for every $\alpha>2$,

$$c_{n,k}\ge0,\qquad \sum_{k=0}^{n}c_{n,k}=1,\qquad c_{n,k}\longrightarrow0\quad\text{for each fixed }k.$$

Thus the sole missing hypothesis of its Theorem 2.10 is the Lorentz condition

$$L_n:=\sum_{k=0}^{\infty}|c_{n,k}-c_{n,k+1}|\longrightarrow0.$$

The key is to apply the source's already-established Fact 1.1 to one universal test operator. Work in the real Hilbert space $\ell^2(\mathbb N_0)$, let

$$Se_k=e_{k+1},$$

and start the iteration at $x_0=e_0$. The unilateral shift $S$ is a linear isometry with $\operatorname{Fix}S=\{0\}$, so Fact 1.1 applies for every real $\alpha>2$ and yields

$$\|x_j-Sx_j\|_2=o(j^{-1}).$$

Equation (43e) gives

$$x_{n+1}=\sum_{k=0}^{n}c_{n,k}e_k,$$

and consequently

$$x_{n+1}-Sx_{n+1} =c_{n,0}e_0 +\sum_{k=1}^{n}(c_{n,k}-c_{n,k-1})e_k -c_{n,n}e_{n+1}.$$

This vector has at most $n+2$ nonzero coordinates, and positivity gives the exact identity

$$\|x_{n+1}-Sx_{n+1}\|_1=c_{n,0}+L_n.$$

Therefore Cauchy–Schwarz and Fact 1.1 imply

$$0\le L_n \le\sqrt{n+2}\,\|x_{n+1}-Sx_{n+1}\|_2 =o(n^{-1/2}).$$

In particular the Lorentz condition holds for every $\alpha>2$, with the stronger quantitative bound $L_n=o(n^{-1/2})$.

All four hypotheses of the source's Theorem 2.10 are now satisfied. Hence, for every real Hilbert space, every linear nonexpansive operator $T$, and every starting point $x_0$,

$$x_{n+1}=\sum_{k=0}^{n}c_{n,k}T^kx_0 \longrightarrow P_{\operatorname{Fix}T}x_0 \qquad\text{in norm}.$$

Thus every assertion of Theorem 5.5 extends from $\alpha=4$ to the entire conjectured range $\alpha>2$.

Source: Bauschke and Gao, Boţ–Nguyen Acceleration, Weighted Mean Ergodic Iteration, and the Beta-Binomial Distribution, revised August 13, 2026, Fact 1.1, Theorems 2.10 and 4.4, and Conjecture 6.1.