Faber polynomials in a deltoid region and power iteration momentum methods

We consider a region in the complex plane enclosed by a deltoid curve inscribed in the unit circle, and define a family of polynomials $P_n$ that satisfy the same recurrence relation as the Faber polynomials for this region. We use this family of polynomials to give a constructive proof that $z^n$ is approximately a polynomial of degree $\sim\sqrt{n}$ within the deltoid region. Moreover, we show that $|P_n| \le 1$ in this deltoid region, and that, if $|z| = 1+\varepsilon$, then the magnitude $|P_n(z)|$ is at least $\frac{1}{3}(1+\sqrt{\varepsilon})^n$, for all $\varepsilon > 0$. We illustrate our polynomial approximation theory with an application to iterative linear algebra. In particular, we construct a higher-order momentum-based method that accelerates the power iteration for certain matrices with complex eigenvalues. We show how the method can be run dynamically when the two dominant eigenvalues are real and positive.

💡 Research Summary

The paper studies a specific complex‑plane region – the deltoid curve (a three‑cusp hypocycloid) inscribed in the unit circle – and develops a family of polynomials (P_n) that obey the same three‑term recurrence as the Faber polynomials associated with this region. By choosing initial conditions (P_0=1), (P_1=z), (P_2=z^2), the authors obtain a basis that is well‑behaved inside the deltoid: for every (n\ge0) and every (z) in the closed deltoid (\Gamma), (|P_n(z)|\le1). Outside the unit disk, the polynomials grow rapidly: if (|z|=1+\varepsilon) then (|P_n(z)|\ge\frac13(1+\sqrt\varepsilon)^n). This mirrors the classic Chebyshev property on (