Stiefel filters

Abstract

The best uniform rational approximation of the Sign function on two intervals separated by zero was explicitly found by E.I. Zolotarëv in 1877. The natural extension of this problem to three bands was solved by E.Stiefel in 1961. We indicate the solutions overlooked by the prominent geometer and study their properties.

Figures (5)

• Figure 1: Left: Big rectangle tiled by the smaller ones, $n=6$ Right: The qualitative graph of Zolotarev fraction.
• Figure 2: Left: Tiled rectangle with a slit, $n=7, m=4$. Right: Qualitative graph of the function $St(x|\dots)$. Bounding boxes of oscillations project to the (maximal) work bands of the filter.
• Figure 3: Left:Tiled rectangle with two slits, $n=8, m=4$. Right: Qualitative graph of the function $St_1(x|\dots)$. Bounding boxes project to the work bands of the filter.
• Figure 4: Left: Tiled rectangular octagon with interior branching $\ast$ Right: Qualitative graph of the function $St_2(x|\dots)$. Bounding boxes of oscillations project to the work bands of the filter.
• Figure 5: Left: Two rectangular decagons $\Pi_3^+$ (top), and $\Pi_3^-$ (bottom) for $n=10$ and $m=5$. Right: Qualitative graphs of the functions $St_{3+}(x|\dots)$ (top) and $St_{3-}(x|\dots)$ (bottom). Bounding boxes project to the work bands of the filter