This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc.) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.

中文翻译：

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分段恒定波速的一维亥姆霍兹方程的逆散射

本文分析了波速分段常数情况下一维亥姆霍兹方程的逆散射问题。显示了为任意小的频率间隔记录的散射数据，以唯一地确定波速，并提出了直接重建算法。如果在足够宽的频率范围内记录数据并且波速的跳跃点相对于传播时间间隔相等，则该算法是精确的。数值示例表明，该算法也适用于任意波速（跳跃或连续变化等）的一般情况，随着记录频率范围的增加，逐渐提供更准确的近似值。一个关键的潜在理论见解是将散射数据与单位圆盘的自同构组合相关联，这又与单位圆上的正交多项式有关。该算法利用正交多项式的三项递归来减少所需的计算。