In this paper, we study the restrictions of both the harmonic functions and the eigenfunctions of the symmetric Laplacian to edges of pre-gaskets contained in the Sierpinski gasket. For a harmonic function, its restriction to any edge is either monotone or having a single extremum. For an eigenfunction, it may have several local extrema along edges. We prove general criteria, involving the values of any given function at the endpoints and midpoint of any edge, to determine which case it should be, as well as the asymptotic behavior of the restriction near the endpoints. Moreover, for eigenfunctions, we use spectral decimation to calculate the exact numbers of the local extrema along edges. This confirms, in a more general situation, a conjecture of Dalrymple et al. (J Fourier Anal Appl 5:203–284, 1999) on the behavior of the restrictions to edges of the basis Dirichlet eigenfunctions, suggested by the numerical data.

中文翻译：

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拉普拉斯本征函数对谢尔宾斯基垫片边缘的限制

在本文中，我们研究了对称拉普拉斯算子的调和函数和本征函数对包含在谢尔宾斯基垫片中的预垫片边缘的限制。对于调和函数，它对任何边的限制要么是单调的，要么是只有一个极值。对于一个特征函数，它可能沿边缘有几个局部极值。我们证明了一般标准，涉及任何给定函数在任何边缘的端点和中点的值，以确定它应该是哪种情况，以及端点附近限制的渐近行为。此外，对于特征函数，我们使用谱抽取来计算沿边缘的局部极值的确切数量。在更一般的情况下，这证实了 Dalrymple 等人的猜想。(J Fourier Anal Appl 5:203–284,