We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces \(h_{\infty}(K)\) of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.

中文翻译：

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带插值小波的带平方拉普拉斯算子的齐次方程边值问题的数值方法。

我们提出了一种有效的数值方法，可使用磁盘中的谐波小波并在二阶有理网格上的边界处进行插值，从这些函数及其正态导数的连续边界值中恢复出这些函数的连续边界值。在小波基础上将边值问题的解扩展为繁琐的插值序列，然后用其部分和的序列代替，这些部分和可紧凑地表示在Hardy空间\（h _ {\ infty}的相应多分辨率分析（MRA）的子空间基中（K）\）谐波功能在磁盘中。根据边界函数的最佳近似，可以得到任意阶次部分和的近似解的有效估计值，该近似函数是阶次较小的三角多项式。结果，为了提供所需的未知双谐波函数表示的精度，可以预先选择相应MRA子空间的缩放参数，以使对该空间的插值投影可以定义该MRA子空间的相应部分和的简单解析表示。根据比例函数的适当压缩和移位来进行插值级数运算，而跳过复杂的迭代过程，以便在插值小波中将边界函数的扩展系数数值构建为级数。我们使用基于改进的Meyer小波的插值和插值正交小波编写解决方案；如果近似地给出了边界值问题的边界值（例如，通过实验发现），则后者的应用很方便。在这种情况下，可以采用惯常的，众所周知的离散正交小波变换过程来分析和细化（校正）边界值。