Quantitative Characteristics of Singular Measures
The main part of this note is a continuation of the work  of M. Sh. Birman and M. Z. Solomyak. In that paper various results in the theory of approximation are obtained. The approach adopted in  is based on a systematic use of the theorem on set functions which is proved there. This theorem is concerned with a certain special characteristic of set functions and estimates are obtained which are uniform with respect to all functions of the class in question. It is clear, moreover, that for a more restricted class of functions the estimates can be improved. In this paper we study singular set functions and prove that for these the behavior of the characteristic mentioned above is qualitatively improved. Quantitative estimates of this improvement for the classes of singular measures considered are also given. The results obtained are used to estimate the eigenvalues of the polyharmonic equation in an L2 space with a singular measure.
KeywordsRestricted Class Singular Function Initial Partition Lebesgue Measure Zero Singular Measure
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