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In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : whose kernel function ''K'' : R''n''×R''n'' → R is singular along the diagonal ''x'' = ''y''. Specifically, the singularity is such that |''K''(''x'', ''y'')| is of size |''x'' − ''y''|−''n'' asymptotically as |''x'' − ''y''| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |''y'' − ''x''| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on ''L''''p''(R''n''). ==The Hilbert transform== (詳細はHilbert transform ''H''. It is given by convolution against the kernel ''K''(''x'') = 1/(π''x'') for ''x'' in R. More precisely, : The most straightforward higher dimension analogues of these are the Riesz transforms, which replace ''K''(''x'') = 1/''x'' with : where ''i'' = 1, …, ''n'' and is the ''i''-th component of ''x'' in R''n''. All of these operators are bounded on ''L''p and satisfy weak-type (1, 1) estimates. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「singular integral」の詳細全文を読む スポンサード リンク
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