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A spherical design, part of combinatorial design theory in mathematics, is a finite set of ''N'' points on the ''d''-dimensional unit n-sphere ''Sd'' such that the average value of any polynomial ''f'' of degree ''t'' or less on the set equals the average value of ''f'' on the whole sphere (that is, the integral of ''f'' over ''Sd'' divided by the area or measure of ''Sd''). Such a set is often called a spherical ''t''-design to indicate the value of ''t'', which is a fundamental parameter. Spherical designs can be of value in approximation theory, in statistics for experimental design (being usable to construct rotatable designs), in combinatorics, and in geometry. The main problem is to find examples, given ''d'' and ''t'', that are not too large. However, such examples may be hard to come by. Spherical t-designs have also recently been appropriated in quantum mechanics in the form of quantum t-designs with various applications to quantum information theory, quantum computing and POVMs. The concept of a spherical design is due to Delsarte, Goethals, and Seidel (1977). The existence and structure of spherical designs with ''d'' = 1 (that is, in a circle) was studied in depth by Hong (1982). == arbitrary-dimensional spherical design == Shortly thereafter, Seymour and Zaslavsky (1984) proved that such designs exist of all sufficiently large sizes; that is, there is a number ''N''(''d'',''t'') such that for every ''N'' ≥ ''N''(''d'',''t'') there exists a spherical ''t''-design of ''N'' points in dimension ''d''. However, their proof gave no idea of how big ''N''(''d'',''t'') is. Good estimates for that were found later on. Besides these "large" sizes, there are many sporadic small spherical designs; many of them are related to finite group actions on the sphere and are of great interest in themselves. Recently, Bondarenko, Radchenko, and Viazovska obtained the optimal asymptotic upper bound 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「Spherical design」の詳細全文を読む スポンサード リンク
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