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In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal processing, the Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. The first few members of the sequence are: : where the second sequence, indicated by ''Q'', is said to be ''complementary'' to the first sequence, indicated by ''P''. ==Construction== The Shapiro polynomials ''P''''n''(''z'') may be constructed from the Golay-Rudin-Shapiro sequence ''a''''n'', which equals 1 if the number of pairs of consecutive ones in the binary expansion of ''n'' is even, and −1 otherwise. Thus ''a''0 = 1, ''a''1 = 1, ''a''2 = 1, ''a''3 = −1, etc. The first Shapiro ''P''''n''(''z'') is the partial sum of order 2''n'' − 1 (where ''n'' = 0, 1, 2, ...) of the power series :''f''(''z'') := ''a''0 + ''a''1 ''z'' + a2 ''z''2 + ... The Golay-Rudin-Shapiro sequence has a fractal-like structure – for example, ''a''''n'' = ''a''2''n'' – which implies that the subsequence (''a''0, ''a''2, ''a''4, ...) replicates the original sequence . This in turn leads to remarkable functional equations satisfied by ''f''(''z''). The second or complementary Shapiro polynomials ''Q''''n''(''z'') may be defined in terms of this sequence, or by the relation ''Q''''n''(''z'') = (1-)''n''''z''2''n''-1''P''''n''(-1/''z''), or by the recursions : : : 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「Shapiro polynomials」の詳細全文を読む スポンサード リンク
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