Rudin-Shapiro-like polynomials in L4
Rudin-Shapiro-like polynomials in L4
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DOI：
10.1090/s0025-5718-00-01221-7
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发表时间：
2000-07
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影响因子：
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通讯作者：
P. Borwein;Michael J. Mossinghoff
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文献类型：
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作者：
P. Borwein;Michael J. Mossinghoff
本文研究了用迭代p（x）→ p（x）± xd +1 p *（-x）构造的具有{+1，-1}系数的多项式序列，其中d是p的次数，p * 是p的倒数多项式，当po = 1时，它们生成Rudin-Shapiro多项式，证明了这些多项式的L4范数是可计算的.我们特别感兴趣的情况下，迭代产生的序列具有最小可能的渐近L4范数（或等价地，具有最大可能的渐近价值因子）。Rudin-Shapiro多项式形成一个这样的序列。我们确定了所有的度小于40的p 0生成的序列下的迭代与此属性。这些序列具有渐近价值因子3。第一个真正不同的例子是p 0的度数为19。
We examine sequences of polynomials with {+1, -1} coefficients constructed using the iterations p(x) → p(x) ± x d+1 p * (-x), where d is the degree of p and p * is the reciprocal polynomial of p. If po = 1 these generate the Rudin-Shapiro polynomials, We show that the L4 norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic L4 norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence. We determine all p 0 of degree less than 40 that generate sequences under the iteration with this property. These sequences have asymptotic merit factor 3. The first really distinct example has a p 0 of degree 19.