Shapiro polynomials

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.[1] In signal processing, the Shapiro polynomials have good autocorrelation properties[2] and their values on the unit circle are small. The first few members of the sequence are:

\( \begin{align} P_1(x) & {} =1 + x \\ P_2(x) & {} =1 + x + x^2 - x^3 \\ P_3(x) & {} =1 + x + x^2 - x^3 + x^4 + x^5 - x^6 + x^7 \\ ... \\ Q_1(x) & {} =1 - x \\ Q_2(x) & {} =1 + x - x^2 + x^3 \\ Q_3(x) & {} =1 + x + x^2 - x^3 - x^4 - x^5 + x^6 - x^7 \\ ... \\ \end{align} \)

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 a₀ = 1, a₁ = 1, a₂ = 1, a₃ = −1, etc.

The first Shapiro P_n(z) is the partial sum of order 2ⁿ − 1 (where n = 0, 1, 2, ...) of the power series

f(z) := a₀ + a₁ z + a₂ z² + ...

The Golay-Rudin-Shapiro sequence {a_n} has a fractal-like structure – for example, a_n = a_2n – which implies that the subsequence (a₀, a₂, a₄, ...) replicates the original sequence {a_n}. 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-)ⁿz^2n-1P_n(-1/z), or by the recursions

\( \( P_0(z)=1; ~~ Q_0(z) = 1 ; \)
P_{n+1}(z) = P_n(z) + z^{2^n} Q_n(z) ; \)
\( Q_{n+1}(z) = P_n(z) - z^{2^n} Q_n(z) . \)

Properties
Zeroes of the polynomial of degree 255

The sequence of complementary polynomials Q_n corresponding to the P_n is uniquely characterized by the following properties:

• (i) Q_n is of degree 2ⁿ − 1;
• (ii) the coefficients of Q_n are all 1 or −1, and its constant term equals 1; and
• (iii) the identity |P_n(z)|² + |Q_n(z)|² = 2^(n + 1) holds on the unit circle, where the complex variable z has absolute value one.

The most interesting property of the {P_n} is that the absolute value of P_n(z) is bounded on the unit circle by the square root of 2^(n + 1), which is on the order of the L² norm of P_n. Polynomials with coefficients from the set {−1, 1} whose maximum modulus on the unit circle is close to their mean modulus are useful for various applications in communication theory (e.g., antenna design and data compression). Property (iii) shows that (P, Q) form a Golay pair.

These polynomials have further properties:^[3]

\( P_{n+1}(z) = P_n(z^2) + z P_n(-z^2) ; \, \)
\( Q_{n+1}(z) = Q_n(z^2) + z Q_n(-z^2) ; \, \)
\( P_n(z) P_n(1/z) + Q_n(z) Q_n(1/z) = 2^{n+1} ; \, \)
\( P_{n+k+1}(z) = P_k(z)P_n(z^{2k+1}) + z^{2k}Q_k(z)P_n(-z^{2k+1}) ; \, \)
\( P_n(1) = 2^{\lfloor (n+1)/2 \rfloor}; {~}{~} P_n(-1) = (1+(-1)^n)2^{\lfloor n/2 \rfloor - 1} . \, \)

See also

Littlewood polynomials

Notes

John Brillhart and L. Carlitz (May 1970). "Note on the Shapiro polynomials". Proceedings of the American Mathematical Society (Proceedings of the American Mathematical Society, Vol. 25, No. 1) 25 (1): 114–118. doi:10.2307/2036537. JSTOR 2036537.
Somaini, U. (June 26, 1975). "Binary sequences with good correlation properties" (PDF). Electronics Letters 11 (13): 278–279. doi:10.1049/el:19750211.

J. Brillhart; J.S. Lomont, P. Morton (1976). "Cyclotomic properties of the Rudin–Shapiro polynomials". J. Reine Angew. Math. 288: 37–65.

References

Borwein, Peter B (2002). Computational Excursions in Analysis and Number Theory. Springer. ISBN 0-387-95444-9. Retrieved 2007-03-30. Chapter 4.
Mendès France, Michel (1990). "The Rudin-Shapiro sequence, Ising chain, and paperfolding". In Berndt, Bruce C.; Diamond, Harold G.; Halberstam, Heini et al. Analytic number theory. Proceedings of a conference in honor of Paul T. Bateman, held on April 25-27, 1989, at the University of Illinois, Urbana, IL (USA). Progress in Mathematics 85. Boston: Birkhäuser. pp. 367–390. ISBN 0-8176-3481-9. Zbl 0724.11010.

Every Eisenstein integer a + bω whose norm a2 − ab + b2 is a rational prime is an Eisenstein prime. In fact, every Eisenstein prime is of this form, or is a product of a unit and a rational prime congruent to 2 mod 3.

This is a discrete Fourier transform.

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