Shapiro polynomials

From Wikipedia, the free encyclopedia

In the branch of mathematics known as Fourier analysis, the Shapiro polynomials are the sequence {Pn} of univariate polynomials uniquely defined as follows: Pn(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 + ...

where an 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.).

This sequence {an} has a fractal-like structure – for example, an = a2n – which implies that the subsequence (a0, a2, a4, ...) replicates the original sequence {an}. This in turn leads to remarkable functional equations satisfied by f(z).

It can be shown that a sequence of complementary polynomials Qn corresponding to the Pn is uniquely characterized by the following properties:

  • (i) Qn is of degree 2n − 1;
  • (ii) the coefficients of Qn are all 1 or −1 , and its constant term equals 1; and
  • (iiii) the identity |Pn(z)|2 + |Qn(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 {Pn} is that the absolute value of Pn(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 Pn. 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).

[edit] References

Languages