Wiener Numbers

A sequence of uncorrelated numbers  developed by Wiener (1926-1927). The numbers are constructed by beginning with

(1)

then forming the outer product with  to obtain

(2)

This row is repeated twice, and its outer product is then taken to give

(3)

This is then repeated four times. The procedure is repeated, and the result repeated eight times, and so on. The sequences from each stage are then concatenated to form the sequence 1, , 1, 1, 1, , , 1, , , 1, 1, 1, , , 1, , , ....

REFERENCES:

Papoulis, A. "The Wiener Numbers." The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 258-259, 1962.

Wiener, N. "The Spectrum of an Array and Its Applications to the Study of the Translation Properties of a Simple Class of Arithmetical Functions." J. Math. Phys. 6, 1926-1927.