Dan, Here’s a hint: the integral of a product of distinct Rademacher functions is 0. To evaluate your integral use the multinomial theorem and the above observation to see that the answer is the sum of all the multinomial coefficients with all even parts. On Mon, May 4, 2020 at 18:38 Dan Asimov <dasimov@earthlink.net> wrote:
The Rademacher function
r_n : [0,1] —> {-1,1}
is your basic square wave of wavelength 1/2^n on [0,1], defined conveniently as
r_n(x) = sign(sin(2^n * 2πx))
I noticed that in Mark Kac's great 1959 book "Statistical Independence in Probability, Analysis, and Number Theory" he states that
Integral_{0 <= t <= 1} ((r_1(t) + ... + r_n(t))^2 dt = n
It's easy to see that if the exponent 2 is replaced by an odd number, this integral is 0, by oddness. But my limited experiments have not suggested what the pattern is for even exponents, other than that it's always an integer.
Question: What is an explicit formula for
f(n, k) = Integral_{0 <= t <= 1} ((r_1(t) + ... + r_n(t))^(2k) dt
???
—Dan
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