MathDB

Let

f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)

, where each

a_j

is areal and

a_N

is not equal to

0

. Let

N_k

denote the number of zeroes (including multiplicites) of

\dfrac{d^k f}{dt^k}

. Prove that

N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N.

[Only zeroes in [0, 1) should be counted.]

Problem Statement