I've been thinking a lot about Fourier transforms lately, specifically how to create Fourier fields where on x and y pairs that touch the Fourier defined curve the height would equal 1 and smoothly fall to zero when moving away from the curve. However while working out how to create these Fourier fields I came across a problem that I haven't been able to work out a solution to which is, in closed form determine if a sinusoidal function has any real zeros or touches the x-axis.
For example lets say you had the sinusoidal periodic function just consisting of the summation of scaled cosines and the frequencies were all odd integers, $Acos(n_1x) + Bcos(n_2x) + ... + K$, {$n_i$ is an odd integer.} Then it's quite easy to work out this closed form evaluation since the max and min of this wave function is simply always the summation of coefficients (A+B+...). Therefore its just a comparison of the coefficient summation to the value K, the periodic function's offset. If S is the summation of the coefficients (A+B+...) then the closed form expression can be written as $$F(K) = | |K| - S| + |K|-S$$ , where if F(K) = 0 then the periodic function has real zeros, and if F(K) is non-zero then there are no real zeros, hence its not touching the x-axis.
This gets more difficult though if the cosines can also have even frequencies, and even more difficult if the cosines can have phase shifts. One possible route that I've looked at was instead of expressing the periodic function as a sum it would be much easier if it were a product. Through experimentation it seems to be the case that any sinusoidal function with integer frequencies can be written as the product of shifted phased cosines with a scalar value.
$$ Acos(n_1x+a)+Bcos(n_2x+b)+...+Icos(n_ix+i) = k(cos(a_2)cos(x+a_3)+sin(a_2)(cos(b_2)cos(x+b_3)+sin(b_2))....( cos(i_2)cos(x+i_3)+sin(i_2)) $$
$n_n$ are all integers. A,B,...I, k are all real numbers. $a_n, b_n, ... i_n$ are all between $[-\pi, \pi]$.
If the sinusoidal function could be expressed as the product above then its quite easy to see where it has real zeros. It would have real zeros if any of its products had $-tan(a_2)$ between [-1, 1] where $a_2$ is used from the notation above.
However finding this product form is not too trivial, I have been able to calculate it numerically but not able to express it through some nice decomposition transformation. I was thinking about the possibility of using a Laplace transform to help act as some sort of scanner to find those certain product signatures but not exactly sure yet how that would work.
Thanks for tuning in. Since I've been working on this problem for months now I thought I would share it and see if anyone had any ideas.
Cheers, Aaron