Similar Fourier Terms with Spline and Original Function

Question

Suppose I am dealing with a periodic, well-behaved, function f(x). Here is the kicker, I do not know what that function is explicitly, but I do have a large amount of data points that fit exactly on that function's graph. I then calculate the interpolating spline through those data points. Since I know that the function is periodic, I calculate the Fourier series for this spline.

Do we know if the Fourier series for f(x) necessarily shares terms with the series for the spline? How can we find the shared terms if they exist?

Answer 1

This is an error bound. I don't believe you can have exact equality (I'll explain that later). First, a theorem from Quarteroni:

Let $f(x) \in C^4([a,b])$, and partition $[a,b]$ with intervals of width $h_i$. Let $h = \max h_i$, and let $s(x)$ be the cubic interpolating spline (i.e., interpolating with the subinterval endpoints as nodes). Then $$ ||f - s||_\infty \leq \frac{5}{384}h^4||f^{(4)}||_\infty $$

Now, we have

$$ a_n = \int_0^1 f(x) \exp(-i2\pi n x) dx $$ and $$ \tilde{a}_n = \int_0^1 s(x) \exp(-i2\pi n x) dx $$ From the above theorem, we have $$ |a_n - \tilde{a}_n| \leq \frac{5}{384}h^4||f^{(4)}||_\infty $$

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Now, on the other hand $$ a_n - \tilde{a}_n = 0 \Rightarrow \langle f(x)-s(x), \exp(-i2\pi n x) \rangle = 0 $$ but this can only happen if $f(x)-s(x) = \exp(i2\pi m x)$ for some $m \neq n$.