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Let $u_k$ be the vector in $\mathbb{R}^n$ whose $i$'th entry is $\sin(\pi ki/n)$. The vectors $u_1,\ldots, u_n$ are orthogonal and correspondingly every vector in $\mathbb{R}^n$ can be decomposed as a linear combination of them.

I have the intuition that the vectors corresponding to low $k$ are "slowly varying'' and if a vector is almost orthogonal to all $u_i$ for $i$ above some value $k_0$ then, in some sense, it should not vary too fast.

Correspondingly, I'd like to ask:

  1. Can this be turned into a formal statement? In particular, assume that a vector $v$ with $||v||_2=1$ satisfies $$|v \cdot \frac{u_i}{||u_i||_2}| \leq \epsilon$$ for all $i$ that are at least some $k_0$. Is there a sense in which $v$ is slowly varying (as a function of $k_0$ and $\epsilon$)?

  2. More specifically, the first eigenvector $u_1$ seems to have the property that $u_1(i+1)-u_1(i)$ is roughly the same size as $u_1(i)-u_1(i-1)$. Can an analogous property be proven in the setting of question 1?

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