A sequence of vectors whose cosines decay exponentially with the distance between indices

Question

$$ \left[ \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \vdots \end{array} \right], \left[ \begin{array}{c} \cos\theta \\ \sin\theta \\ 0 \\ 0 \\ 0 \\ 0 \\ \vdots \end{array} \right], \left[ \begin{array}{c} \cos^2\theta \\ \sin\theta\cos\theta \\ \sin\theta \\ 0 \\ 0 \\ 0 \\ \vdots \end{array} \right], \left[ \begin{array}{c} \cos^3\theta \\ \sin\theta\cos^2\theta \\ \sin\theta\cos\theta \\ \sin\theta \\ 0 \\ 0 \\ \vdots \end{array} \right], \left[ \begin{array}{c} \cos^4\theta \\ \sin\theta\cos^3\theta \\ \sin\theta\cos^2\theta \\ \sin\theta\cos\theta \\ \sin\theta \\ 0 \\ \vdots \end{array} \right], \, \ldots \ldots $$ Call the sequence above $(\mathbf v_n)_{n\,=\,1}^\infty.$ Then for $n,m=0,1,2,3,\ldots$ we have $\mathbf v_n\cdot\mathbf v_m = \cos^{\left|n-m\right|}\theta.$ (This is the usual dot-product.) (In particular, all are unit vectors.)

• Is there a natural way to extend this to nonnegative real (not necessarily integer) indices?
• Where has this been written about?

Answer 1

This is an expanded collection of the thoughts and examples in my comments.

My first idea $\ell_2(\ell_2)$ was to introduce another countable orthonormal sequence at each nonintegral positive dyadic rational $\mathbf{e}_\lambda$ (grouping them by the denominator of $\lambda$ gives the $\ell_2(\ell_2)$) and construct $\mathbf{v}_q$, $q\in\mathbb{Z}[\frac12]\cap\mathbb{R}^+$ inductively according to the denominator $2^n$, similar to how one construct Brownian paths with infinitely many Gaussians:

• base case: $\mathbf{v}_n$ for all $n=0,1,2,\dots$ already given (shifting the index a little bit).
• assuming $\mathbf{v}_{k/2^n}$ has been constructed for all $k$, let $$\mathbf{v}_{(m+\frac12)/2^n}=(\cos\theta)^{1/2^{n+1}}\mathbf{v}_{m/2^n}+\sqrt{1-(\cos\theta)^{1/2^n}}\,\mathbf{u}_{(m+\frac12)/2^n},$$ where $\mathbf{u}_{(m+\frac12)/2^n}$ is a unit vector spanned by $\mathbf{v}_{(m+1)/2^n}-(\cos\theta)^{1/2^{n+1}}\mathbf{v}_{m/2^n}$ and $\mathbf{e}_{(m+\frac12)/2^n}$ such that the inner product $\langle\mathbf{v}_{(m+1)/2^n},\mathbf{v}_{(m+\frac12)/2^n}\rangle=(\cos\theta)^{1/2^{n+1}}$.

It is easy to check we have $\langle\mathbf{v}_{m/2^{n+1}},\mathbf{v}_{m'/2^{n+1}}\rangle=(\cos\theta)^{\lvert m-m'\rvert/2^{n+1}}$. And $q\mapsto\mathbf{v}_q$ is easily seen to be a uniformly continuous map $\mathbb{Z}[\frac12]\cap\mathbb{R}^+\to\ell_2(\ell_2)$, so there is a unique continuous extension to $\mathbb{R}^+\to\ell_2(\ell_2)$ and we have $\langle\mathbf{v}_s,\mathbf{v}_t\rangle=(\cos\theta)^{\lvert s-t\rvert}$.

This is of course rather unnatural in the introduction of the $\mathbf{e}_\lambda$s.

The more natural way was motivated by the $MA(\infty)$ representation of a stationary AR(1) process $X_t=\phi X_{t-1}+\epsilon_t$ as $\sum_{n=-\infty}^{t}\phi^{t-n}\epsilon_n$, adapting it to this case. So the sum (or rather, the direct sum of the subspaces $\mathbb{C}\epsilon_n$) becomes a (direct) integral and we have the exponential handed in golden plate. Of course a direct integral $\int^\oplus_X H_x\,\mathrm{d}\mu(x)$ with a global trivialisation $H_s\cong\mathbb{C}$ is just the usual $L^2(\mu)$. So we arrive at the family $$ f_x\colon t\mapsto C(\cos\theta)^{\frac12(x-t)}1_{t<x}, $$ in $L^2(\mathbb{R})$ with some normalisation constant $C$.