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Hi I was attempting the following question and I am stuck and I don't know how to proceed. From My understanding I first substituted $\hat{y} = Ax$ where A is T x P selector matrix , in the minimizing equation. the will have to minimize $\|A x - y\|^2$ over $x$. which will give us $\hat{x} = (A^{T}A)^{-1}A^Ty$

The Problem is problem 15.10 in Introduction to Applied Linear Algebra Vectors, Matrices, and Least Squares by Stephen Boyd

Estimating a periodic time series. (See $\{15.3 .2 .)$ Suppose that the $T$ -vector $y$ is a measured time series, and we wish to approximate it with a $P$ -periodic $T$ -vector. For simplicity, we assume that $T=K P$, where $K$ is an integer. Let $\hat{y}$ be the simple least squares fit, with no regularization, $i . e .,$ the $P$ -periodic vector that minimizes $\|\hat{y}-y\|^{2} .$ Show that for $i=1, \ldots, P-1,$ we have $\hat{y}_{i}=\frac{1}{K} \sum_{k=1}^{K} y_{i+(k-1) P}$ ] In other words, each entry of the periodic estimate is the average of the entries of the original vector over the corresponding indices.

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