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Exercise 11.4.3

Avg difficulty 3.0

In the inner product space $V=C[-1,1]$ , consider the function $f\in V$ given by

f(x)~{}=~{}\begin{cases}1&\text{if $x<0$,}\\ 1-x&\text{if $x\geq 0$.}\end{cases}

Find the closest approximation to $f$ by a polynomial of degree at most 0, 1, 2, 3, and 4. Graph both $f$ and the approximating polynomials.

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Orthogonal projections and Fourier series, Exercise 11.4.3

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.