(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[itex]P_{2} \subset L_{2} [/itex] is the set of all polynomials of degree [itex]n \leq 2[/itex]. Complete the following approximation. In other words find the polynomial of degree 2 that minimises the following expression:

[itex]\int \left|cos(\frac{\pi t}{2}) - p(t)\right|^{2}dt = min[/itex] with -1 <= t >= 1

2. Relevant equations

[itex]x(t) = cos(\frac{\pi t}{2}) [/itex] this is real so [itex]\overline{x} = x [/itex]

[itex]p(t) = \sum a_{n}t^{n}[/itex] because polynomial max 2nd degree: 0 <= n >= 2 (we do not know whether it has imaginary terms or not)

3. The attempt at a solution

[itex]\left|cos(\frac{\pi t}{2}) - p(t) \right|^{2} = \left|x(t) - p(t) \right|^{2} [/itex]

[itex]= \left\langle x(t) - p(t),x(t) - p(t)\right\rangle [/itex]

[itex]= \int (x(t) - p(t))( \overline{x(t) - p(t)})dt [/itex] scalar product defined in set of polynomial functions

[itex]= \int (x\overline{x} - x\overline{p} - \overline{x}p + p\overline{p}) dt [/itex]

[itex]= \int (x^{2} - x\overline{p} - xp + p\overline{p}) dt [/itex]

[itex]= \int (x^{2} - x\sum\overline{a_{n}}t^{n} - x\sum a_{n}t^{n} + \sum a_{n}\overline{a_{n}}t^{2n}) dt[/itex]

[itex]= \int (x^{2} - x\sum(\overline{a_{n}} + a_{n})t^{n} + \sum \left|a_{n} \right|^{2}t^{2n}) dt[/itex]

[itex]= \int (x^{2} - 2x\sum Real(a_{n})t^{n} + \sum \left|a_{n} \right|^{2}t^{2n}) dt[/itex]

now I don't know what to do. what conditions make this minimal?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Minimisation in Hilbert space

**Physics Forums | Science Articles, Homework Help, Discussion**