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Approximate $f(t) = 1-|2t-5|$ in $[2,3]$ by $p\in P_2$ by using the least squares method in respect to $\langle f,g\rangle = \int_2^3 f(x)g(x) \ dx$ (use the Legendre polynomials)

I think I need to project $p$ onto the subspace generated by $f(t)$ but I don't know how to do this because I should do

$$\langle p-f(t),something\rangle$$

Or should I do $\langle f(t)-p,1\rangle=0, \langle f(t)-p,t\rangle=0, \langle f(t)-p,t²\rangle=0$?

Can somebody give me a ligth?

Guerlando OCs
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1 Answers1

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If you are due to use Legendre polynomials, you have to

  • change the working interval :

Instead of $t \in [2,3]$, take variable $T \in [-1,1]$

using coordinates change

$$T=2t-5 \tag{1}$$

Meanwhile, expression $f(t)$ is changed into expression $F(T)=1-|T|.$

  • compute coefficients $$a_k=\dfrac{2k+1}{2}\int_{-1}^{1}F(T)L_k(T)dT \ \ k=0,1,2$$

Then, the looked for quadratic $Q(T)$ is the beginning of the infinite expansion limited to its 3 first terms : $$F(T)=\underbrace{a_0L_0(T)+a_1L_1(T)+a_2L_2(T)}_{Q(T)}+...$$

(see https://en.wikipedia.org/wiki/Legendre_polynomials).

You should find $a_0=\tfrac12, a_1=0, a_2=-\tfrac58$, giving

$$Q(T)=\tfrac12-\tfrac{5}{16}(3T^2-1)$$

which is indeed very satisfactory (see Fig. 1).

  • It remains to do the "return path", i.e., express the result as a quadratic $q(t)$ with respect to initial variable using (1) :

$$q(t)=\tfrac12-\tfrac{5}{16}(3(2t-5)^2-1)$$

(best quadratic approximation).

enter image description here

Fig. 1 : In red (resp. blue), curve with equation $Y=F(T)$ (resp. $Y=Q(T)$, its best quadratic approximation) on reference interval $[-1,1]$.

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Jean Marie
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  • Where did you get that formula for the coefficients? – Guerlando OCs Apr 06 '19 at 19:26
  • ... in paragraph "Orthonormality and completeness" of the Wikipedia reference I gave. – Jean Marie Apr 06 '19 at 20:21
  • Is this method of yours possibly related to https://math.stackexchange.com/questions/3176464/find-a-b-c-that-minimize-fa-b-c-int-11-t2-a-bt-c-cos-t2-dt? My teacher only talked about projections, so I should know how to arrive at those formulas by this method – Guerlando OCs Apr 06 '19 at 21:25
  • No, the question you mention doesn't use the Legendre polynomials $L_0(t)=1, L_1(t)=t, L_2(t)=\tfrac12(3t^2-1)...$. I used them 1) because you mention them in your question 2) It is the most efficient method. – Jean Marie Apr 06 '19 at 21:49
  • When you develope a periodic function into Fourier series, and you truncate it after its 2nd terms, you do a projection. It is exactly the same for the decomposition $\sum a_kL_k$ : if you truncate at after the term n°2, you do a projection. It is a theorem, you haven't to prove it again... – Jean Marie Apr 06 '19 at 22:18
  • No more comments to my solution ? – Jean Marie Apr 08 '19 at 14:31