MHB Orthonormal basis for the poynomials of degree maximum 2

AI Thread Summary
The discussion focuses on calculating an orthonormal basis for polynomials of degree up to 2 using the inner product defined by the integral of the product of functions weighted by \(1 - x^2\). The user initially applied the Gram-Schmidt algorithm but made an error in calculating the norm, mistakenly equating it to the inner product instead of its square root. After receiving clarification, the user acknowledged the mistake and successfully obtained the correct results. The conversation highlights the importance of accurate calculations in the Gram-Schmidt process for achieving orthonormality. Overall, the thread emphasizes the significance of proper mathematical procedures in polynomial orthonormalization.
mathmari
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Hey! 😊

We consider the inner product $$\langle f,g\rangle:=\int_{-1}^1(1-x^2)f(x)g(x)\, dx$$ Calculate an orthonormal basis for the poynomials of degree maximum $2$.

I have applied the Gram-Schmidt algorithm as follows:

\begin{align*}\tilde{q}_1:=&1 \\ q_1:=&\frac{\tilde{q}_1}{\|\tilde{q}_1\|}=\frac{1}{\langle \tilde{q}_1, \tilde{q}_1\rangle}=\frac{1}{\int_{-1}^1(1-x^2)\cdot 1\cdot 1\, dx}=\frac{1}{\int_{-1}^1(1-x^2)\, dx}=\frac{1}{\left [x-\frac{x^3}{3}\right]_{-1}^1}=\frac{1}{\left [1-\frac{1^3}{3}\right]-\left [(-1)-\frac{(-1)^3}{3}\right ]}\\ =&\frac{1}{1-\frac{1}{3}+1-\frac{1}{3}}=\frac{1}{2-\frac{2}{3}}=\frac{1}{\frac{4}{3}}=\frac{3}{4}\end{align*}

\begin{align*}\tilde{q}_2:=&x-\langle x, q_1\rangle q_1=x-\left (\int_{-1}^1(1-x^2)\cdot x\cdot \frac{3}{4}\, dx\right )\cdot \frac{3}{4}=x-\frac{3}{4}\cdot \left (\int_{-1}^1(x-x^3)\, dx\right )\cdot \frac{3}{4}=x-\frac{9}{16}\cdot \left [\frac{x^2}{2}-\frac{x^4}{4}\right ]_{-1}^1 \\ = & x-\frac{9}{16}\cdot 0=x\\ q_2:=&\frac{\tilde{q}_2}{\|\tilde{q}_2\|}=\frac{x}{\langle \tilde{q}_2, \tilde{q}_2\rangle}=\frac{x}{\int_{-1}^1(1-x^2)\cdot x\cdot x\, dx}=\frac{x}{\int_{-1}^1(x^2-x^4)\, dx}=\frac{x}{\left [\frac{x^3}{3}-\frac{x^5}{5}\right]_{-1}^1}=\frac{x}{\left [\frac{1}{3}-\frac{1}{5}\right]-\left [-\frac{1}{3}+\frac{1}{5}\right ]}\\ =&\frac{x}{\frac{1}{3}-\frac{1}{5}+\frac{1}{3}-\frac{1}{5}}=\frac{x}{\frac{2}{3}-\frac{2}{5}}=\frac{x}{\frac{4}{15}}=\frac{15x}{4}\end{align*}

\begin{align*}\tilde{q}_3:=&x^2-\langle x^2, q_1\rangle q_1-\langle x^2, q_2\rangle q_2=x^2-\left (\int_{-1}^1(1-x^2)\cdot x^2\cdot \frac{3}{4}\, dx\right )\cdot \frac{3}{4}-\left (\int_{-1}^1(1-x^2)\cdot x^2\cdot \frac{15x}{4}\, dx\right )\cdot \frac{15x}{4}\\ =&x^2-\frac{9}{16}\cdot \int_{-1}^1(x^2-x^4)\, dx-\frac{225x}{16}\cdot \int_{-1}^1(x^3-x^5)\, dx =x^2-\frac{9}{16}\cdot \frac{4}{15}-\frac{225x}{16}\cdot 0=x^2-\frac{3}{20}\\ q_3:=&\frac{\tilde{q}_3}{\|\tilde{q}_3\|}=\frac{x^2-\frac{3}{20}}{\langle \tilde{q}_3, \tilde{q}_3\rangle}=\frac{x^2-\frac{3}{20}}{\int_{-1}^1(1-x^2)\cdot \left (x^2-\frac{3}{20}\right )\cdot \left (x^2-\frac{3}{20}\right )\, dx}=\frac{x^2-\frac{3}{20}}{\int_{-1}^1\left (-x^6+\frac{13x^4}{10}-\frac{129x^2}{400}+\frac{9}{400}\right )\, dx}=\frac{x^2-\frac{3}{20}}{\frac{9}{140}}\\ =&\frac{140}{9}\left (x^2-\frac{3}{20}\right )\end{align*} I wanted to check if I have the correct answers and I noticed (if I am not mistaken) that the polynomials that I found are not orthonormal in respect to the other terms. Have I applied a wrong formula? :unsure:
 
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Hey mathmari!

I see you wrote $\|\tilde q_1\|=\langle\tilde q_1,\tilde q_1\rangle$.
Shouldn't that be $\|\tilde q_1\|=\sqrt{\langle\tilde q_1,\tilde q_1\rangle}$? šŸ¤”
 
Klaas van Aarsen said:
I see you wrote $\|\tilde q_1\|=\langle\tilde q_1,\tilde q_1\rangle$.
Shouldn't that be $\|\tilde q_1\|=\sqrt{\langle\tilde q_1,\tilde q_1\rangle}$? šŸ¤”

Oh yes, you 're right! Now I get the correct result! :giggle:
 
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