Which polynomial has similar properties with Legendre?

Question

I am looking for an kind of polynomial such as Legendre properties that polynomial sequence of orthogonal polynomials such as bellow image. Could you suggest to me one polynomial? Is B-spline correct? Thanks

Answer 1

We can actually invent infinitely many! From what i gather you are interested in orthogonal polynomials, meaning polynomials $P_i$ such that there exists $a,b$ such that

$$ \int_a^bP_i(x)P_j(x) dx = 0 \forall i \ne j$$

To generate such polynomial sequences we can start with some initial Polynomial $P_0$ it can be any polynomial of your choice (An example is $x^2+3x$) and then pick a suitable a,b (lets say 0,1).

Then to get the next polynomial in teh sequence note it will have degree one greater than our current one. So $P_0 = x^2 +3x$ $P_1$ that we are trying to calculate will have the form $a_0 + a_1x + a_2x^2 + a_3x^3 $. Then we can write the integration law:

$$ \int_a^b P_0 P_1 dx = 0 $$

Which for us is

$$ \int_0^1 (x^2 + 2x)(a_0 + a_1x + a_2x^2 + a_3x^3) dx = 0 $$

And this yields a single equation in 4 unknowns after integration

$$ \frac{1}{3}a_0 + \frac{1}{4}a_1 + \frac{1}{5}a_2 + \frac{1}{6}a_3 + \frac{2}{2}a_0 + \frac{2}{3}a_1 + \frac{2}{4}a_2 + \frac{2}{5}a_3 = 0 $$

That can be cleaned up to

$$ \frac{4}{3}a_0 + \frac{11}{12}a_1 + \frac{14}{20}a_2 + \frac{17}{30}a_3 = 0 $$

Any four numbers $a_0 ... a_3$ you pick to satisfy that generates an orthogonal polynomial to the $P_0$ we selected.

Like so $P_2$ can be found by putting conditions on $P_0$ and $P_1$ such that it is orthogonal to both. Like this we can repeatedly generate one orthogonal polynomial after another to make whatever orthogonal sequence we choose.