Which polynomial has similar properties with Legendre?

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

• What does "similar properties" mean? – Martin R Jul 2 '15 at 18:37
• It is "polynomial sequence of orthogonal polynomials" – Jame Jul 2 '15 at 18:38
• Does this help? en.wikipedia.org/wiki/Orthogonal_polynomials – eigenchris Jul 2 '15 at 18:39

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.

• Thank you. Do you hear about Haar basis function. Could you based on your explanation and find first few Haar basis function as above image in my question?en.wikipedia.org/wiki/Haar_wavelet – Jame Jul 2 '15 at 19:14
• Your link gives there general formula right there. Anyone should be able to find them. The proof of orthonormality is also given – frogeyedpeas Jul 2 '15 at 19:16
• The tools above are overkill if that's your intention. – frogeyedpeas Jul 2 '15 at 19:16
• It is does not look like your tool. What is P(x) in the Haar case. Could you put some example in your answer? – Jame Jul 2 '15 at 19:24
• Looking a bit more into it. Those aren't even polynomials. So the technique above doesn't apply. They are much more artificial (perhaps specially invented for some real life application). Do you want a proof of their orthogonality? Etc... They are orthogonal but they aren't polynomials and so asking how to construct them from polynomials won't make too much sense – frogeyedpeas Jul 2 '15 at 19:28