3.1 Orthogonality
Definition 3.1.1. Let \(f(x)\) and \(g(x)\) be continuous on a closed interval \([a,b]\). Let \(w(x)\) be a positive function that is Riemann integrable on \([a,b]\) i.e \[\int _a^bw(x)\, dx < \infty .\] We define the inner product or dot product of \(f\) and \(g\) with respect to the weight function \(w(x)\) by \[\langle f,g\rangle _w = \int _a^b f(x)\, g(x)\, w(x)\, dx.\]
The norm of \(f(x)\) with respect to \(w(x)\), denoted by \(\lVert f\rVert _w\) is defined by \[\lVert f\rVert _w = \left [\int _a^b(f(x))^2\, w(x)\, dx\right ]^{\frac {1}{2}}.\] i.e \[\lVert f\rVert _w = \sqrt {\langle f,g\rangle _w}.\]
Definition 3.1.2. Two functions \(f(x)\) and \(g(x)\) are said to be orthogonal with respect to the weight function \(w(x)\) if the inner product is zero.
In general, a sequence \(\{P_n(x)\}^{\infty }_{n = 0}\) of continuous functions defined on \([a,b]\) are said to be orthogonal with respect \(w(x)\) if \[\langle f_m,f_n\rangle _w = 0 \hspace {0.3cm} \text {for all}\hspace {0.3cm} m\neq n.\]
If \(\lVert P_n\rVert = 1\), then the set \(\{P_n(x)\}^{\infty }_{n = 0}\) is said to be orthogonal.
A sequence of orthogonal polynomials can be constructed using the Gram-Schmidt orthogonalisation process.
Theorem 3.1.3. The polynomials \(\{P_n(x)\}^{\infty }_{n=0}\) which are defined according to the following recurrence relation are orthogonal: \begin {align*} P_0(x) & = 1,\\ P_1(x) & = \frac {x - \langle x P_0, P_0\rangle _w}{\lVert P_0\rVert _w^2}\\ \vdots & \\ P_n(x) & = (x-a_n)P_n(x) - b_nP_{n-2}(x), \hspace {0.5cm} n = 1, \, 2, \, 3,\, \cdots \end {align*}
where \[a_n = \frac {\langle x P_{n-1}, P_{n-1}\rangle _w}{\lVert P_{n-1}\rVert ^2_w}\, , \, \, \, b_n = \frac {\langle x P_{n-1}, P_{n-2}\rangle _w}{\lVert P_{n-2}\rVert ^2_w}.\]
Proof. By induction on \(n\), we show that the inner product \(\langle P_n , P_i \rangle _w = 0\) for \(i < n\) when \(n = 1\), \begin {align*} \langle P_1, P_0\rangle _w & = \int _a^bP_1(x)\, P_0(x)\, w(x)\, dx\\ & = \int _a^b\left [x - \frac {\langle x,1\rangle _w}{\lVert 1\rVert _w^2}\right ]\, w(x)\, dx\\ & = \int _a^b x\, w(x)\, dx - \frac {\langle x,1\rangle _w}{\lVert 1\rVert _w^2}\, \int _a^b w(x)\, dx\\ & = \langle x,1\rangle _w - \frac {\langle x,1\rangle _w}{\lVert 1\rVert _w^2}\, \cdot \, \lVert 1\rVert ^2_w\\ & = 0. \end {align*}
Now suppose that the assertion is true for \(n - 1\) \(\, (n\geq 2)\). We show that it is true for \(n\). We have \begin {align*} \langle P_n, P_i\rangle _w & = \int _a^b\left [(x-a_n)P_{n-1}(x) - b_nP_{n-2}(x)\right ]P_i(x)\, w(x)\, dx\\ & = \int _a^bxP_{n-1}(x)P_i(x)w(x)\, dx - \int _a^ba_n P_{n-1}(x)P_i(x)w(x)\, dx - b_n\int _a^bP_{n-2}(x)P_i(x)w(x)\, dx\\ & = \langle xP_{n-1}(x), P_i(x)\rangle _w - a_n\langle P_{n-1},P_i\rangle _w - b_n\langle P_{n-2},P_i\rangle _w. \end {align*}
Thus for \(i = n-1\), \begin {align*} \langle P_n, P_{n-1}\rangle & = \langle x P_{n-1}, P_{n-1}\rangle _w - a_n \langle P_{n-1}, P_{n-1}\rangle - b_n \langle P_{n-2}, P_{n-1}\rangle _w\\ & = \langle x P_{n-1} , P_{n-1}\rangle _w - \frac {\langle x P_{n-1},P_{n-1}\rangle _w}{\lVert P_{n-1}\rVert _w^2}\, \cdot \, \lVert P_{n-1}\rVert _w^2 - b_n \langle P_{n-2},P_{n-1}\rangle _w\\ & = 0, \end {align*}
by the induction hypothesis.
Similarly, for \(i = n-2\), \begin {align*} \langle P_n, P-i\rangle _w & = \langle x P_{n-1}, P_{n-2}\rangle _w - a_n\langle P_{n-1}, P_{n-2}\rangle _w - b_n \langle P_{n-2}, P_{n-2}\rangle _w\\ & = \langle x P_{n-1}, P_{n-2}\rangle _w - b_n\lVert P_{n-2}\rVert _w^2\\ & = b_n \lVert P_{n-2}\rVert _w^2 - b_n\lVert P_{n-2}\rVert _w^2\\ & = 0, \end {align*}
by definition of \(b_n\).
Finally, for \(i < n-2\), we have \begin {align*} \langle P_n, P_i\rangle _w & = \langle x P_{n-1}, P_i\rangle _w - a_n\langle P_{n-1},P_i\rangle _w - b_n\langle P_{n-2},P_i\rangle _w\\ & = \int _a^b x\, P_{n-1}(x)\, P_i(x)\, w(x)\, dx. \end {align*}
But now, from the recurrence relation, we have \[P_{i+1}(x) = (x - a_{i+1})P_i(x) - b_{i + 1}P_{i-1}(x),\] so that \[x P_i(x) = P_{i + 1}(x) + a_{i + 1} P_i(x) + b_{i + 1} P_{i-1}(x).\] It follows that \begin {align*} \langle P_n, P_i\rangle _w & = \int _a^bx\, P_{n-1}(x)\, P_i(x)\, w(x)\, dx\\ & = \int _a^b\left [P_{i+1}(x) + a_{i + 1}P_i(x) + b_{i+1}(x)\right ]P_{n-1}(x)\, w(x)\, dx\\ & = \langle P_{n-1},P_{i+1}\rangle _w + a_{i+1} \langle P_{n-1},P_i\rangle _w + b_{i+1}\langle P_{n-1}, P_{i-1}\rangle _w\\ & = 0, \end {align*}
by orthogonality of the \(P_i\)’s for \(i < n\).
Thus the set \(\{P_n\}^{\infty }_{n=0}\) is orthogonal. □
Particular orthogonal polynomials can be derived depending on the choice of the interval \([a,b]\) and the weight function \(w(x)\) for example, we have
| Orthogonal polynomials | \(a,\,b\) | \(w(x)\) |
| Legendre | -1, 1 | 1 |
| Jacobi | -1, 1 | \((1 - x)^{\alpha }(1 + x)^{\beta }, \, \hspace {0.2cm} \alpha ,\, \beta >-1\) |
| Chebyshev of \(1^{\text {st}}\) kind | -1, 1 | \((1 - x^2)^{-\frac {1}{2}}\) |
| Chebyshev of \(2^{\text {nd}}\) kind | -1, 1 | \((1 - x^2)^{\frac {1}{2}}\) |
| Hermite | \(-\infty , \, \infty \) | \(e^{-\frac {x^2}{2}}\) |
| Laguerre | \(0,\, \infty \) | \(x^{\alpha }\, e^{-x}, \,\hspace {0.2cm} \alpha > -1\) |
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