orthogonality of Chebyshev polynomials from recursion
In this entry, we shall demonstrate the orthogonality relation of the Chebyshev polynomials from their recursion relation. Recall that this relation reads as
with initial conditions and . The relation we seek to demonstrate is
when .
We start with the observation that is an even function when is even and an odd function when is odd. That this is true for and follows immediately from their definitions. When , we may induce this from the recursion. Suppose that when . Then we have
From this observation, we may immediately conclude half of orthogonality. Suppose that and are nonnegative integers whose difference is odd. Then , so we have
because the integrand is an odd function of .
To cover the remaining cases, we shall proceed by induction. Assume that is orthogonal to whenever and and . By the conclusions of last paragraph, we know that is orthogonal to . Assume then that . Using the recursion, we have
By our assumption, each of the three integrals is zero, hence is orthogonal to , so we conclude that is orthogonal to when and and .
Title | orthogonality of Chebyshev polynomials from recursion |
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Canonical name | OrthogonalityOfChebyshevPolynomialsFromRecursion |
Date of creation | 2013-03-22 18:54:46 |
Last modified on | 2013-03-22 18:54:46 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 6 |
Author | rspuzio (6075) |
Entry type | Proof |
Classification | msc 33C45 |
Classification | msc 33D45 |
Classification | msc 42C05 |