Fourier Series and Orthogonal Functions by Harry F. Davis

Author:Harry F. Davis
Language: eng
Format: epub
ISBN: 9780486140735
Publisher: Dover Publications
Published: 2012-10-07T16:00:00+00:00

(25)

where Pn and Qn are linearly independent. It will be recalled, however, that the existence theorem in question is valid only if the leading coefficient of the differential equation is always nonzero.

The leading coefficient in (20) is nonzero when −1 < x < 1, but is zero at the endpoints. Therefore the existence theorem guarantees a solution of the form (25) valid in the open interval −1 < x < 1, but not necessarily in the closed interval −1 x 1. It turns out, in this case, that every solution of (21) for λ ≠ n(n + 1) is unbounded at one of the endpoints, and if λ = n(n + 1) the “other” solution Qn(x) is unbounded at both endpoints. Thus, Theorem 6 does not contradict the existence theorem. This also explains why we can prove orthogonality more easily here than in most Sturm-Liouville equations; the only “boundary condition” needed is boundedness at the endpoints (compare Example 6, Section 2.4).

Let us return to the expansion (12). If a function ƒ is even, f(x) = ƒ(−x), we will have

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