Lebesgue

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11: 18.18 Sums

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Expansion of $L^2$ functions

►In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. …

12: 1.17 Integral and Series Representations of the Dirac Delta

… ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. …

13: 2.4 Contour Integrals

… ►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) …

14: 1.5 Calculus of Two or More Variables

… ►A more general concept of integrability (both finite and infinite) for functions on domains in ${ℝ}^n$ is Lebesgue integrability. …

15: 2.10 Sums and Sequences

… ►Hence by the Riemann–Lebesgue lemma (§1.8(i)) …

16: 2.5 Mellin Transform Methods

… ►(The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) …