Delta Function Representation by Orthogonal Functions

Dirac's delta function can be expanded in terms of any basis of real orthogonal functions ${ \phi_n(x), n = 0, 1, 2, \ldots }$. Such functions will occur in Chapter 10 of Mathematical Methods for Physicists 6th Edition by George Arfken as solutions of ordinary differential equations of the Sturm–Liouville form.

They satisfy the orthogonality relations:

$$\int_{a}^{b} \phi_m(x) \phi_n(x) dx = \delta_{mn}, \tag{1.187}$$

where the interval $(a, b)$ may be infinite at either end or both ends. We use the $\phi_n$ to expand the delta function as

$$\delta(x - t) = \sum_{n=0}^{\infty} a_n(t) \phi_n(x), \tag{1.188}$$

where the coefficients $a_n$ are functions of the variable $t$. Multiplying by $\phi_m(x)$ and integrating over the orthogonality interval (Eq. (1.187)), we have

$$a_m(t) = \int_{a}^{b} \delta(x - t) \phi_m(x) dx = \phi_m(t), \tag{1.189}$$

or

$$\delta(x - t) = \sum_{n=0}^{\infty} \phi_n(t) \phi_n(x) = \delta(t - x). \tag{1.190}$$

This series is not uniformly convergent, but it may be used as part of an integrand in which the ensuing integration will make it convergent.

Suppose we form the integral $\int F(t) \delta(t - x) dx$, where it is assumed that $F(t)$ can be expanded in a series of orthogonal functions $\phi_p(t)$, a property called completeness. We then obtain

$$\int F(t) \delta(t - x) dt = \int \sum_{p=0}^{\infty} a_p \phi_p(t) \sum_{n=0}^{\infty} \phi_n(x) \phi_n(t) dt = \sum_{p=0}^{\infty} a_p \phi_p(x) = F(x), \tag{1.191}$$

the cross products $\int \phi_p \phi_n dt$ ($n \neq p$) vanishing by orthogonality (Eq. (1.187)). Our series representation, Eq. (1.190), satisfies the defining property of the Dirac delta function and therefore is a representation of it. This representation of the Dirac delta function is called closure. The assumption of completeness of a set of functions for expansion of $\delta(x - t)$ yields the closure relation. The converse, that closure implies completeness, is the topic of Exercise 1.15.16 in [MMP].