User:Brews ohare/Identity

Resolutions of the identity

Given a complete orthonormal basis set of functions {$\varphi _{n}$} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as:

$f=\sum _{n=1}^{\infty }\ \alpha _{n}\varphi _{n}\ .$

The coefficients {α_n} are found as:

$\alpha _{n}=\langle \varphi _{n},\ f\rangle \ ,$

which may be represented by the notation:

$\alpha _{n}=\varphi _{n}^{\dagger }\ f\ ,$

a form of the bra-ket notation of Dirac. Adopting this notation, the expansion of f takes the dyadic form:

$f=\left(\sum _{n=1}^{\infty }\ \varphi _{n}\varphi _{n}^{\dagger }\right)\ f.$

The expression:

$I=\sum _{n=1}^{\infty }\ \varphi _{n}\varphi _{n}^{\dagger },$

is called a resolution of the identity I. When the Hilbert space is the space L²(D) of square-integrable functions on a domain D, the quantity:

$\varphi _{n}\varphi _{n}^{\dagger },$

is an integral operator, and the expression for f can be rewritten as:

$f(x)=\int _{D}\,d\xi \ w(\xi )\left(\sum _{n=1}^{\infty }\ \varphi _{n}(x)\varphi _{n}^{*}(\xi )\right)f(\xi )\ ,$

where allowance is made that a weighting function w(x) occurs in the inner product. The right-hand side converges to f in the L² sense. It need not hold in a pointwise sense, even when f is a continuous function. Nevertheless, it is common to abuse notation and write the inner product of f with the δ-function as:

$f(x)=\int \ d\xi \ w(\xi )\ \delta (x-\xi )f(\xi ),$

resulting in the representation of the delta function:

$\delta (x-\xi )=\sum _{n=1}^{\infty }\ \varphi _{n}(x)\varphi _{n}^{*}(\xi ).$

With a suitable rigged Hilbert space (Φ,L²(D),Φ^∗) where Φ⊂L²(D) contains all compactly supported smooth functions, this summation may converge in Φ^*, depending on the properties of the basis φ_n. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the distribution sense.

Example: This formalism encompasses much of generalized Fourier series. For example, the Fourier-Bessel series:

$f(x)=\sum _{n=0}^{\infty }c_{n}J_{0}(\lambda _{n}x/b),$

for x in the range 0 ≤ x ≤ b, where the {λ_n} are the zeros of the zero-order Bessel function J₀, with coefficients:

$c_{n}={\frac {2}{b^{2}J_{1}(\lambda _{n})^{2}}}\int _{0}^{b}\ \xi \ d\xi \ f(\xi )\ J_{0}\left({\frac {\lambda _{n}\xi }{b}}\right)\ ,$

converges in the norm L_w²(0, b) with weight w = x. The basis functions satisfy the orthonormality condition based upon the weight function w(x) = x:

$\int _{0}^{1}J_{0}(x\lambda _{m})\,J_{0}(x\lambda _{n})\,x\,dx={\frac {\delta _{mn}}{2}}[J_{1}(\lambda _{n})]^{2}\ .$

If the coefficient expression is substituted back into the series expansion, the result is:

$f(x)=\int _{0}^{b}\ d\xi \ \xi f(\xi )\ \left(\sum _{n=0}^{\infty }{\frac {2}{b^{2}J_{1}(\lambda _{n})^{2}}}\ J_{0}(\lambda _{n}\xi /b)J_{0}(\lambda _{n}x/b)\right)\ ,$

which may be viewed as the inner product of f with the δ-function, based upon the weight function w(x) = x:

$f(x)=\int _{0}^{b}\ d\xi \ \xi f(\xi )\delta (x-\xi )=\langle f(\xi ),\ \delta (x-\xi )\rangle \ ,$

resulting in the representation of the delta function as:

$\delta (x-\xi )=\ \sum _{n=0}^{\infty }\ {\frac {2}{b^{2}J_{1}(\lambda _{n})^{2}}}J_{0}(\lambda _{n}x/b)J_{0}(\lambda _{n}\xi /b)\ .$