User:Brews ohare/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:
 * Resolutions of the identity
 * $$f = \sum_{n=1}^\infty \ \alpha_n \varphi_n \ . $$

The coefficients {αn} 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 (&Phi;,L2(D),&Phi;&lowast;) where &Phi;&sub;L2(D) contains all compactly supported smooth functions, this summation may converge in &Phi;*, depending on the properties of the basis &phi;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 {λn} are the zeros of the zero-order Bessel function J0, with coefficients:
 * $$c_n

=\frac{2}{b^2J_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 Lw2(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:


 * $$f(x) =\int_0^b \ d\xi \ \xi  f(\xi)\ \left(  \sum_{n=0}^\infty \frac{2}{b^2J_1(\lambda_n)^2}\ J_0(\lambda_n \xi / b) J_0(\lambda_n x/b)\right) \ ,$$

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^2J_1(\lambda_n)^2}

J_0 ( \lambda_n x / b) J_0 ( \lambda_n \xi/b) \. $$