Como ustedes saben, los armónicos esféricos$$Y_\ell^m(\theta, \phi) = \sqrt{{2\ell + 1(\ell + |m|)!}\over{4\pi (\ell + |m|)!}} \cdot P_\ell^m(\cos \theta)e^{im\phi}$$(here $\ell = 0, 1, 2, \ldots$ and $m = 0, \pm1, \pm2, \ldots, \pm\ell$) are orthonormal on the sphere$$\int_0^{2\pi} \int_0^\pi (Y_{\ell'}^{m'} (\theta, \phi))^* (Y_{\ell^{\prime\prime}}^{m^{\prime\prime}}(\theta, \phi)) \sin\theta\,d\theta\,d\phi = \delta^{m'm^{\prime\prime}}\delta_{\ell' \ell^{\prime\prime}}$$y están supuestas a ser completa.
https://dspace.mit.edu/bitstream/handle/1721.1/90373/8-07-fall-2005/contents/readings/completeness.pdf
$$\sum_{\ell = 0}^\infty \sum_{m = -\ell}^\ell (Y_\ell^m(\theta, \phi))^* (Y_\ell^m(\theta', \phi')) = {1\over{\sin \theta}} \delta(\theta - \theta') \delta(\phi - \phi')$$Set $m = 0$ and get$$Y_\ell^0(\theta, \phi) = \sqrt{{2\ell + 1}\over{4\pi}} \cdot P_\ell(\cos\theta),$$$$2\pi \int_0^\pi (Y_{\ell'}^0(\theta, \phi))^* (Y_{\ell^{\prime\prime}}^0(\theta, \phi)) \sin\theta\,d\theta = \delta_{\ell'\ell^{\prime\prime}},$$or$${{2\ell + 1}\over2} \int_0^\pi P_{\ell'}(\cos\theta)P_{\ell^{\prime\prime}}(\cos\theta)\sin\theta\,d\theta = \delta_{\ell' \ell^{\prime\prime}},$$which by $x = \cos \theta$ becomes the orthogonality statement$${{2\ell + 1}\over2} \int_0^\pi P_{\ell'}(x)P_{\ell^{\prime\prime}}(x)\,dx = \delta_{\ell' \ell^{\prime\prime}}.$$The Wikipedia article on Legendre polynomials reports the completeness relation$$\sum_{\ell = 0}^\infty {{2\ell + 1}\over2} P_\ell(x) P_\ell(y) = \delta(x - y),$$que he de improviso ni idea de cómo probar.
https://en.wikipedia.org/wiki/Legendre_polynomials
Deduzco de lo que usted ha escrito que usted está bien enterado de la teoría de los armónicos esféricos es central para la teoría cuántica de momento angular. De la $3$-dimensiones de los análogos de la fundamental relación de conmutación$$[\textbf{x}, \textbf{p}] \equiv \textbf{xp} - \textbf{px} = i\hbar \textbf{I},$$it follows that that the operators$$\textbf{L}_x = \textbf{yp}_z - \textbf{zp}_y, \quad\textbf{L}_y = \textbf{zp}_x - \textbf{xp}_z, \quad\textbf{L}_z = \textbf{xp}_y - \textbf{yp}_x$$(inherited from the classical construction $\textbf{L}= \textbf{r} \times \textbf{p}$) fail to commute:$$[\textbf{L}_x, \textbf{L}_y] = i\hbar \textbf{L}_z, \quad [\textbf{L}_y, \textbf{L}_z] = i\hbar\textbf{L}_x, \quad [\textbf{L}_z, \textbf{L}_x] = i\hbar\textbf{L}_y.$$But from this it follows that $\textbf{L}^3 \equiv \textbf{L}_x^2 + \textbf{L}_y^2 + \textbf{L}_z^2$ does commute with each of those operators, so $\{\textbf{L}^2, \textbf{L}_z\}$ (one could replace $\textbf{L}_z$ with any real linear combination of $\{\textbf{L}_x \textbf{L}_y, \textbf{L}_z\}$; it is by standard convention that we select $\textbf{L}_z$) comprise a commuting set of operators, and it becomes possible to ask for simultaneous eigenvectors/functions of these operators, objects $f$ such that$$\textbf{L}^2f = \lambda f \text{ and }\textbf{L}_zf = \mu f.$$An algebraic argument-that makes use of the operators$$\textbf{J}_+ = \textbf{L}_x + i\textbf{L}_y, \quad \textbf{J}_- = \textbf{L}_x - i\textbf{L}_y$$and that is as pretty as it is elementary-leads quickly to the conclusion that necessarily$$\begin{align}
\lambda = \hbar^2\ell(\ell + 1) & : \ell = 0, {1\over2}, 1, {3\over2}, 2, \ldots \\
\mu = \hbar m & : m = -\ell, -\ell + 1, \ldots, \ell - 1, \ell
\end{align}$$and accounts for the $(2\ell + 1)$-fold degeneracy of the states with a given $\ell$. For details, see i.e. §4.3.1 in David Griffiths's "Introduction to Quantum Mechanics". Elaboration of the argument leads in the cases $\ell = 0, 1, 2,\ldots$ explicit constructions of the eigenfunctions $f_\ell^m(r, \theta, \phi)$, whence to the spherical harmonics. But those arguments fail when $\ell = {1\over2}, {3\over2}, {5\over2}, \ldots$; one is led to $(2\ell + 1)$-dimensiones de los vectores; los operadores asociados están representados no por los operadores diferenciales, pero por matrices; y uno es llevado a spinors y a la vuelta de las representaciones de la rotación del grupo.
Los gráficos vectoriales están disponibles también para todos los fijos valor integral de $\ell$. "Integridad" adquiere entonces el significado estándar de la teoría de finito-dimensional espacios vectoriales, como cuando se dice de una base $\{\textbf{e}_1, \textbf{e}_2, \ldots, \textbf{e}_{2\ell + 1}\}$ que es completa (abarca el espacio vectorial).
Mueve ahora más cerca de su problema, como yo lo entiendo: se dice de las funciones de $\{u_1(x), u_2(x), \ldots\}$ que son ortogonales en el intervalo de $[a, b]$$$\int_a^b u_i(x)u_j(x)\,dx = \delta_{ij}$$that they are complete if every well-behaved function $f(x)$ on $[a, b]$ can be developed$$f(x) = \sum_i f_i u_i(x) \text{ with }f_i = \int_a^b f(y)u_i(y)\,dy.$$Then$$\begin{align}
f(x) & = \sum_i f_i u_i(x) \\
& = \sum_i \int_a^b f(y)u_i(y)u_i(x)\,dy \\
& = \int_a^b \left\{\sum_i u_i(x)u_i(y)\right\}f(y)\,dy : \text{all nice }f(x),
\end{align}$$whence formally $\sum_i u_i(x)u_i(y) = \delta(x - y)$. If one or several of the elements of the set $\{u_i(x)\}$ are deleted then the remaining functions are complete only with respect to the space of functions $g(x)$ that are orthogonal to the deleted basis functions:$$\int_a^b g(y)u_{\text{deleted}}(y)\,dy = 0.$$You propose to delete all spherical harmonics except those with some prescribed value of $\ell$. You look, therefore, to the restricted class $\mathcal{G}$ of functions of the form$$g(\theta, \phi) = P_\ell(\cos\theta) \sum_{m = -\ell}^{m = +\ell} g_m e^{im\phi}.$$That the functions $e^{im\phi} : m = -\ell, \ldots, +\ell$ are orthogonal and complete in $\mathcal{G}$ es, yo la tome, evidente. Así que el problema, como yo lo entiendo, parece haberse evaporado. Tengo por esta razón que no se trató de trabajo a través de los detalles de los argumentos esbozado en sus párrafos finales.
