Fourier–Bessel series

Remarks

The functions $J_\nu(j_{\nu,k} x)$ are orthogonal in the weighted Lebesgue space $L^2((0, 1), x \, dx)$, with $$\int_0^1 x \, J_\nu(j_{\nu,k} x) \, J_\nu(j_{\nu,m} x) \, dx = \frac{\delta_{km}}{2} J_{\nu+1}(j_{\nu,k})^2.$$ So the series is the generalized Fourier series of $f$ with respect to this orthogonal family, and the formula for $c_k$ is the usual quotient of inner products. The family is complete in $L^2((0, 1), x \, dx)$, so the series converges to $f$ in this space. On an interval $[0, b]$ one uses $J_\nu(j_{\nu,k} x / b)$ instead.

Example: $\nu = 0$ and $f = 1$. Since $\frac{d}{dt}\big(t J_1(t)\big) = t J_0(t)$, the substitution $t = j x$ gives $\int_0^1 x J_0(jx) \, dx = J_1(j)/j$. Hence $c_k = \frac{2}{j_{0,k} J_1(j_{0,k})}$ and $$1 = \sum_{k=1}^{\infty} \frac{2}{j_{0,k} \, J_1(j_{0,k})} \, J_0(j_{0,k} x) \quad \text{in } L^2((0,1), x\,dx).$$

Non-example: the functions $J_0(k\pi x)$, $k = 1, 2, \dots$, do not give a Fourier–Bessel series. The numbers $k\pi$ are not zeros of $J_0$; for instance $J_0(\pi) \approx -0.304 \ne 0$.