Fourier-Bessel series

The expansion of a function $ f $ in a series

$$ \tag{* } f ( x) = \ \sum _ {m = 1 } ^ \infty c _ {m} J _ \nu \left ( x _ {m} ^ {( \nu ) } \cdot { \frac{x}{a} } \right ) ,\ \ 0 < x < a, $$

where $ f $ is a function given on the interval $ ( 0, a) $, $ J _ \nu $ is the Bessel function of order $ \nu > \- 1/2 $( cf. Bessel functions), and the $ x _ {m} ^ {( \nu ) } $ are the positive zeros of $ J _ \nu $ taken in increasing order; the coefficients $ c _ {m} $ have the following values:

$$ c _ {m} = \ { \frac{2}{a ^ {2} J _ {\nu + 1 } ^ {2} ( x _ {m} ^ {( \nu ) } ) } } \int\limits _ { 0 } ^ { a } rf ( r) J _ \nu \left ( x _ {m} ^ {( \nu ) } \cdot { \frac{r}{a} } \right ) dr. $$

If $ f $ is a piecewise-continuous function given on an interval $ ( 0, a) $ and if the integral

$$ \int\limits _ { 0 } ^ { a } \sqrt r | f ( r) | dr < \infty , $$

then the Fourier–Bessel series converges and its sum is equal to $ [ f ( x + ) + f ( x - )]/2 $ at each interior point $ x $ of $ ( 0, a) $ at which $ f $ locally has bounded variation.