Difference between revisions of "Fourier-Bessel series"
From Encyclopedia of Mathematics
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| − | + | 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|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: | ||
| − | + | $$ | |
| + | c _ {m} = \ | ||
| + | { | ||
| + | \frac{2}{a ^ {2} J _ {\nu + 1 } ^ {2} | ||
| + | ( x _ {m} ^ {( \nu ) } ) } | ||
| + | } | ||
| + | \int\limits _ { 0 } ^ { a } | ||
| + | rf ( r) J _ \nu \left ( x _ {m} ^ {( \nu ) } | ||
| + | \cdot { | ||
| + | \frac{r}{a} | ||
| + | } \right ) dr. | ||
| + | $$ | ||
| − | then the Fourier–Bessel series converges and its sum is equal to | + | 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. | ||
Latest revision as of 19:39, 5 June 2020
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.
How to Cite This Entry:
Fourier-Bessel series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Bessel_series&oldid=46960
Fourier-Bessel series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Bessel_series&oldid=46960
This article was adapted from an original article by L.N. Karmazina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article