Fourier series of function of bounded variation
If the real function is of bounded variation on the interval , then its Fourier series expansion
(1) |
with the coefficients (http://planetmath.org/FourierCoefficients)
(2) |
converges at every point of the interval. The sum of the series is at the interior points equal to the arithmetic mean of the left-sided (http://planetmath.org/OneSidedLimit) and the right-sided limit of at and at the end-points of the interval equal to .
Remark 1. Because of the periodicity of the terms of the terms, the expansion (1) converges for all real values of and it represents a periodic function with the period .
Remark 2. If the function is of bounded variation, instead of , on the interval the equations (1) and (2) may be converted via the change of variable to
(3) |
and
(4) |
Correspondingly, the sum of (3) at the points of is expressed via the left-sided and righr-sided limits of .