A half-range Fourier series is a Fourier series representing a function given on 
by extending it to 
as either an odd function
or an even function, then extending periodically
with period 
. The odd extension gives the Fourier
sine series
 |
(1)
|
where
 |
(2)
|
The even extension gives the Fourier cosine series
 |
(3)
|
where
 |
(4)
|
for 
,
1, .... The symbol 
indicates a Fourier series
representation, not an asymptotic expansion.
For a piecewise smooth extension, either series converges at an interior point to the average of the left and right
limits. Consequently, both represent 
at its points of continuity inside
even though they use different periodic extensions. The sine extension can have jumps
at the endpoints, so equality there requires separate consideration.
