hat happens if one takes Fourier series
of nonperiodic function? To put the question in exact terms, suppose the
function
is such that the restriction of it to the interval
belongs to
.
We form the
function
with the functions
being the Fourier exponentials normalized to
.
Such function will be the
periodic
repetition of the restriction
to the intervals
,
:
This is so because the function
is
periodic
and the sum
is the Fourier series of
.
Then by uniqueness of Fourier series for
periodic
functions we must
have
