Re: Fourier series expansion doubt

Nimo <azeez541@xxxxxxxxx> writes:

I've to expand any given polynomial of degree n i.e P(n)
as Fourier series, then "I want the polynomial in purely
Cosine and sine terms "


infinity
------|
\ [ An cos (nx) + Bn Sin (nx) ] + ao/2
/
------|
n =3D 1

and again transforming the series would bring back
My_Original Polynomial.

To achieve this what conditions I've to follow.
Please help it in clear, I'll make out the notes
of it.

(a) How and what to select, interval values [ , ] or ( , )

The interval you select is up to you. But it may be slightly
simpler to use a symmetric interval such as [-pi,pi]. Then
for odd n, the Fourier series of x^n on this interval involves
only sines, and for even n only cosines.

(b) Problem with converges, and
'
'
'
if there any conditions to achieve this
please provide them.

As has been stated, the Fourier series will converge to the polynomial
on the open interval, and to the average of the two endpoint values at the
endpoints. If you use the interval [-pi,pi], then an even polynomial
has the same values at the two endpoints, an odd function does not.
However, convergence will be quite slow: the coefficients will be asymptotic
to constant/n in general (in which case the series will only converge
conditionally, not absolutely), or constant/n^2 for most even polynomials.

[ OR ]
Fourier integral transforms are any useful here.
F(x) =3D integral -infinity_to_+infinity P(n) (e)^-2pi ixE dx
P(n) =3D f(x) =3D integral -infinity_to_+infinity F(x)(e)^2pi ixE dx

Polynomials have no Fourier transform in the usual sense, because they
are not square-integrable. They do have Fourier transforms in the
sense of distributions, but these involve Dirac deltas and their derivatives.
Thus the Fourier transform of x^n is 2 pi i^n Dirac^(n)(k),
corresponding to the fact that the n'th derivative of exp(i k x)
with respect to k at k = 0 is i^n x^n.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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