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## Main Question or Discussion Point

Hello,

I was wondering if such a thing even exists, so here it goes... Let's say I have a function x(s) (it is real, smooth, differentiable, etc.) defined on (0,1). In addition, dx/ds = 0 on the boundary (s=0 and s=1). I can compute its Fourier transform (?) as

[tex] a_p = \int_0^1 x(s) \cos(sp\pi)\, ds [/tex]

and now I have a set of numbers [itex]a_p[/itex] which contain the same information as the original function x(s).

The good news is that if I compute the same Fourier transform on the derivatives of x, i.e.

[tex] \int_0^1 \frac{d^2 x}{ds^2} \cos(sp\pi)\, ds = -p^2 \pi^2 a_p[/tex]

I get an answer in terms of the [itex]a_p[/itex] which I already know.

What if I want to find the Fourier transform of higher powers of x?

[tex] \int_0^1 x^2 \cos(sp\pi)\, ds =\, ? [/tex]

Can it be expressed in terms of, let's say, a power series in [itex]a_p[/itex]?

[tex] \int_0^1 x^2 \cos(sp\pi)\, ds =\, ?\, \sum_{n=0}^{\infty} c_n a_p^n[/tex]

And what if I want to find the Fourier transform of a generic functional f[x(s)] (smooth, differentiable, etc.)? Is it related somehow to the Fourier transform of the original function x(s)?

I was wondering if such a thing even exists, so here it goes... Let's say I have a function x(s) (it is real, smooth, differentiable, etc.) defined on (0,1). In addition, dx/ds = 0 on the boundary (s=0 and s=1). I can compute its Fourier transform (?) as

[tex] a_p = \int_0^1 x(s) \cos(sp\pi)\, ds [/tex]

and now I have a set of numbers [itex]a_p[/itex] which contain the same information as the original function x(s).

The good news is that if I compute the same Fourier transform on the derivatives of x, i.e.

[tex] \int_0^1 \frac{d^2 x}{ds^2} \cos(sp\pi)\, ds = -p^2 \pi^2 a_p[/tex]

I get an answer in terms of the [itex]a_p[/itex] which I already know.

**So here's my question:**What if I want to find the Fourier transform of higher powers of x?

[tex] \int_0^1 x^2 \cos(sp\pi)\, ds =\, ? [/tex]

Can it be expressed in terms of, let's say, a power series in [itex]a_p[/itex]?

[tex] \int_0^1 x^2 \cos(sp\pi)\, ds =\, ?\, \sum_{n=0}^{\infty} c_n a_p^n[/tex]

And what if I want to find the Fourier transform of a generic functional f[x(s)] (smooth, differentiable, etc.)? Is it related somehow to the Fourier transform of the original function x(s)?