(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

show that A_{k}will satisfy:

[tex]\left| A_{k} \right| \leq Mk^{-4} [/tex]

2. Relevant equations

[tex] A_{k} = \frac{2}{L}\int^{L}_{0} \phi(x) sin \left( \frac{k \pi x}{L} \right) dx [/tex]

given

[tex] \phi(x) \in C^{4} ([0,L]) \and\ \phi^{(p)}(0) = \phi^{(p)}(L) = 0, p = 0, 1, 2, 3 [/tex]

in this notation, \phi is continuous and differentiable up to 4th derivative over [0,L], and (p) denotes the order of the derivative.

3. The attempt at a solution

I differentiated 4 times using substitution, and came up with

[tex] \frac{L^{4}}{k^{4}\pi^{4}} \int^{L}_{0} \phi^{(4)}(x)sin \left( \frac{k \pi x}{L}\right) dx [/tex].

I was told using the Weierstrass M-test (or Comparison test?) would help find solution, but I'm not sure how to proceed.

Any comments or suggestions?

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# Finding a bound for a Fourier coefficient

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