# Finding a bound for a Fourier coefficient

## Homework Statement

show that Ak will satisfy:

$$\left| A_{k} \right| \leq Mk^{-4}$$

## Homework Equations

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

given

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

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

## The Attempt at a Solution

I differentiated 4 times using substitution, and came up with

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

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

Office_Shredder
Staff Emeritus
$$\frac{L^{4}}{k^{4}\pi^{4}} \left| \int^{L}_{0} \phi^{(4)}(x)sin \left(\frac{k \pi x}{L} \right) dx \right| \leq M$$