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stripes

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## Homework Statement

Third question of the day because this assignment is driving me crazy:

Suppose that [itex]\left\{ f_{k} \right\} ^{k=1}_{\infty}[/itex] is a sequence of Riemann integrable functions on the interval [0, 1] such that

[itex]\int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow \infty[/itex].

Show that [itex]\hat{f} _{k} (n) \rightarrow \hat{f} (n) [/itex] uniformly in n as [itex]k \rightarrow \infty[/itex]

## Homework Equations

## The Attempt at a Solution

I can't seem to do this rigorously. I can only approach it intuitively. Since the integral of the

**absolute value**tends to zero, I want to say that [itex]f_{k}(x) \rightarrow f(x)[/itex]. But I'm not sure how to show that. If [itex]f_{k}(x) \rightarrow f(x)[/itex] is indeed true, then is it not trivial that [itex]\hat{f} _{k} (n) \rightarrow \hat{f} (n) [/itex]? Furthermore, how would I show the convergence is uniform? Do I just have to use the epsilon definition? I also want to say for all epsilon greater than zero, there exists f

_{k}(x) such that [itex]|f_{k}(x) - f(x)| < \epsilon[/itex] since f

_{k}converges to f. But I need to first show that f

_{k}converges to f, using the fact that the integral of the absolute value of the difference between the two converges to zero! I am piecing it together but I don't know how to write it down in the form of an answer. Thanks in advance.

Edit: f hat is the Fourier coefficient, I guess in complex form.