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f is null

Markov

Member
Feb 1, 2012
149
Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$

I know this has to do with the uniqueness of the Fourier coefficients, but I don't know how to solve it.
Thanks!
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$

I know this has to do with the uniqueness of the Fourier coefficients, but I don't know how to solve it.
Thanks!
Sounds an awful lot like the Riemann-Lebesgue Lemma. Are the tools of Lebesgue integration available to you?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$

I know this has to do with the uniqueness of the Fourier coefficients, but I don't know how to solve it.
Thanks!
This requires some fairly heavy machinery. One method is to use Fejér's theorem, which says that $f$ is the uniform limit of the Cesàro sums $s_n(f)$ of its Fourier series. If all the Fourier coefficients of $f$ are zero then $s_n(f)=0$ for all $n$, and hence $f=0.$