Proving Uniqueness of Fourier Coefficients for Continuous Periodic Functions

In summary, if the Fourier coefficients of a continuous periodic function are all zero, then the function itself must be zero.
  • #1
Markov2
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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!
 
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  • #2
Markov said:
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?
 
  • #3
Markov said:
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 $f=0.$
 

FAQ: Proving Uniqueness of Fourier Coefficients for Continuous Periodic Functions

What is the Fourier series and why is it used?

The Fourier series is a mathematical tool used to represent a periodic function as a sum of sinusoidal functions. It is commonly used in signal processing, physics, and engineering to analyze and manipulate periodic functions.

How do you prove the uniqueness of Fourier coefficients for continuous periodic functions?

To prove the uniqueness of Fourier coefficients, we use the Fourier inversion theorem which states that if a function is both continuous and periodic, its Fourier series converges to the original function. This means that the coefficients of the Fourier series must be unique for a given continuous periodic function.

What is the significance of proving the uniqueness of Fourier coefficients?

Proving the uniqueness of Fourier coefficients is important because it guarantees that the Fourier series accurately represents the original function. This allows us to use the Fourier series for various applications such as signal processing and solving differential equations.

What conditions must be met for the Fourier series to have unique coefficients?

The Fourier series has unique coefficients for a continuous periodic function if the function is square integrable and has a finite number of discontinuities in one period. Additionally, the function must have a finite number of maxima and minima in any finite interval within one period.

Are there any alternative methods for proving the uniqueness of Fourier coefficients?

Yes, there are alternative methods such as the Dirichlet conditions, which state that a function must be absolutely integrable and have a finite number of extrema in one period for the Fourier series to have unique coefficients. Another method is the Riemann-Lebesgue lemma, which states that the Fourier coefficients of a square integrable function tend to zero as the frequency increases.

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