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Version 10 |
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Let $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ be the $n$-dimensional torus, let $\{\phi_k(x)\}_{k\in\mathbb{Z}^n}$ be an orthornormal basis for $L^2(\mathbb{T}^n)$, and suppose that $f(x)\in L^2(\mathbb{T}^n)$.
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Let $\mathbb{T}^n\subset\mathbb{R}^n$ be the $n$-dimensional torus, let $\{\phi_k(x)\}_{k\in\mathbb{Z}^n}$ be an orthornormal basis for $L^2(\mathbb{T}^n)$, and suppose that $f(x)\in L^2(\mathbb{T}^n)$.
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| We can expand $f$ as a Fourier series |
We can expand $f$ as a Fourier series |
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| \begin{align*} |
\begin{align*} |
| \sum_{k\in\mathbb{Z}^n}\hat{f}(k)\phi_k, |
\sum_{k\in\mathbb{Z}^n}\hat{f}(k)\phi_k, |
| \end{align*} |
\end{align*} |
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| and we call the numbers $\hat{f}(k)$ the \emph{Fourier coefficients} of $f$ with respect to the given basis. In particular, the Fourier series for $f$ converges to $f$ in the $L^2$ norm. |
and we call the numbers $\hat{f}(k)$ the \emph{Fourier coefficients} of $f$ with respect to the given basis. In particular, the Fourier series for $f$ converges to $f$ in the $L^2$ norm. |
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| The most basic incarnation of this is finding the Fourier coefficients of a Riemann integrable function with respect to the orthonormal basis given by the trigonometric functions: |
The most basic incarnation of this is finding the Fourier coefficients of a Riemann integrable function with respect to the orthonormal basis given by the trigonometric functions: |
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| Let $f$ be a Riemann integrable function from $[-\pi,\pi]$ to $\mathbb{R}$. Then the numbers |
Let $f$ be a Riemann integrable function from $[-\pi,\pi]$ to $\mathbb{R}$. Then the numbers |
| $$a_0 = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx,$$ |
$$a_0 = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx,$$ |
| $$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx,$$ |
$$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx,$$ |
| $$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx$$ |
$$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx$$ |
| are called the Fourier coefficients of the function $f.$ |
are called the Fourier coefficients of the function $f.$ |
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| The trigonometric series |
The trigonometric series |
| $$ a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx))$$ is called the trigonometric series of the function $f$, or Fourier series of the function $f.$ |
$$ a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx))$$ is called the trigonometric series of the function $f$, or Fourier series of the function $f.$ |
