Fourier Series

Definition

Continuous Fourier Series

Sine-Cosine Form

$f(x)\sim A_0+\sum _{n=1}^{\infty }\left(A_n\cos (nx\frac{\pi }{L})+B_n\sin (nx\frac{\pi }{L})\right)$ where $L$ is half of the period of the $f(x)\in C_p(-L,L)$¹, $A_0=\frac{1}{2L}{\int }_{-L}^{L}f(x)dx$, $A_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\cos (nx\frac{\pi }{L})dx$, and $B_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\sin (nx\frac{\pi }{L})dx$

Exponential Form

$f(x)\sim \sum _{n=-\infty }^{\infty }\left(c_n{e}^{inx\frac{\pi }{L}}\right)$ where $L$ is half of the period of the $f(x)\in C_p(-L,L)$¹, $c_n=\frac{1}{L}{\int }_{-L}^{L}f(x)e^{-inx\frac{\pi }{L}}dx$

Discrete Fourier Series

$x[n]=\sum _{k}X[k]e^{i2\pi \frac{k}{N}n},n\in \mathbb{Z}$ where $\frac{1}{N}$ is a fundamental frequency, and for some positive $N$, the practical range of $k$ is $[0,N-1]$, and $X[k]=\frac{1}{N}\sum _{k}x[n]e^{ik\frac{2\pi }{N}n}$

Facts

Suppose that

• $f$ is continuous on $(-\pi ,\pi )$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then ${\sum }_{n=1}^{\infty }\sqrt{B_n^2+B_n^2}$ converges where $A_n,B_n$ are the coefficients of Fourier series It is stronger than Bessel’s Inequality

Suppose that

• $f$ is continuous on $[-\pi ,\pi ]$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then the Fourier series converges uniformly and absolutely to $f(x)$

Differentiation of Fourier series

Suppose that

• $f$ is continuous on $[-\pi ,\pi ]$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then the Fourier series is differentiable at $(-\pi <x<\pi )$ and if $f^{\prime \prime }(x)$ exist $f^{\prime }(x)=\sum _{n=1}^{\infty }(-n{A}_n\sin (nx)+n{B}_n\cos (nx))$

Integration of Fourier series

Suppose that

• $f$ is continuous on $(-\pi ,\pi )$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then the Fourier series representation for $f(x)$ on $(-\pi ,\pi )$ $F(x)={\int }_{-\pi }^{\pi }f(t)dt-\frac{A_0}{2}x$ continuous for $(-\pi <x<\pi )$ and
$F^{\prime }(x)=f(x)$ if $F$ is differentiable at $x$ We obtain that $F$ is continuous on $(-\pi <x<\pi )$ and $F^{\prime }\in C_p(-\pi ,\pi )$¹

Footnotes

1. Piecewise Continuous ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷