Fourier and Wavelet Transforms

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2.1 Fourier Series and Fourier Transforms

\displaystyle{(1)\quad <f(x),g(x)>=\int_a^b f(x)\bar{g}(x)dx }

\displaystyle{(2)\quad {\bf f}=\left[\begin{array}{c} f_(x_1)\\ \vdots\\ f_(x_n)\\ \end{array}\right],\ {\bf g}=\left[\begin{array}{c} g_(x_1)\\ \vdots\\ g_(x_n)\\ \end{array}\right] \Rightarrow <\bf f},{\bf g}>=\sum_{k=1}^{n}f(x_k)\bar{g}(x_k) }

\displaystyle{(2)\quad ||f||_2=\sqrt{<f(x),f(x)>}=\int_a^b f(x)\bar{f}(x)dx }

Fourier Series
●f(x):\pi-periodic
\displaystyle{(3)\quad f(x)=\frac{a_0}{2}+\sum_{k=1}^\infty(a_k\cos(kx)+b_k\sin(kx)) }

\displaystyle{(4a)\quad a_k=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(kx)dx=\frac{1}{||\cos(kx)||^2}<f(x),\cos(kx)> }

\displaystyle{(4b)\quad b_k=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(kx)dx=\frac{1}{||\sin(kx)||^2}<f(x),\sin(kx)> }

2.2 Discrete Fourier Transform (DFT) and Fast Fourier Transform FFT)

2.3 Transforming Partial Differential Equations

2.4 Gabor Transform and the Spectrogram

2.5 Laplace Transform

2.6 Wavelets and Multi-Resolution Analysis

2.7 Two-Dimensional Transforms and Image Processing