Convolution and Good Kernels
Definition 1.1 Let $f,g:\mathbb{R}\to\mathbb{C}$ be $2\pi$-periodic functions. The convolution $f*g$ of $f$ and $g$ is the function defined by $[-\pi, \pi]$ ...
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Introduction to Fourier Series
Basic Knowledge We focus on a class of functions
Moore-Penrose Pseudoinverse
Definition Let $V$ and $W$ be finite dimensional inner product spaces over the same field $F$ and let $T:V\to W$ be a linear transformation. Let $L:\ker T^\p...
Lebesgue Integration
Proposition 8.2.6 Let $\Omega$ be a measurable set, and let $f: \Omega\rightarrow [0,\infty]$ and $g: \Omega\rightarrow [0,\infty]$ be non-negative measurabl...
Dimension Theorem and Rank Theorem
Definition Let $A\in \mathfrak{M}_{m\times n}(\mathbb{R})$ be a matrix. We define the column rank of $A$ as dimension of $\langle [A]^1, \ldots, [A]^n \rangl...