Finite Frames

  Let us look at linear functionals briefly, these are the foundation of frame theory... DEFINITION:  Let $\mathcal{V}$ be a finite-dimensional vector space. A linear functional is linear transformation $\varphi :\mathcal{V}⟶\mathbb{F}$, that is a scalar-valued function that takes as input a vector. Now let us take a look at the Riesz Representation Theorem, this will require some preliminaries... First, let $A=[a_{ij}]$ be an $m\times n$ matrix over the field $\mathbb{F}$, the ADJOINT matrix $A^{\ast }$ of $A$ is the matrix $B=[b_{ij}]\in {\mathbb{F}}^{n\times m}$ such that $b_{ij}=\overline{a_{ji}}\mathrm{\forall }i,j\in \mathbb{N}$. Simply put, the adjoint is the conjugate transpose of the matrix... PROPOSITION  Let $A,B$ be two matrices over the field $\mathbb{F}$, and let $\alpha ,\beta \in \mathbb{F}$. Assume that $A,B\in {\mathbb{F}}^{m\times n}$, that is the same sizes. Then we have the following properties... i) $(\alpha A+\beta B{)}^{\ast }=\bar{a}{A}^{\ast }+\bar{b}{B}^{\ast }$ ii...

Okay here is an attempt to writing an equation... I used an exchange post to modify the html Example 1: $x^2$ Example 2: $$x^2$$  This was the link I used:  https://support.google.com/blogger/thread/234344761/inserting-latex-equations?hl=en DEFINITION (FRAME)  Let $V$ be an inner product space $\{e_k{\}}_{k\in \mathbb{N}}$ be a set of vectors in $V$. The set $\{e_k{\}}_{k\in \mathbb{N}}$ is a frame  of $V$ if it satisfies the frame condition . This is, if there exists constants $A,B$ such that $0<A\le B<\mathrm{\infty }$ where for each $v\in V$ we have   $$A\Vert v{\Vert }^2\le \sum _{k\in \mathbb{N}}|⟨v,e_k⟩{|}^2\le B\Vert v{\Vert }^2\mathrm{\forall }v\in V$$ where $A,B$ are the frame bounds. This is the other definition I used from the book "Frames for undergraduates"... DEFINITION (FRAME)  A frame for a Hilbert space $\mathcal{H}$ is a sequence of vectors $\{x_i\}\subset \mathcal{H}$ for which there exist constant $0<A\le B<\mathrm{\infty }$ such that for ...

This is the first entry of the Finite Frames blog. This blog will be dedicated to showing my mathematical analysis of finite frames...more to be added.. Below is a great reference book that I will be following frequently, there are other books that I will also mention for reference...