graphon shift operator

[[concept]]

The graphon shift operator

Let $W$ be a graphon and $x$ a graphon signal. The graphon shift operator is the integral linear operator $T_W:L^2([0,1])\to L^2([0,1])$ which maps $x\mapsto T_{W}x$ $T_{W}x={\int }_0^{1}W(u,\cdot )x(u)dx$

NOTE

A graphon shift operator is a Hilbert-Schmidt integral operator (continuous and compact) because graphons are bounded $⟹W\in L^{\infty }⟹W\in L^2$ And has Hilbert-Schmidt norm $||T_W|{|}_{HS}^2=||W|{|}_2^2={\int }_0^1{\int }_0^1{W}^2(u,v)dudv$

Review

dsg

A graphon shift operator is a {==ass||Hilbert-Schmidt integral operator||type}(ie it is {hah||continuous and compact}) because {hha||graphons are bounded on closed subset of ${\mathbb{R}}^2$||why type==}

Graphon shift operators are also {==as||self-adjoint||special property} because graphons are {ha||symmetric||why==}.

Mentions

Mentions

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