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Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in ...

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb R}...

Let $k :[0,1]^2 \rightarrow \bf{R} $ be a kernel function definded by $ k(x,y)= (1- max(x,y))^2 .$ Now, let $ L $ be a linear operator defined on $ L^2 [0,1] $ by $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...

Disclaimer. Please, be patient, I'm here to learn functional analysis... Let $X$ be the unit sphere in $\mathbb R^n$ and let $\sigma$ be the uniform measure on $X$. Consider a positive definite ...

Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by $$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$ What kind of assumption might I impose on $K$ such that $\lambda=1$ will be ...