# Questions tagged [integral-kernel]

The tag has no usage guidance.

60 questions
Filter by
Sorted by
Tagged with
87 views

120 views

### Injectivity of an integral operator

Consider the operator $$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$ with $k\in L^2((0,1)\times(0,1)).$ I want to know under what assumption the kernel is reduced to zero. i....
85 views

36 views

### Existence of continuous integral kernel

Let $(X_i,\mathcal{F}_i,\mu_i)$, $i\in \{1,2\}$, be a $\sigma$-finite measure space and $E_i$ an ideal of measurable functions on $X_i$ with full carrier (for example $E_i=L^p(X_i,\mu_i)$). A ...
311 views

### Eigenfunctions and eigenvalues of an operator defined by a certain integral

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 $$Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
55 views

96 views

### Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!)

Consider the following Volterra integral equation $g(t)=∫_0^tK_n(t,s)w_n(s)ds$ where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)≥K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to ...
I have the following problem at hand. Define the kernel $$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$ Now, if \$R(y_1,...