# Questions tagged [integral-kernel]

The tag has no usage guidance.

86 questions
Filter by
Sorted by
Tagged with
501 views

### Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$

My question seems elementary, yet I could not find the solution after working on and searching for several days... I'd like to find the eigenfunctions of a simple integral kernel: \begin{equation} \...
234 views

### How to find the inverse of a product of two integral equations

Problem I am trying to invert an equation of the form: $R(l_0)=(\int_{0}^{l_0} \rho(x) \, dx)(\int_{l_0}^{l} \rho(x) \, dx)$ where $0\leq l_0 \leq l$ I.e. I want to find $\rho(x)$ given $R(l_0)$ via ...
142 views

### Rate of convergence of Fejer kernel to the Dirac delta function

This seems like something one might find in a book so I would be grateful for any references you think may be helpful. I am interested in the rate at which of a function integrated against the $N$th ...
64 views

### Numerical methods for integral eigenvalue equation

I have an integral equation which is not exactly an eigenvalue type equation, but similar: $$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$ Here $\lambda$ can be thought of as an eigenvalue, so it is ...
1 vote
155 views

1 vote
50 views

1 vote
70 views

### Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls

Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
191 views

1 vote
98 views

### Is this a positive definite kernel?

Under which conditions on the function : \begin{array}{l|rcl} K : & \mathbb R^+ & \longrightarrow & (0, 1)\\ &t & \longmapsto & K(t) \end{array} is the symmetric ...
1 vote
75 views

67 views

### Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R}$ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
53 views

### Integral equation with kernel defined in a rectangle

Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$ Observe that the kernel is not defined on a square. My question: Can I apply the classical ...
Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation: $$f(t)\int_0^t a(s)\,ds + \int_0^t a(t - s) f(s) \, ds = 0$$ My question is : under what condition ...