Questions tagged [integral-operators]

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Understanding relation of 2 dependent, integral equations which are nested in a Bayesian Expectation

I'm trying hard to try understand the recursive nature between two equations in a recent macroeconomics paper, but my question mainly relates to how mathematically such recursive equations can depend ...
1 vote
0 answers
48 views

About Fourier integral operators

Consider the operator $$(A_\Delta\psi)(p) = \int_\Delta \int_{\mathbb{R}^n} e^{ix\cdot(p-k) + i\phi(x,p,k)} a(x,p,k) \psi(k) dk\: dx$$ where $\Delta$ is a Borel or Lebegue set in $\mathbb{R}^n$, in ...
0 votes
1 answer
123 views

Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$

I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$ I've tried Mathematica, but it does not converge to a solution....
1 vote
0 answers
23 views

Characterization of the Picard's condition for integral equation

Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
2 votes
0 answers
103 views

Existence of solutions to n-dimensional integral equation with solutions into [0,1]

I have a research problem I am working on where a step involves proving the existence of solutions to a certain kind of integral equation. A statement of this problem is as below. I would appreciate ...
0 votes
2 answers
4k views

Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...
2 votes
0 answers
61 views

Derivative of a functional involving integral and level set

Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional $$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$ where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
3 votes
1 answer
268 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 ...
0 votes
0 answers
143 views

Why is this function in $L^1$?

I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
3 votes
0 answers
116 views

Reference request: trace norm estimate

In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$ then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
0 votes
1 answer
127 views

When integrating by part produces a singularity

I'm currently interesting in the following operator: $$O[f](x):=f(x)-2xe^{x^2}\int_x^{+\infty}dt \, e^{-t^2} f(t)$$ for all $x\in\mathbb{R}$ and $f$ smooth and decaying at infinity fast enough with ...
2 votes
1 answer
426 views

Structure of the inverse of a Fredholm integral operator of the second kind

NOTE: Cross-posted on Mathematics Stack Exchange I am trying to solve an equation of the form $$ (\mathbb{I} + K)\phi = f $$ where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\...
0 votes
0 answers
68 views

$ \int_{\mathbb{R}^n} f R_1 f=0$ if $f\in L^2$ Riesz transform

How can I see that? It seems that it has to do with the adjoint of the Riesz transform $R_1^*=-R_1$, but here we do not have the complex $L^2$ scalar product.
2 votes
3 answers
386 views

Delta-distribution composed with a function from the Fourier representation

A well known representation of Dirac's delta-distribution is via the Fourier transform of distributions: \begin{equation} \delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{...
1 vote
0 answers
109 views

When is the solution to a Fredholm integral equation a PDF?

I have two questions about inhomogenous Fredholm integral equations of the first kind: $$f(x) = \int_a^b K(x,t) g(t) dt$$ where $f, K$ are known and $g$ is not. If a unique solution for $g$ exists, ...
5 votes
1 answer
218 views

An inequality for polynomials

I have been thinking about the validity of the following inequality: if $P(z)=\sum_{k=0}^na_kz^k, a_n\neq 0$ and $P(z)$ is non-zero in $|z|<1, $ then for $\theta \in [0, 2\pi],$ and $p>0$ \...
3 votes
0 answers
139 views

Extrapolated Integral operator (compactness)

I am studying the compactness of some convolution operators. Let the convolution with extrapolation $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a ...
4 votes
1 answer
218 views

Integral operator (compactness)

I am studying the compactness of some convolution operators. Let the convolution $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a $C_0$-semigroup on some ...
2 votes
0 answers
58 views

Translation request: Boundedness of Cauchy integral on Lipschitz boundary

The reference: "L'intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes" (https://annals.math.princeton.edu/1982/116-2/p04) is written in French. Can we ...
8 votes
2 answers
788 views

Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by $$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$ Is it possible to decompose this into two separate "...
4 votes
0 answers
149 views

Roots of smoothing operators

Suppose that $(M,g)$ is a smooth, compact Riemann manifold and $K:M\times M\to\mathbb{R}$ is a smooth, symmetric nonnegative function. We regard is as the Schwartz kernel of a smoothing operator. In ...
0 votes
0 answers
77 views

Integral over $S^{n-1}$ [duplicate]

What is the values of the following integral: $$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw.$$ where $\lambda\in\Bbb R, i^2=-1,x\in\Bbb R^n;<,>$ the inner product scalar on $\Bbb R^n$ ...
4 votes
1 answer
310 views

Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras. Suppose that I want to define a direct integral on either Banach spaces or ...
1 vote
0 answers
58 views

Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$

Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
0 votes
1 answer
140 views

What are the steps involved in the solution to $\int{x^{-a} (b -cx^{-d})^e }dx$? [closed]

Mathematica gives me the following solution to $\int{x^{-a} (b -cx^{-d})^e }$: $$\int{x^{-a} (b -cx^{-d})^e dx} = -\frac{b^{e}x^{1-a} \, _2F_1\left(\frac{a-1}{d},-e;\frac{a+d-1}{d};\frac{c x^{-d}}{b}\...
1 vote
1 answer
89 views

How to prove that $\int (1-z)^{u} z^{v} dz$ is equal to $\frac{z^{v+1}}{v+1}_2F_1(-u, v+1; v+2; z)$?

How to prove that $$\int (1-z)^{u} z^{v} dz = \frac{z^{v+1}}{v+1} \, _2F_1(-u, v+1; v+2; z)?$$
0 votes
1 answer
85 views

Solution or approximation to $\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx$?

I'm looking for a closed solution or an approximation to $$\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx,$$ where $a, b, c, d > 0$.
3 votes
0 answers
197 views

Extended adjoint of Volterra operator

Let $V$ be a Volterra operator on $L^2 [0,1]$. Does there exist a nonzero operator $X $ satisfying the following system $VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator? $$ V(f) (x) =\...
1 vote
1 answer
107 views

Solution to $\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx$

I'm looking for a solution to $$\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx.$$ Mathematica gives me the following solution, but I'd like to know/understand the steps involved in finding it. $$\int ...
2 votes
1 answer
264 views

Numerical methods for IDE [closed]

I would like to read a popular literature on the topic "Numerical methods for integro-differential equations". Could you recommend me any articles or book with a brief overview of some methods (maybe ...
1 vote
1 answer
177 views

Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?

I'm looking for a solution to the following integral. However, it seems it doesn't have a solution. $$\int\limits_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{\displaystyle1}{\...
1 vote
3 answers
212 views

Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$

Is there a solution to this integral? $$\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx,$$ where $a > 0$ and $d > 0$.
3 votes
1 answer
413 views

Hilbert-Schmidt integral operator with missing eigenfunctions

I'm having some issues with the spectral decomposition of the integral operator \begin{equation} (Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}. \end{equation} Since \begin{equation} ...
3 votes
0 answers
113 views

Schatten norm estimate of spatially truncated resolvent of Laplacian

Consider the operator $-\Delta$ on $L^2(\mathbb R^d)$. In my studies I stumbled upon operators of the form $$1_{\Gamma_n} (-\Delta -z)^{-1} 1_{\Gamma_m},$$ where $1_{\Gamma_m}$ denotes multiplication ...
2 votes
0 answers
131 views

Optimization of functionals with constraints

I have a minimization problem as follows: $\min\left( \int_0^1\int_0^1\beta(t)\beta(s)G_1(t, s)dtds\right)^{1/2}+\left( \int_0^1\int_0^1\beta(t)\beta(s)G_2(t, s)dtds\right)^{1/2} $ $\texttt{s.t.}\;\;\;...
2 votes
1 answer
621 views

General strategy for studying the decay of eigenvalues of kernel integral operators

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 ...
4 votes
1 answer
463 views

Calculation of an inverse Mellin transform

Let $z \in C$ and consider the following integral equation: $$-\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,b-a;z)}= \int_{0}^{+ \infty}x^{z-1}K(x) \mathrm{d}x$$...
1 vote
1 answer
166 views

Unique solution for 2$\times$2 Fredholm integral equations system

Consider the following system of Fredholm integral equations with constant kernel matrix $$ f(x)=K(x)\int_{0}^{1}f(s)ds $$ where $K(x)\in C([0,1];M_{2\times 2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \...
21 votes
5 answers
9k views

Can an integral equation always be rewritten as a differential equation?

Given an integral equation is there always a differential equation which has the same (say smooth) solutions? It seems like not but can one prove this in some example? Edit: Naively I'm hoping for ...
-1 votes
1 answer
58 views

Seek help to formalize an argument to positiveness of function defined inductively by integral [closed]

I have on domain $[0,\infty)$ a known and positive function $f(x)$ and two unknown functions $g(x), h(x)$ that start positive when $x=0$. I also know that if $h(x)$ is positive, then $g(x)$ is also ...
2 votes
0 answers
125 views

Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

I have a system of nonlinear Volterra integral equations of form $$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$ and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
0 votes
1 answer
275 views

Solution set of integral equation/ Kernel of linear operator

I am interested in understanding the solutions $\phi$ of the following integral equation: $$0=\int_0^1 \int_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy.$$ Equivalently, I am interested in ...
4 votes
1 answer
218 views

Nonlinear system of integral equations

I have encountered a system of nonlinear integral equations in my work. They take the form $$\int_{0}^{1} \frac{1}{g(y)}e^{f(x)/g(y)}(x+f(x)/g(y)-f(x))dy=0$$ $$\int_{0}^{1}\frac{f(x)}{g(y)^2} e^{f(x)/...
9 votes
1 answer
310 views

Cauchy path integral as a linear operator: kernel and image?

Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
2 votes
2 answers
573 views

Conditions for continuity of an integral functional

Let $X$ be a metric space, $\nu,\mu$ be Borel measures on $X$, $f:X\times \mathbb{R}\rightarrow [0,\infty)$ be a measurable function. Under what conditions is the integral functional $F_f$, defined ...
4 votes
1 answer
136 views

Boundedness of Riesz potential on Hardy space

I encounter the following claim in one paper: If $(-\Delta)^{\frac14}u\in L^{2,\infty}(\mathbb{R})$, then $u\in BMO(\mathbb{R})$. Equivalently in its dual version, if $u\in \mathcal{H}^1(\mathbb{R})$,...
3 votes
0 answers
96 views

Boundedness of Calderon-Zygmund type operator

I am trying to prove the following fact. Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define $$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$ where P.V. means ...
2 votes
0 answers
67 views

Unique continuation for integral operator

I accidentally met such question. Let's start from easy ones. Let $\Omega$ be an open convex domain in $\mathbb{R}^2$ and $u(x)$ satisfies that $$u(x) = \int_{\partial\Omega} \nabla_y G(|x-y|)\cdot ...
1 vote
0 answers
122 views

A Fredholm equation with non-separable kernel [closed]

I'm trying to solve this form of Fredholm equation: $$ g(v)=f_1(v)+\int\limits_{0}^{v_\mathrm{th}} g(v_s)\frac{e^{-\tfrac{\big[(v-v_\mathrm{init})-(1-v_\mathrm{leak})(v_s-v_\mathrm{init})\big]^2}{2v_\...
1 vote
0 answers
70 views

Convolution Integral Equation on a compact subset of the real line

I am dealing with the following equation: $$ f(x) = g(x) + \intop_{X} dt K(x-t)f(t) \;,\qquad \left\lbrace \begin{array}{c}f(x)>0\;,\;x\in X \\ f(x)<0\;,\;x\notin X \end{array}\right.$$ where $X$...