Questions tagged [singular-integrals]
Singular integrals are integral operators with kernel $K:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ that is not defined on the diagonal $x=y.$ This tag is for questions related to singular integrals and their applications.
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Why are kernels often singular on the diagonal?
Many kernels/integral operators are given in terms of a function that is singular near the origin:
For example, the heat kernel on $\mathbb{R}^d$:
$$
\operatorname{K}\left(t,x,y\right) =
\frac{1}{\...
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Is the Riemann Liouville fractional integral compact operator? [closed]
I am about to figured out that is the Riemann Liouville fractional integral compact operator or not? where f is continuous function in [0, b].
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Complex integral with fractional singularities
If one consider the complex value function
$$
f(z)=\frac{1}{\sqrt{z-1}\sqrt{z-2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$. Could someone please explain why
$$
2\int_1^2 f(x)dx=\oint f(...
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Convolution between $L^1$ function and a singular integral kernel
I meet a problem when reading Modern Fourier Analysis(3rd. Edition) written by L.Grafakos. On pg.82 he writes:
Fix $L\in\mathbb{Z}^+$. Suppose that $\{K_j(x)\}_{j=1}^L$ is a family of functions ...
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Limiting behavior of integral representation of $(\sqrt{\alpha^2-\partial_x^2}-\alpha)f(x)$
While studying pseudo-differential operators of type $\left(\sqrt{\alpha^{2} - \partial_{x}^{2}}-\alpha\right)\operatorname{f}\left(x\right)$, I came across the following integral representation of ...
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Elliptic integral singular expansion
The question. Consider the Elliptic Integral
$$
F(x;k)=\int_0^x \frac{dx}{\sqrt{(1-x^2)(1-k^2x^2)}}.\tag{1}\label{1}
$$
I am interested in the singular series expansion of $F(1;k)$ about $k=1$. I was ...
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A doubt in the proof of the spectrum of convolution operator
The following is a link showing the spectrum of convolution operator $A f(x)= \int_{-\pi}^{\pi} h(x-y) f(y) dy$ $L^2( {-\pi},{\pi})->L^2( {-\pi},{\pi})$ is range of $h$ where $h$ is continuous ...
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Solving the Poisson equation $-\Delta u=f$ on a domain $G$
Let $G$ be a domain and $f\in C_c^1(\mathbb{R}^n)$. We want to solve
$$-\Delta u =f \text{ in } G$$
without any boundary condition. Can we just take $\Delta^{-1}$ the inverse Laplacian on $\mathbb{R}^...
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Integral in terms of Hypergeometric function
Consider the integral $$ I = \int_0^1 \int_0^1 (1+c^2v^2)^{-s}u^{1-2s}(1-uv)^{s-1}dudv$$
where $c>0$ is some constant and $0<s<1$. Clearly the integral is absolutely integrable (Two ...
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An Estimate in Calderon Zygmund for Periodic Function
I am reading the following paper of Calderon and Zygmund http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm14122.pdf
On page 3 (252) they provide an estimate (2.3) which is following
$$ |K(x-x_{\nu}) - K(-...
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Measurability issue in the definition of maximal singular integral operator
I'm learning harmonic analysis with Grafakos' book Classical Fourier Analysis. I'm having a problem understanding the definition 5.2.1 of maximal singular integral operators.
Let me recall the ...
- 554
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Calderon Zygmund decomposition - Empty set
The statement of the Calderon-Zygmund decomposition as given in Duoandikoetxea's book is the following.
For $f$ integrable on $\mathbb{R}^n$ and non-negative and given $\lambda>0$, there exist a ...
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Oscillatory Integrals near the Riemann singularity
The question comes from E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, we concerned about the highly oscillatory distribution
$$
D(x)=\mathrm{p.v.} \...
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Evaluate 2D trigonometric integration of $\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\frac{\cos(x)^3}{\cos(x)-\cos(y)}dydx$
I have the following example of an integral:
$$ I = \frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi}\frac{\cos^3 x}{\cos x - \cos y} \ dy \ dx $$
I can determine the exact value of this integral by ...
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Existence of the Limit in Defining the Riesz Transform
Let
$$K(x) := \lim_{R \to \infty}{\int_{\mathbb{R}^d}{e^{2\pi i \xi \cdot x}\frac{\xi_j}{|\xi|}\phi(\xi/R)d\xi}}$$
Where $\phi$ is a $C^\infty_c(\mathbb{R}^d,[0,1])$ such that $\phi$ is $1$ when $|x| \...
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