# Tagged Questions

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### Find an harmonic function in $\mathbb R^n$ which is a polynomial of degree 4 and is 1 at the origin

Find an harmonic function in $R^n$ which It is a polynomial of degree 4 and is =1 at the origin. It is a polynomial of degree 5 and its partial derivatives are both =0 at the origin. Important ...
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### Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
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### Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
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### How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)=$The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu$; and ...
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### What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
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### Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
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### Extracting Harmonic series components

I have a number which is made up of a Harmonic series. 1/2 + 1/3 + 1/4 etc. Some of the components may not be in the number.. 1/2 + 1/7 + 1/11 etc. Is it possible to recover the individual ...
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### Fourier transform of $|x|^{-t}$

In $\mathbb{R}^d$, let $f(x)=|x|^{-t}$, its Fourier tranform $F(f)(ξ):=(2\pi)^{-\frac{d}{2}}∫_{\mathbb{R}^d} e^{ix\cdot ξ}f(x)dx$, is there any fast way to see that this integral converge at $ξ \neq0$ ...
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### Dirichlet Problem: Uniqueness of solution

Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$. Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$? ...
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### Dirichlet problem: Obtaining the harmonic measure through Riesz representation theorem

For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$ $$\Delta u=0, \text{ on } D, \\ \left. u\right|_{\partial D}=f \in C\left( \partial D\right).$$ With a fix $x$ in $D$, an ...
Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$. Is the Poisson integral of $f$,  Pf(x)=\int_{\partial D} f(t) ...