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To get polynomials $P_i(x,y)$ such that $P_i(z_j)=\delta_{ij}$, you can write $$P_i(x,y)=\frac{\prod\limits_{j:j\ne i}[(x-x_j)^2+(y-y_j)^2]}{\prod\limits_{j:j\ne i}[(x_i-x_j)^2+(y_i-y_j)^2]}.$$ So, polynomials of minimal degree with such an interpolation property do exist.


$\newcommand\R{\mathbb R}\newcommand{\Ga}{\Gamma}$The integral in question is \begin{equation*} \begin{aligned} \int_0^1 dr\,\int_{rS_{n-1}}dx\,f(x) &=\int_0^1 dr\,r^{2m+n-1}\int_{S_{n-1}}du\,f(u) \\ &=\frac1{2m+n}\,|S_{n-1}|Ef(U_n), \end{aligned} \end{equation*} where \begin{equation*} f(x):=\Big(\sum_{j=1}^n a_j x_j^2\Big)^m \end{...


Let me separate the radial integration from the angular integration, $$\int_{|\mathbf{x}|\leq 1}f(\mathbf{x})d\mathbf{x}=\frac{2\pi^{n/2}}{\Gamma(n/2)}\int_0^1 r^{n-1}\bar{f}(r)\,dr,$$ where $\bar{f}(r)$ is the average of $f$ over the surface of the $n$-dimensional hypersphere of radius $r$. In our case $$f(\mathbf{r})=\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^...


This was carried out in Koppelman, W.; Pincus, J. D., Spectral representations for finite Hilbert transformations, Math. Z. 71, 399-407 (1959). ZBL0085.31701. The spectrum is purely continuous on $[-\pi,\pi]$ (with the OP's normalisation) and Theorem 3.1 gives the explicit Fourier-Plancherel theorem that diagonalises this operator (see Theorem 3.2). Note ...

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