We prove that, for every $n\ge1$, the degree-$n$ Bernstein operator on $[0,\pi]$ preserves positive-definiteness on the circle $S^1$. Equivalently, if a continuous function on $[0,\pi]$ defines a positive-definite isotropic kernel on $S^1$, then its Bernstein polynomial approximation of any fixed degree does as well. The proof reduces the problem to the nonnegativity of the cosine coefficients of the Bernstein images $Q_{n,m}=B_n[\cos(mx)]$, which we prove using an explicit coefficient formula and a two-regime positivity argument. We also discuss the higher-dimensional sphere analogue and show that the naive affine Bernstein operator fails to preserve the positive-definite cone already on $S^2$.
Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harm...
We show that if an upper semi-continuous function $\varphi$ on $\Omega\in\mathbb C^n$
satisfies some natural conditions, and for any non-negative psh function $\psi$ on
$\mathbb D\times]Omega$, the fiberwise Bergman kernel associated to $p_2^*\varphi+\psi$ is log-psh, then
$\varphi$ must be psh. This offers a version o...
Zhi Li, Wang Xu, Xiangyu Zhou· Известия Российской академии...· 0 citations
We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified for every $n\geq...
Dominique Guillot, Himanshu Gupta, P. K. Vishwakarma et al.· 0 citations
We study the cone $\mathcal{M}_{n,2d}$ of nonnegative mean polynomials---real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M_{q,p}(Y,w)$ with $q>p$. This cone simultaneously generalises the cone of sums of squares $\Sigma_{n,2d}$ and the cone of sums of nonnegative circuit polynomials...
Let $\omega$ be a concave modulus of continuity that is weaker than Lipschitz, meaning $\omega(t)/t$ diverges as $t$ approaches $0$. We construct a diffeomorphism of the circle with irrational rotation number, in the regularity class $C^{1+\omega}$, with a wandering interval. This construction implies that Denjoy's 193...
Let $X\subset{\mathbb P}^{n-1}$ be a hypersurface of degree $d\ge3$ with ordinary double points, where $n\ge3$. The roots of Bernstein-Sato polynomial of its defining polynomial $f$ are given up to sign by 1, $(n-1)/2$, and $j/d$ for $j\in{\mathbb Z}\cap[n,nd-n-p_f]$ with $p_f$ a positive integer. Here $p_f$ is bounded...
Seung-Jo Jung, Morihiko Saito· 0 citations
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