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Yes, indeed, Birkhoff-James orthogonality is an orthogonality in the sense of Rätz. A proof appears on p.36 of J. Rätz, On orthogonally additive mappings, Aequations Math. 28 (1985), 35-49. Th …

The equality is true, independently of $d$. By Theorem 2.1.3 of Ransford's book "Potential theory in the complex plane", since $u$ is bounded above on the compact set $K$, there is a decreasing seque …

So, I write this as an answer rather than a comment to close the question. This is problem VI.103 in volume 2 of Polya and Szego. For the interval $[-1,1]$, the extremal polynomial is $$\frac{P_{n}( …

Let $n\geq2$. Is it true that the minimax problem: $$ \min_{p\in\mathcal{P}}\max_{H\in\mathcal H}p^tH\bar{p}, $$ where $\mathcal H\subset\mathcal{M}(n)$ is a strictly convex bounded subset of hermitia …

Almost everywhere is too strong, but a Lusin-like theorem holds true : Let $u$ be a subharmonic function in a domain $D\subset\mathbb{R}^n$, $K\subset D$ a compact set, and $\epsilon>0$. There exist …

One possible reference among many many others... : H. Brezis and F. Browder, Partial differential equations in the 20th century, Advances in Mathematics 135 (1998), 76-144.

When $[a,b]=[-1,1]$, the $\inf$ on the right-hand side is attained (uniquely) by the monic Chebyshev polynomial $T_{n+1}$. It is well known that its roots belong to $[-1,1]$ and are simple. For a ge …

The paper, by C. Dellacherie himself, C. Dellacherie, Capacities and analytic sets, Cabal seminar 77-79, Proc., Caltech-UCLA logic Semin. 1977-79, Lect. Notes Math. 839, 1-31 (1981). covers, …

It is a classical result that the Lagrange interpolants to $f$ at the zeros of the orthogonal polynomials $q_n$ with respect to an arbitrary measure $\mu$ converge to $f$ in $L^2(\mu)$. For a finite …

The answer to the question is yes. The property even holds for any $0<p<\infty$. The first such result is due to Erdös and Turan (1936) : let $f$ be a continuous function and $w(x)$ a weight on $[-1 …

Yes, it follows from the following result of S. Bernstein (Quelques remarques sur l'interpolation, Math. Ann. 79 (1918), 1-12): For an arbitrary scheme $X=\cup_{n} A_{n}$ of points in $[-1,1]$, …

This question was recently studied in that paper: F. Charles, M. Campos-Pinto, B. Després, Algorithms for positive polynomial approximation, hal-01527763, assuming that the function $f$ is Lipschitz …

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set, and let $(x_{n})_{n\geq1}$ be a sequence of all points of $\Omega$ with rational coordinates. Consider the discrete measure of finite mass, $$\ …

Let $E$ be the (incomplete) subspace of sequences in $\ell^{2}(\mathbb{R})$ having at most finitely many nonzeros terms. In [1], a subset $S$ of $E$ is constructed (by a long induction argument), whic …

As you said, the inequality holds for the orthonormal polynomials $P_{k}$, $k\geq0$, of any positive measure $\mu$, with support $K$, a compact subset of $\mathbb{C}$. It follows from a basic result a …