Dimension-free discretizations of the uniform norm by small product sets

Abstract Let $f$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> be an analytic polynomial of degree at most $K-1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:math> . A classical inequality of Bernstein compares the supremum norm of $f$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> over the unit circle to its supremum norm over the sampling set of the $K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -th roots of unity. Many extensions of this inequality exist, often understood under the umbrella of Marcinkiewicz–Zygmund-type inequalities for $L^{p},1\le p\leq \infty $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:math> norms. We study dimension-free extensions of these discretization inequalities in the high-dimension regime, where existing results construct sampling sets with cardinality growing with the total degree of the polynomial. In this work we show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of $\deg (f)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>deg</mml:mo> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>)</mml:mo> </mml:math> and is instead governed by the maximum individual degree of $f$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> ; i.e. , the largest degree of $f$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> when viewed as a univariate polynomial in any coordinate. For example, we find that for $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -variate analytic polynomials $f$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> of degree at most $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> and individual degree at most $K-1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:math> , $\|f\|_{L^{\infty }(\mathbf{D}^{n})}\leq C(X)^{d}\|f\|_{L^{\infty }(X^{n})}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> <mml:mi>f</mml:mi> <mml:mo>∥</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> <mml:mi>f</mml:mi> <mml:mo>∥</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> </mml:math> for any fixed $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> in the unit disc $\mathbf{D}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> with $|X|=K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:mi>X</mml:mi> <mml:mo>|</mml:mo> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> </mml:math> . The dependence on $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> in the constant is tight for such small sampling sets, which arise naturally for example when studying polynomials of bounded degree coming from functions on products of cyclic groups. As an application we obtain a proof of the cyclic group Bohnenblust–Hille inequality with an explicit constant $\mathcal{O}(\log K)^{2d}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup>