Point evaluation in Paley–Wiener spaces

Abstract We study the norm of point evaluation at the origin in the Paley–Wiener space PW p for 0 &lt; p &lt; ∞, i.e., we search for the smallest positive constant C , called $${\mathscr{C}}_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> , such that the inequality $$\vert f(0)\vert ^{p}\leq C \Vert f\Vert _{p}^{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>|</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> <mml:mo>∥</mml:mo> <mml:mi>f</mml:mi> <mml:msubsup> <mml:mo>∥</mml:mo> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> holds for every f in PW p . We present evidence and prove several results supporting the following monotonicity conjecture: The function $$p \mapsto {\mathscr{C}}_{p}/p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> <mml:mo>↦</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:math> is strictly decreasing on the half-line (0, ∞). Our main result implies that $${\mathscr{C}}_{p} &lt; p/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:math> for 2 &lt; p &lt; ∞, and we verify numerically that $${\mathscr{C}}_{p} &gt; p/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:math> for 1 ≤ p &lt; 2. We also estimate the asymptotic behavior of $${\mathscr{C}}_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> as p → ∞ and as p → 0 + . Our approach is based on expressing $${\mathscr{C}}_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> as the solution of an extremal problem. Extremal functions exist for all 0 &lt; p &lt; ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p &lt; ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 &lt; p &lt; ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to $${\mathscr{C}}_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> and the extremal functions.