Hilbert points in Hardy spaces

A Hilbert point in H p ( T d stretchy="false">) H^p(\mathbb {T}^d) , for alttext="d greater-than-or-equal-to 1"> ≥ 1 encoding="application/x-tex">d\geq 1 and alttext="1 less-than-or-equal-to normal infinity"> ≤ encoding="application/x-tex">1\leq \leq \infty is a nontrivial function alttext="phi"> φ<!-- φ encoding="application/x-tex">\varphi such that alttext="double-vertical-bar phi double-vertical-bar Subscript Sub right-parenthesis plus f fence="false" stretchy="false">‖<!-- ‖ <mml:msub> + f encoding="application/x-tex">\| \varphi \|_{H^p(\mathbb {T}^d)} \|\varphi + f\|_{H^p(\mathbb {T}^d)} whenever alttext="f"> encoding="application/x-tex">f orthogonal to the usual L squared"> L 2 encoding="application/x-tex">L^2 sense. When alttext="p not-equals 2"> ≠<!-- ≠ encoding="application/x-tex">p\neq 2 {T}) if only nonzero multiple of an inner function. An on alttext="double-struck d"> encoding="application/x-tex">\mathbb {T}^d any spaces but there are other points as well when . The case alttext="1"> encoding="application/x-tex">1 -homogeneous polynomials studied depth and, byproduct, new proof given sharp Khinchin inequality Steinhaus variables range alttext="2 greater-than &gt; encoding="application/x-tex">2&gt;p&gt;\infty Briefly, dynamics certain nonlinear projection operator treated. This characterizes its fixed points. example exhibited cubed 3 {T}^3) equals 2 comma 4"> = , 4 encoding="application/x-tex">p=2, 4 not alttext="p"> encoding="application/x-tex">p ; this verified rigorously encoding="application/x-tex">p&gt;4 numerically p&gt;4