Subdifferential Analysis of the Van der Waerden Function
Abstract
<jats:p> A concise and direct proof is given that Hölder subdifferentials of the (continuous but nowhere differentiable) Van der Waerden function <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">H(\cdot)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> exhibits the same behaviour as the Weierstrass function: There exists a countable dense set <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Gamma \subset R</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> (the dyadic rationals) such that each Hölder subdifferential <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\partial_\alpha H(x)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is all of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb R</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> for every <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">x\in\Gamma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , while <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\partial_\alpha H(x)=\emptyset</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∅</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">x\notin \Gamma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo mathvariant="normal">∉</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . </jats:p>