On the invariance of certain classes of functions under composition

Abstract

Certain classes of functions are mapped into themselves by any change of variable. For some classes of this type which are of interest in the study of Fourier series, it is shown that the necessary and sufficient condition that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g ring f"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo> ∘ </mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">g \circ f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be in the class for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of that class is that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g element-of upper L i p 1"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>Lip</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">g \in \operatorname {Lip} 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .