The Kechris-Woodin rank is finer than the Zalcwasser rank

Abstract

For each differentiable function <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> on the unit circle, the Kechris-Woodin rank measures the failure of continuity of the derivative function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f prime"> <mml:semantics> <mml:msup> <mml:mi>f</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">f’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> while the Zalcwasser rank measures how close the Fourier series of <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> is to being a uniformly convergent series. We show that the Kechris-Woodin rank is finer than the Zalcwasser rank. Roughly speaking, small ranks mean the function is well behaved and big ranks imply bad behavior. For each countable ordinal, we explicitly construct a continuous function with everywhere convergent Fourier series such that the Zalcwasser rank of the function is bigger than the ordinal.

Locations

• Transactions of the American Mathematical Society - View - PDF