### Abstract

For each differentiable function f on the unit circle, the Kechris Woodin rank measures the failure of continuity of the derivative function f' while the Zalcwasser rank measures how close the Fourier series of f 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.

Original language | English |
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Pages (from-to) | 4471-4484 |

Number of pages | 14 |

Journal | Transactions of the American Mathematical Society |

Volume | 347 |

Issue number | 11 |

DOIs | |

Publication status | Published - 1995 Nov |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Transactions of the American Mathematical Society*, vol. 347, no. 11, pp. 4471-4484. https://doi.org/10.1090/S0002-9947-1995-1321581-2

**The kechris–woodin rank is finer than the zalcwasser rank.** / Ki, Haseo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The kechris–woodin rank is finer than the zalcwasser rank

AU - Ki, Haseo

PY - 1995/11

Y1 - 1995/11

N2 - For each differentiable function f on the unit circle, the Kechris Woodin rank measures the failure of continuity of the derivative function f' while the Zalcwasser rank measures how close the Fourier series of f 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.

AB - For each differentiable function f on the unit circle, the Kechris Woodin rank measures the failure of continuity of the derivative function f' while the Zalcwasser rank measures how close the Fourier series of f 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.

UR - http://www.scopus.com/inward/record.url?scp=33646878852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646878852&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1995-1321581-2

DO - 10.1090/S0002-9947-1995-1321581-2

M3 - Article

AN - SCOPUS:33646878852

VL - 347

SP - 4471

EP - 4484

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -