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 |
|---|---|
| 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
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The kechris–woodin rank is finer than the zalcwasser rank. / Ki, Haseo.
In: Transactions of the American Mathematical Society, Vol. 347, No. 11, 11.1995, p. 4471-4484.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 -