A boundary value problem for

Author:
Rotraut Goubau Cahill

Journal:
Proc. Amer. Math. Soc. **67** (1977), 241-247

MSC:
Primary 30A72; Secondary 26A24

DOI:
https://doi.org/10.1090/S0002-9939-1977-0457728-4

MathSciNet review:
0457728

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Abstract: Let be an subset of the unit circle of measure 0 and let , be a decreasing sequence with and . There exists an *H* in of norm whose modulus has radial limit along every radius which has radial limit of modulus on and on . If *W* is simultaneously a set, *H* may be chosen to have no zeros on *C*. It follows that for *W* countable, say , there is such an *H* of norm 1 for which . The proof of the theorem depends on the existence of a special collection of closed sets , real, for which the function *h*, defined by , is such that the function du has the required properties. Some of the techniques used are similar to those developed in an earlier paper [**1**].

**[1]**R. Cahill,*On bounded functions satisfying averaging conditions*, Trans. Amer. Math. Soc.**206**(1975), 163-174. MR**0367208 (51:3450)****[2]**Z. Zahorski,*Über die Menge der Punkte in welchen die Ableitung unendlich ist*, Tôhoku Math. J.**48**(1941), 321-330. MR**0027825 (10:359h)**

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0457728-4

Keywords:
Zahorski,
,
radial limit

Article copyright:
© Copyright 1977
American Mathematical Society