Analytic functions characterized by their means on an arc

Authors:
Chin Hung Ching and Charles K. Chui

Journal:
Trans. Amer. Math. Soc. **184** (1973), 175-183

MSC:
Primary 30A72

MathSciNet review:
0338375

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is known that a function *f*, holomorphic in the open unit disc *U* with boundary data for some , is uniquely determined by its arithmetic means over equally spaced points on . By using different techniques, we weaken the hypothesis to functions with derivatives, . We also prove that a function is determined by its averages over an arc *K* if *f* is holomorphic in a neighborhood of , and that this result is false for some functions *f* in . On the other hand, we can capture a function from its means and shifted means on *K*.

**[1]**Chin Hung Ching and Charles K. Chui,*Uniqueness theorems determined by function values at the roots of unity*, J. Approximation Theory**9**(1973), 267–271. MR**0367211****[2]**Chin Hung Ching and Charles K. Chui,*Mean boundary value problems and Riemann series*, J. Approximation Theory**10**(1974), 324–336. MR**0382661****[3]**Chin Hung Ching and Charles K. Chui,*Recapturing a holomorphic function on an annulus from its mean boundary values*, Proc. Amer. Math. Soc.**41**(1973), 120–126. MR**0320326**, 10.1090/S0002-9939-1973-0320326-3**[4]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, Oxford, at the Clarendon Press, 1954. 3rd ed. MR**0067125****[5]**E. Landau,*Elementare Zahlentheorie*, Teubner, Leipzig, 1927; English transl., Chelsea, New York, 1958. MR**19**, 1159.**[6]**D. J. Patil,*Representation of 𝐻^{𝑝}-functions*, Bull. Amer. Math. Soc.**78**(1972), 617–620. MR**0298017**, 10.1090/S0002-9904-1972-13031-3**[7]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30A72

Retrieve articles in all journals with MSC: 30A72

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0338375-2

Keywords:
Arithmetic means on an arc,
Möbius function,
holomorphic functions,
uniqueness and nonuniqueness,
representation formulas

Article copyright:
© Copyright 1973
American Mathematical Society