A Quantitative Version of the Young Test for the Convergence of Conjugate Series

✍ Scribed by F. Moricz

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 196 KB
• Volume: 81
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1995.1045

✦ Synopsis

The classical Young test says that if (f) is a (2 \pi)-periodic function of bounded variation on ([-\pi, \pi]), then the conjugate series to the Fourier series of (f) converges at (x) if and only if the conjugate function (f) exists at (x). Our main goal is to give estimates of the rate of this convergence in terms of the oscillation of (\psi_{x}(t):=) (f(x+t)-f(x-t)) over appropriate subintervals. In particular, we obtain a conjugate version of the well-known Dini-Lipschitz test. As a byproduct, we obtain the rate of convergence in (L^{1})-norm. 1995 Academic Press. Inc.