Jackson's theorem for compact connected Lie groups (Q916956)

The following theorem is obtained. Th: Let G be a CCLG (compact connected Lie group), \(E=C(G)\), \(L^ p(G)\), \(1\leq p<\infty\), \(r\in \{1,2,..\}\). Then for each \(n\in Z^+\) there exists a central trigonometric polynomial \(K_ n\) of degree \(\leq n\), such that \(\| f-K_ n*f\|_ E\leq C_ r\omega_ r(n^{-1},f)\), where \(\omega_ r(t,f)\) is the r-th modulus of continuity of f in E. The case \(r=2\) was known before [see \textit{H. Johnen}, Linear Operators and Approximation, ISNM 20, 254-272 (1972; Zbl 0266.43005)]. As a result, combining this result with the sharp estimates of the Lebesgue constants of Fourier partial sums [see \textit{S. Giulini}, \textit{G. Travaglini}, J. Funct. Anal. 68, 106-116 (1986; Zbl 0664.22008)], it is shown that: Let G be a semisimple CCLG, \(E=C(G)\), \(L^ 1(G)\), and \(f\in E\) such that \(\omega_ r(t,f)=o(t^{(d-1)/2})\), \(t\to 0\), for some \(r>(d-1)/2\), where \(d=\dim (G)\). Then the spherical partial sum \(S_ nf\to f\) in E. This might be said to be a ``best possible'' criterion.