ON HOPF GALOIS HIRATA EXTENSIONS
Let H be a finite-dimensional Hopf algebra over a field K, H* the dual Hopf algebra of H, and B a right H*-Galois and Hirata separable extension of BH. Then B is characterized in terms of the commutator subring VB(BH) of BH in B and the smash product VB(BH)#H. A sufficient condition is also given for B to be an H*-Galois Azumaya extension of BH.
On Hopf DeMeyer-Kanzaki Galois extensions
Let H be a finite-dimensional Hopf algebra over a field k , B a left H -module algebra, and H ∗ the dual Hopf algebra of H . For an H ∗ -Azumaya Galois extension B with center C , it is shown that B is an H ∗ -DeMeyer-Kanzaki Galois extension if 
and only if C is a maximal commutative separable subalgebra of 
the smash product B # H . Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.
Points of strong summability of Fourier series
In this paper we present a new solution of Hardy and Littlewood's problem concerning strong summability of Fourier series; we also present a property of points of strong summability.