In the present paper we consider periodic spline systems in order to obtain SCHAUDEB bases for the real HARDY spaces Hp(T) (0 < p 5 1) defined on the one-dimensional torus T .
In a recent note [la] we have shown that the periodic FFLANKLIN system forms a basis in H J T ) if 112 < p < 1. Obviously, this statement does not hold for 0 < p < 112. Thus to cover the case 0 < p 5 112 we shall deal with orthogonal spline systems P(ffl) of arbitrary degree m 2 0 introduced by Z. CIESIELSKI (cf. [2]-[6]) for both periodic and non-periodic cases. These systems possess nice properties for most of the function spaces used in analysis.
The main result is the following Theorem. The periodic orthogonal spline system F(") of degree m 2 0 forms a
SOHAUDER basis in the real HARDY space H p ( T ) if (m + l)-l
As in [la] the proof of the theorem given in section 3 is based on some special properties of B-splines and the atomic approach to HL~RDY spaces (see sections 1 and 2, respectively). As a corollary one can also obtain SCHAUDER bases for the classical €€ARDY spaces H J D ) , 0 < p < 1, of analytic functions in the unit disc D. This and further questions will be considered in section 4. p 5 1.