ISRAEL JOURNAL OF MATHEMATICS, Vol. 45, Nos. 2-3, 1983 SPLINE SYSTEMS AS BASES IN HARDY SPACES BY PER SJOLIN AND JAN-OLOV STR()MBERG ABSTRACI" It is known that the spline system of order m is an unconditional basis for He[0, 1] when p > 1/(m + 2) and a Schauder basis when p >= 1/(m + 2). We show that these results are sharp. I. Introduction Set I = [0, 1] and let (f~))~=o denote the Franklin system and (f~m))7 . . . . m _-> 1, spline systems of higher order on I (for a definition see e.g.Z. Ciesielski [1]). We shall write f, instead of [~') and set /.(t)=0 for tER\L We let H p = H p (R), 0 < p < ~, denote the usual Hardy spaces on R (cf. [2]). For a > 0 we set N = [a], where [ ] denotes the integral part, and 6 = a - N. If a is not an integer set =/q~ E CN(R); sup IIAhDNq~ll~/lh 18 <~1 h~0 J (here AhF(X)= F(x + h)-F(x)) and if a is an integer set :/~ • CN-~(R); sup IIA~D N-lq~ IIJI hi< oo]. Ao J Also set A~ = A~ [pN, where pN denotes the class of polynomials of degree =< N. The projection from ~.~ to A~ is denoted 7r. For 0 < p =< 1 set a = 1/p - 1. It is then well-known that for 0 < p < 1, A~ is the dual space of H p. If ~o ~ A~ set q~(f) = (Tr(qQ)(f) for f e H p. Also set HP(I) = {f E HP(R); suppf C I and ~(f) E R for every real-valued ,p E A~}, 0<p<l. Received April 9, 1982 and in revised form November 22, 1982 147