By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^p[0,1]$, 1 < p < ∞.

We give a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order k is an unconditional basis in the atomic Hardy space H¹[0,1].

We prove that given any natural number k and any dense point sequence (tₙ), the corresponding orthonormal spline system is an unconditional basis in reflexive $L^p$.

Exact conditions for α, β, a, b > −1 and 1 ≤ p ≤ ∞ are determined under which the inclusion property $$L_{w^{(a,b)}}^p[-1,1]$$ ⊂ $$L_{w^{(\alpha ,\beta )}}^1[-1,1]$$ is valid. It is shown that the conditions characterize the inclusion property. The paper concludes with some results, in which the inclusion property can be detected in relation with estimates of Jacobi differential operators and with Muckenhoupt’s transplantation theorems and multiplier theorems for Jacobi series.

We establish conditions on the partial moduli of continuity which guarantee uniform convergence of the N-dimensional Walsh-Fourier series of functions f from the class $C_W\left(I^N\right)\cap {\bigcap }_{i=1}^{N}B{V}_{i,p\left(n\right)}$, where p(n)↑ ∞ as n → ∞.