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AbstractWe study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in $L^p$, 1 < p < ∞, and $H^p$, 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in $L^p$ for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces BMO and...

We show that each general Haar system is permutatively equivalent in $L^p\left([0,1]\right)$, 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^p\left([0,1]\right)$, 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^p\left({[0,1]}^d\right)$, 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases in $L^p\left({[0,1]}^d\right)$ for 1...

J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $\Omega \subset {ℝ}^d$ is a smooth bounded domain, 1 ≤ p < ∞ and $f\in W^{1,p}\left(\Omega \right)$, then $li{m}_{s\nearrow 1}(1-s){\int }_{\Omega }{\int }_{\Omega }{\left(\right|f\left(x\right)-f\left(y\right)|}^p{)/(\left|\right|x-y\left|\right|}^{d+sp})dxdy=K{\int }_{\Omega }{\left|\nabla f\left(x\right)\right|}^{p}dx$, where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_p^{s,p}\left(\Omega \right)$. The purpose of this paper is to obtain analogous asymptotic formulae for some...

We prove unconditionality of general Franklin systems in $L^p\left(X\right)$, where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

We investigate when the trigonometric conjugate to the periodic general Franklin system is a basis in C(𝕋). For this, we find some necessary and some sufficient conditions.

By a general Franklin system corresponding to a dense sequence of knots 𝓣 = (tₙ, n ≥ 0) in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots 𝓣, that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is a characterization of sequences 𝓣 for which the corresponding general Franklin system is a basis or an unconditional basis in H¹[0,1].

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 study the problem of consistent and homogeneous colourings for increasing families of dyadic intervals. We determine when this problem can be solved and when it cannot.

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].