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General Franklin systems as bases in H¹[0,1]

Gegham G. Gevorkyan, Anna Kamont (2005)

Studia Mathematica

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].

Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

Let p = m k k = 0 p - 1 be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements μ N V N ( N ) , where V N are p N -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of p and N V N ¯ = L ( ) . The results involve mutual absolute continuity or singularity of such Riesz products extending previous results on...

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