[Paper Review] Haar system as Schauder basis in Besov spaces: The limiting cases for 0 < p <= 1

Peter Oswald|arXiv (Cornell University)|Aug 24, 2018

Advanced Harmonic Analysis Research5 references4 citations

TL;DR

This paper establishes the Schauder basis property of the d-dimensional Haar system in classical and difference-based Besov spaces for the critical limiting case s = d(1/p − 1) when 0 < p < 1 and 0 < q ≤ p. Using atomic decompositions and local means, it proves that the Haar system forms a Schauder basis in Bs_p,q,1(Id) and Bs_p,q(Id) under these conditions, resolving the last open case from 1979. The result completes the characterization of Haar basis properties in the full parameter range for 0 < p ≤ 1.

ABSTRACT

We show that the d-dimensional Haar system H^d on the unit cube I^d is a Schauder basis in the classical Besov space B_{p,q,1}^s(I^d), 0<p<1, defined by first order differences in the limiting case s=d(1/p-1), if and only if 0<q\le p. For d=1 and p<q, this settles the only open case in our 1979 paper [4], where the Schauder basis property of H in B_{p,q,1}^s(I) for 0<p<1 was left undecided. We also consider the Schauder basis property of H^d for the standard Besov spaces B_{p,q}^s(I^d) defined by Fourier-analytic methods in the limiting cases s=d(1/p-1) and s=1, complementing results by Triebel [7].

Motivation & Objective

• To resolve the open problem of whether the d-dimensional Haar system is a Schauder basis in Besov spaces Bs_p,q,1(Id) at the critical parameter s = d(1/p − 1) for 0 < p < 1 and p < q < ∞.
• To complete the characterization of the Schauder basis property of the Haar system in both difference-based and Fourier-analytic Besov spaces for the full range 0 < p ≤ 1.
• To settle the limiting cases for the Haar system in Besov spaces Bs_p,q(Id) at s = d(1/p − 1) and s = 1, where earlier results left gaps.
• To extend earlier results from univariate to multivariate settings, particularly for d > 1, using atomic decompositions and local means techniques.

Proposed method

• Uses atomic decompositions of functions in Besov spaces to analyze the behavior of Haar coefficients and partial sum operators.
• Applies local means characterization of Besov norms via smooth kernels to control the quasi-norm of partial projections.
• Employs dyadic decomposition and moment conditions on kernels to estimate the Lp quasi-norms of local means operators.
• Establishes uniform boundedness of partial sum operators by controlling the Lp norms of differences between functions and their dyadic projections.
• Adapts univariate counterexamples from [4] to the multivariate case to show failure of boundedness and coefficient functional extension in critical parameter ranges.
• Applies comparison between difference-based and Fourier-analytic Besov norms to show equivalence in the range d/(d+1) < p ≤ 1.

Experimental results

Research questions

• RQ1 Does the d-dimensional Haar system form a Schauder basis in Bs_p,q,1(Id) when s = d(1/p − 1) and 0 < p < 1, 0 < q ≤ p?
• RQ2 What happens to the Schauder basis property of the Haar system in Bs_p,q,1(Id) when s = d(1/p − 1) and q > p?
• RQ3 Is the Haar system a Schauder basis in the standard Besov space Bs_p,q(Id) at the endpoint s = d(1/p − 1) for d/(d+1) < p < 1 and 0 < q ≤ p?
• RQ4 Why does the Haar expansion fail to converge in Bs_p,q(Id) when s = 1 and d/(d+1) ≤ p < 1, 0 < q < ∞?
• RQ5 Can the coefficient functionals of the Haar expansion be extended to bounded linear functionals on Bs_p,q(Id) when s = d(1/p − 1) and q > 1?

Key findings

• The d-dimensional Haar system is a Schauder basis in Bs_p,q,1(Id) if and only if s = d(1/p − 1) and 0 < q ≤ p, for d ≥ 1 and d/(d+1) < p < 1.
• For s = d(1/p − 1) and p < q ≤ 1, the partial sum operators of the Haar expansion are not uniformly bounded on Bs_p,q,1(Id), indicating failure of basis property.
• When s = d(1/p − 1) and 1 < q < ∞, the coefficient functionals on the span of the Haar system cannot be extended to bounded linear functionals on Bs_p,q,1(Id), so the system is not a Schauder basis.
• The Haar system fails to be a Schauder basis in Bs_p,q(Id) at s = 1 for d/(d+1) ≤ p < 1 and 0 < q < ∞, as demonstrated by divergence of the expansion of f(x) = x1 + ... + xd.
• For the standard Besov space Bs_p,q(Id), the Haar system is a Schauder basis at s = d(1/p − 1) if d/(d+1) < p < 1 and 0 < q ≤ p, completing the characterization.
• The results extend to the full space Rd, with the same conditions on p, q, s, and the Haar system remains a Schauder basis under the same parameter constraints.