[Paper Review] An example of an almost greedy uniformly bounded orthonormal basis for $L_p([0,1])$

Morten Nielsen|ArXiv.org|Nov 28, 2006

Mathematical Analysis and Transform Methods6 references4 citations

TL;DR

This paper constructs a uniformly bounded orthonormal basis for $L_p([0,1])$, $1<p<\ olimits\infty$, that is almost greedy, demonstrating that Orlicz's theorem on the nonexistence of uniformly bounded unconditional bases for $L_p$, $p\neq2$, does not extend to the broader class of almost greedy bases. The construction uses a modified Rademacher-like system with block-structured, normalized Haar-like functions and relies on Khintchine's inequality and democratic properties to ensure quasi-greedy convergence.

ABSTRACT

We construct a uniformly bounded orthonormal almost greedy basis for $L_p([0,1])$, $1

Motivation & Objective

• To construct a uniformly bounded orthonormal basis for $L_p([0,1])$, $1<p<\infty$, that is almost greedy, thereby challenging the possibility of extending Orlicz's theorem to this class of bases.
• To demonstrate that uniformly bounded orthonormal bases for $L_p$, $p\neq2$, can be quasi-greedy and even almost greedy, despite failing to be unconditional.
• To provide a concrete example of a democratic, uniformly bounded, orthonormal system in $L_p$ that supports convergent greedy approximation.
• To show that the failure of classical systems like the trigonometric and Walsh systems to be quasi-greedy does not preclude the existence of such bases in $L_p$.

Proposed method

• The construction uses a block-structured orthonormal system based on Haar functions and Rademacher-like functions, with normalization tailored to $L_p$-boundedness.
• The basis is formed by combining normalized block functions $\phi_k / \sqrt{N_k}$ with coefficients derived from inner products $\langle f, \psi_j^{(k)} \rangle$.
• Khintchine's inequality is applied to control $L_p$-norms of random sums involving Rademacher functions, ensuring democratic behavior.
• The system is shown to be democratic by comparing $\ell_p$-norms of partial sums over sets of equal size, using estimates on coefficient decay and block structure.
• Convergence of the greedy algorithm is established by decomposing the approximation error into three parts: $T$, $T'$, and $T''$, each shown to converge in $L_p$ under decreasing rearrangement.
• The proof leverages duality: for $1<q<2$, $p$ is chosen such that $1/p + 1/q = 1$, and bi-democratic properties are used to extend quasi-greedy convergence to $L_q$.

Experimental results

Research questions

• RQ1 Can a uniformly bounded orthonormal basis for $L_p([0,1])$, $1<p<\infty$, $p\neq2$, be almost greedy despite the failure of unconditional bases?
• RQ2 Does the non-existence of uniformly bounded unconditional bases in $L_p$, $p\neq2$, as per Orlicz's theorem, extend to the class of almost greedy bases?
• RQ3 Can a democratic, uniformly bounded orthonormal system in $L_p$ support convergent greedy approximation?
• RQ4 Is it possible to construct such a basis using a modified Rademacher or Haar system with controlled $L_p$-norms?

Key findings

• A uniformly bounded orthonormal basis for $L_p([0,1])$, $1<p<\infty$, exists that is almost greedy, directly contradicting a potential extension of Orlicz's theorem.
• The constructed basis is democratic, as shown by uniform bounds on $\ell_p$-norms of partial sums over sets of equal size.
• The greedy approximation algorithm converges in $L_p$ for all $f \in L_p$, establishing the basis as quasi-greedy.
• The convergence is proven via decomposition into three parts: $T$, $T'$, and $T''$, each shown to converge absolutely or conditionally in $L_p$.
• The system is bi-democratic in $L_p$ and $L_q$ for $1<q<2$, $1/p + 1/q = 1$, enabling extension of quasi-greedy convergence to $L_q$.
• The construction uses a block-structured system with $N_k$ elements per block, where $N_k$ grows rapidly, ensuring coefficient decay and democratic behavior.