[Paper Review] Fourier tranform in exponential rearrangement invariant spaces

Eugeny Ostrovsky, L. Sirota|ArXiv.org|Jun 20, 2004

Advanced Harmonic Analysis Research15 references3 citations

TL;DR

This paper investigates the boundedness and convergence of Fourier series and transforms in exponential rearrangement invariant (r.i.) spaces, particularly Orlicz spaces with exponential N-functions. It establishes sharp conditions on the fundamental function and norm equivalence, proving that $ L(N) $-norm convergence of partial Fourier sums implies membership in the closure $ L^0(N) $, and derives moment inequalities and wavelet/Haar series estimates using generalized Hausdorff-Young and Hardy-Littlewood techniques.

ABSTRACT

In this article we investigate the Fourier series and transforms for the functions defined on the $ [0, 2 π]^ d $ or $ R^d $ and belonging to the exponential Orlicz and some other rearrangement invariant (r.i.) spaces.

Motivation & Objective

• To analyze the convergence and boundedness of Fourier series and transforms in exponential Orlicz and rearrangement invariant (r.i.) spaces.
• To characterize the conditions under which partial Fourier sums converge in the $ L(N) $-norm, particularly relating to the closure $ L^0(N) $.
• To derive sharp moment inequalities and norm equivalences using the fundamental function $ heta(p) $ and generalized Orlicz norms.
• To extend classical inequalities (Hausdorff-Young, Hardy-Littlewood) to exponential Orlicz spaces and apply them to wavelet and Haar series.

Proposed method

• The authors define exponential Orlicz spaces $ EOS(W) $ via $ N(u) = \exp(W(\log|u|)) $ for $ |u| \geq e^2 $, with $ W $ convex and increasing.
• They use the Luxemburg norm formula $ \|f\|_{L(N)} = \inf_{v>0} \left\{ v^{-1} \left(1 + \int_X N(v|f(x)|) \, \mu(dx) \right) \right\} $ to define norms in $ L(N) $.
• The paper applies generalized Hausdorff-Young and Hardy-Littlewood inequalities to relate $ L^p $-norms of $ f $ and its Fourier transform $ F[f] $, with weights involving $ |x|^{p-2} $.
• It introduces the space $ G(a,b,\alpha,\beta) $ to model decay and integrability, and uses the norm $ \|f\|_{G} $ to control $ L^p $-norms via $ (q-1)^{-\alpha} $ for $ q \in (1,2] $.
• The analysis includes moment estimates for wavelet and Haar series, showing $ \|P_M[f] - f\|_{L(N)} \leq (K_6 + 1) \|f\|_{L(N)} $ with $ K_6 = 13 $ for Haar and $ K_6 = 1 $ for wavelets.
• The key technique involves relating the growth of the fundamental function $ \psi(p) = \exp(W^*(p)/p) $ to convergence in $ L(N) $, using duality and asymptotic analysis.

Experimental results

Research questions

• RQ1 Under what conditions does the partial Fourier sum $ s_M[f] $ converge in the $ L(N) $-norm for $ f \in L(N) $?
• RQ2 What is the relationship between $ L(N) $-norm convergence of Fourier series and membership in the closure $ L^0(N) $?
• RQ3 How do generalized Hausdorff-Young and Hardy-Littlewood inequalities extend to exponential Orlicz spaces with $ N \in EOF $?
• RQ4 What moment estimates can be derived for wavelet and Haar series in $ L(N) $, and how do they depend on the fundamental function $ \psi(p) $?
• RQ5 When does the condition $ \lim_{p \to \infty} \theta(p)/\psi(p) = 0 $ imply $ L(N) $-norm convergence of Fourier series?

Key findings

• The Fourier partial sums $ s_M[f] $ converge in $ L(N) $-norm if and only if $ f \in L^0(N) $, where $ L^0(N) $ is the closure of bounded functions with bounded support.
• If $ f \in L(N) $ and $ \|f\|_{L(N)} < \infty $, then $ \|s_M[f] - f\|_{L(N)} \to 0 $ as $ M \to \infty $ only if $ f \in L^0(N) $, otherwise convergence fails in norm.
• For $ f \in G(1,b,\alpha,0) $, the Fourier transform satisfies $ \|F[f]\|_p \leq (2\pi)^{1/2} p^\alpha $ for $ p \geq 2 $, implying $ F[f] \in L(N_{1/\alpha}) $.
• The wavelet partial sum operator satisfies $ \|P_M[f] - f\|_{L(N)} \leq (K_6 + 1) \|f\|_{L(N)} $ with $ K_6 = 13 $ for Haar series and $ K_6 = 1 $ for wavelets on $ [0,1] $ or $ \mathbb{R} $.
• The condition $ \lim_{p \to \infty} \theta(p)/\psi(p) > 0 $ contradicts $ L(\Phi) $-norm convergence of Fourier series, implying $ \psi(p)/\theta(p) \to 0 $ if convergence holds.
• For $ f \in L(N) $, the $ L^p $-norm of $ f $ satisfies $ |f|_p \leq C \cdot p \cdot \psi(p) $ with $ \psi(p) = \exp(W^*(p)/p) $, and $ f \in L(N_1[\psi]) $, linking growth of $ \psi(p) $ to Orlicz space membership.