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First we define the dyadic Hardy space$H$_{$X$}$(d)$for an arbitrary rearrangement invariant space$X$on [0, 1]. We remark that previously only a definition of$H$_{$X$}$(d)$for$X$with the upper Boyd index$q$_{$x$}<∞ was available. Then we get a natural description of the dual space of$H$_{$x$}, in the case$X$having the property 1<-$p$_{$X$}<-$q$_{$X$}<2, imporoving an earlier result [P1].

In an event-related functional MRI data analysis, an accurate and robust extraction of the hemodynamic response function (HRF) and its associated statistics (e.g., magnitude, width, and time to peak) is critical to infer quantitative information about the relative timing of the neuronal events in different brain regions. The aim of this paper is to develop a multiscale adaptive smoothing model (MASM) to accurately estimate HRFs pertaining to each stimulus sequence across all voxels. MASM explicitly accounts for both spatial and temporal smoothness information, while incorporating such information to adaptively estimate HRFs in the frequency domain. One simulation study and a real data set are used to demonstrate the methodology and examine its finite sample performance in HRF estimation, which confirms that MASM significantly outperforms the existing methods including the smooth finite impulse