The fast Fourier transform (FFT) is used widely in signal processing for efficient computation of the FT of finite-length signals over a set of uniformly spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e., a nonuniform FT. Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

Cerebral white matter (WM) degeneration occurs with increasing age and is associated with declining cognitive function. Research has shown that cardiorespiratory fitness and exercise are effective as protective, even restorative, agents against cognitive and neurobiological impairments in older adults. In this study, we investigated whether the beneficial impact of aerobic fitness would extend to WM integrity in the context of a one‐year exercise intervention. Further, we examined the pattern of diffusivity changes to better understand the underlying biological mechanisms. Finally, we assessed whether training‐induced changes in WM integrity would be associated with improvements in cognitive performance independent of aerobic fitness gains. Results showed that aerobic fitness training did not affect group‐level change in WM integrity, executive function, or short‐term memory, but that greater aerobic fitness derived from the walking program was associated with greater change in WM integrity in the frontal and temporal lobes, and greater improvement in short‐term memory. Increases in WM integrity, however, were not associated with short‐term memory improvement, independent of fitness improvements. Therefore, while not all findings are consistent with previous research, we provide novel evidence for correlated change in training‐induced aerobic fitness, WM integrity, and cognition among healthy older adults. Hum Brain Mapp 34:2972–2985, 2013.

Neuroimaging data emphasize that older adults often show greater extent of brain activation than younger adults for similar objective levels of difficulty. A possible interpretation of this finding is that older adults need to recruit neuronal resources at lower loads than younger adults, leaving no resources for higher loads, and thus leading to performance decrements [Compensation-Related Utilization of Neural Circuits Hypothesis; e.g., Reuter-Lorenz, P. A., & Cappell, K. A. Neurocognitive aging and the compensation hypothesis. Current Directions in Psychological Science, 17, 177–182, 2008]. The Compensation-Related Utilization of Neural Circuits Hypothesis leads to the prediction that activation differences between younger and older adults should disappear when task difficulty is made subjectively comparable. In a Sternberg memory search task, this can be achieved by assessing brain activity as a function of load relative to the individual's memory span, which declines with age. Specifically, we hypothesized a nonlinear relationship between load and both performance and brain activity and predicted that asymptotes in the brain activation function should correlate with performance asymptotes (corresponding to working memory span). The results suggest that age differences in brain activation can be largely attributed to individual variations in working memory span. Interestingly, the brain activation data show a sigmoid relationship with load. Results are discussed in terms of Cowan's [Cowan, N. The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24, 87–114, 2001] model of working memory and theories of impaired inhibitory processes in aging.

In magnetic resonance imaging, magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast Fourier trasform (FFT) methods. Several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities, but these methods assume that the field map that characterizes the off-resonance frequencies is spatially smooth. Recently, iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off-resonance effects. However, straightforward implementations of such iterative methods suffer from inconveniently long computation times. This paper describes a tool for accelerating iterative reconstruction of field-corrected MR images: a novel time-segmented approximation to the MR signal equation. We use a min-max formulation to derive the temporal interpolator. Speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast Fourier transform with normalized root mean squared approximation errors of 0.07%. The proposed method provides fast, accurate, field-corrected image reconstruction even when the field map is not smooth.