We consider the variation space corresponding to a dictionary of functions in $L^2(\Omega)$ and present the basic theory of approximation in these spaces. Specifically, we compare the definition based on integral representations with the definition in terms of convex hulls. We show that in many cases, including the dictionaries corresponding to shallow ReLU$^k$ networks and a dictionary of decaying Fourier modes, that the two definitions coincide. We also give a partial characterization of the variation space for shallow ReLU$^k$ networks and show that the variation space with respect to the dictionary of decaying Fourier modes corresponds to the Barron spectral space.

翻译：我们考虑了与用美元计算的函数字典相对应的变异空间,并提出了这些空格中近似的基本理论。具体地说,我们比较了基于整体表示的定义和对柔性船体的定义。我们表明,在许多情况下,包括浅ReLU$k$网络的词典和Fourier模式衰减词典在内的许多词典,这两个定义是一致的。我们还对浅ReLU$QK$网络的变异空间作了部分定性,并表明Fourier模式衰变词典的变异空间与Barron光谱空间相对应。