中文摘要：

从理论上可以证明标准的城市人口密度负指数距离衰减模型本质上是一种空间相关函数，基于这种思想对Clark模型进行Fourier变换，可以导出城市人口密度的幂次频谱分布，且功率谱指数理应为β=2^±。负指数与幂指数的这种变换关系暗示了城市地理系统简单与复杂的辩证关系。借助中国杭州市4年的人口普查资料转换的平均人口密度分布数据对上述推论进行检验，发现口渐进式趋近于2但并不约等于2。将β值进一步换算为人口过程的分维D和Hurst指数H，结果表明：城市人口具有长程负相关作用，但这种空间作用显示明确的局域化倾向。目前的城市形态演化模拟几乎无一例外地引入了长程作用，根据杭州人口分布的局域化特征，有关地理长程作用的假设和应用有必要重新探讨。

英文摘要：

The standard negative exponential distance-decay relationship of urban population density, namely the Clark empirical law, p（r） =p0exp（-br）, is proved to be a correlation function reflecting the relation between the center of the city and the point at a given distance. Therefore the Fourier transformation of the urban density should give an index/exponent of power spectra such as β≈2 theoretically. However, when the fast Fourier transformation （FFT） is applied to the urban densities of the Hangzhou metropolis in 1964, 1982, 1990 and 2000 （according to Census time）, it turns out to be that the β values vary from 1.44 to 1.80, not often approximating to 2. If the exponential function with power, p（r） =p0exp（-br^σ） , is employed to fit the urban density data of Hangzhou instead of Clark＇ s model, the results are the restraint parameter σ values vary from 0. 45 to 0.78, not often approximating to 1. A semi-log relation between α and β values can be expressed as β≈2 ＋ 0.7 In σ . Going a further step, the fractal dimension of self-similar curves, D, and Hurst exponent, H, can be given by means of the formulae β = 5 - 2D and D = 2 - H. The conclusions can be drawn from the mathematical transformations and computations as follows： First, Urban density has a long-distance dependence, but it is passive/negative and tends to become weak with the lapse of time; Second, the action of urban population in space is localized as a city grows. Third, since the Clark model can be derived using entropy-maximizing method, the calculated results imply that the so-called entropy-maximization is only a developing tendency instead of a realistic state. Last but not least, the idea of action-at-a-distance has been introduced into the various urban simulations based on cellular automata （CA） and all that, but the localization of urban population activity means that the long-range effect should be reflected and revised in the light of amplitude-spectra analyses of urban density equation as an a