Classical Kramers-Krönig (K–K) relations connect real and imaginary parts of the frequency-domain response of a system. The K–K relations also hold between the logarithm of modulus and the argument of the response, e.g. between the attenuation and the phase shift of a solution to a wave-propagation problem. For square-integrable functions of frequency, the satisfaction of classical K–K relations implies causality in the time domain. On the other hand, when the K–K relations are checked for the logarithm of the system response, the function is not a square integrable one. Then one can employ classical K–K relations with subtractions, but their satisfaction for the logarithm of the system response does not imply causality of the original function. In this paper, the K–K relations are generalized towards functions which are not square-integrable, also allowing for causality evaluation when the logarithm of the system response is considered. That is, we propose generalization of the K–K relations with subtractions, whose validity for the logarithm of the system response and the satisfaction of additional assumptions imply causality of the originally considered function. The derived theory is then applied to electromagnetic media characterized by power-law frequency dispersion, i.e. the media which are described by fractional-order models (FOMs). In this case, the subtraction procedure generates functions which may be not square integrable, or even not locally integrable. However, we can rigorously analyse causality of the media described by FOM using the derived theory, as well as the parameter ranges for which such models are causal.

经典的Kramers-Krönig（K–K）关系将系统的频域响应的实部和虚部连接起来。Ｋ－Ｋ关系在模的对数和响应参数之间也成立，例如在波传播问题的解的衰减和相移之间。对于频率的平方可积函数，经典K–K关系的满足意味着时域的因果关系。另一方面，在检查K-K关系以获取系统响应的对数时，该函数不是平方可积函数。然后可以采用经典的K–K关系与减法，但是它们对系统响应的对数的满意并不意味着原始函数的因果关系。在本文中，将K–K关系推广到不是平方可积的函数，当考虑系统响应的对数时，还可以进行因果关系评估。就是说，我们建议用减法来推广K–K关系，其对系统响应的对数的有效性和对其他假设的满足暗示了最初考虑的函数的因果关系。然后将派生的理论应用于以幂律频率色散为特征的电磁介质，即由分数阶模型（FOM）描述的介质。在这种情况下，减法过程生成的函数可能不是平方可积分的，甚至是局部不可积分的。但是，我们可以使用派生理论来严格分析FOM描述的媒体的因果关系，以及此类因果关系的参数范围。