Optimum Achievable Rates in Two Random Number Generation Problems with $f$-Divergences Using Smooth R\'enyi Entropy

Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth R\'enyi entropy. Recently, optimum achievable rates with respect to $f$-divergences have been characterized using the information spectrum quantity. The $f$-divergence is a general non-negative measure between two probability distributions on the basis of a convex function $f$. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to $f$-divergences. However, optimum achievable rates with respect to $f$-divergences using the smooth R\'enyi entropy have not been clarified yet in both of two problems. In this paper we try to analyze the optimum achievable rates using the smooth R\'enyi entropy and to extend the class of $f$-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to $f$-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on $f$-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions $f$. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth R\'enyi entropy.