The radio-loud Active Galactic Nuclei (AGNs) are one of the most intriguing powerful radio sources. Usually they are classified as steep-spectrum and flatspectrum sources. According to the unified model of AGN, their difference can be ascribed to the angle of inclination. Both of them can be roughly described as a picture of “compact core + jet + extended radio lobes”. Traditionally, the steep-spectrum and flat-spectrum sources are studied respectively by radio luminosity functions (RLFs) to obtain the information of their evolution. However, due to the peculiarity of radio radiation and activity, e.g., extension, inclination dependent, and intermittence, the information obtained by traditional methods is limited. With the improvement of radio observation accuracy, more and more radio cores are observed. The existence of radio core is a good indicator of radio activity. In this thesis, we utilized the public observational data to study the RLF of radio cores. As the radio cores are located at the centers of AGNs, research on their RLF can helps to explore the relationship between matter accretion and jet more directly.Firstly, a big sample of 1063 steep-spectrum radio-loud AGNs was established by consulting literature and database retrieval. Based on the sample, the RLF at 408 MHz of steep-spectrum radio sources was calculated by the 1/Va method. The results support a luminosity-dependent density evolution. Most of the radio cores in the sample have flux densities at 5 GHz. Using these data, the core RLF at 5 GHz was also estimated. The results show that the comoving number density of radio cores displays a persistent decline with redshift, implying a negative density evolution. A possible explanation to this result is that in the earlier epoch of AGNs, the relatively higher accretion rate may adversely affect the radio emission of jet. The so-called ‘soft’ and ‘hard’ states in X-ray Binaries may also exist in AGNs. The above results were also consistent with previous researches from different point of views. Also, we noticed that the core RLF is obviously different from the total RLF at 408 MHz band which is mainly contributed by extended lobes, implying that the cores and extended lobes could not be co-evolving at radio emission.Becides, the classical 1/Va and PC methods of constructing binned luminosity functions (LFs) are revisited and compared by graphical analysis. Using both theoretical analysis and illustration with an example, we show why the two methods give different results for the bins which are crossed by the flux limit curves L = Llim(z). Based on a combined sample simulated by a Monte Carlo method, the estimate ? of two methods are compared with the input model LFs. The two methods give identical and ideal estimate for the high luminosity points of each redshift interval. However, for the low luminosity bins of all the redshift intervals both methods give smaller estimate than the input model. We conclude that once the LF is evolving with redshift, the classical binned methods will unlikely give an ideal estimate over the total luminosity range. Previous researches noticed that for objects close to the flux limit ?1/Va nearly always to be too small. We believe this is due to the arbitrary choosing of redshift and luminosity intervals. Because ?1/Va is more sensitive to how the binning are chosen than ?PC. If the binning are wisely chosen according to a simple rule of thumb suggested by us, it is hard to say the estimate of PC method is markedly better than the 1/Va method.
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