Out of ~ 500 individual source images we have constructed a mosaic map of the K band surface brightness in an area DELTA alpha * DELTA delta ~ 650''*710'' (R_equiv ~ 15.8 pc for R0 = 8.5 kpc) centered approximately on Sgr A*. An observing technique was used which allows us to recover an extended background emission. To separate sources from an unresolved background continuum we fitted Lorentzian distributions to the sources and find that about one half of an integrated, not dereddened K band flux density of 752 Jy is contributed by ~ 6*104 stars with flux densities SK' >~ 100 mu Jy and the remainder is contributed by an extended continuum provided by about 6*108 stars too weak to be observed as individual sources. We estimate that >~ 80% of the integrated flux density of the mosaic is contributed by stars in the Nuclear Bulge (NB; R <~ 300 pc); the remaining <~ 20% come from stars located along the line of sight in the Galactic Bulge (GB; 0.3 <~ R/kpc <~ 3) and Galactic Disk (GD; R > 3 kpc). We determine the K band luminosity functions (KLF) of the mosaic and of subareas dominated by Nuclear Bulge, Galactic Bulge and Disk stars, respectively, and construct difference KLFs which relate to the specific stellar populations of these regions. The detection limit is SK' ~ 100 mu Jy, for the completeness limit we estimate SK' ~ 2 000 mu Jy. We find that the stellar population of the Nuclear Bulge contains considerably more bright stars (i.e. with reddened K band flux densities SK' >~ 5*103 mu Jy), most of which are probably early O stars, Giants and Supergiants. The stellar population of the Galactic Bulge on the other hand is dominated by stars which appear to be lower mass (<~ 6 Mo) Main Sequence (MS) stars. A model KLF constructed with a Salpeter Initial Mass Function (IMF) for stars of spectral type O9 or later (SK' <~ 2 000 mu Jy) explains the observations satisfactorily and connects well with the observed KLF of more luminous stars. About 6*108 stars with masses ranging from 0.06 to 6 Mo account for the unresolved continuum. Combining observed and model KLF we obtain a mosaic KLF which increases \propto S_K' - 1 for 106 >~ SK'/ mu Jy >~ 103 and \propto S_K' - 0.6 for 103 >~ SK'/ mu Jy >~ 3*10-3. ... shortened ... We present and discuss the radio/IR spectrum of the central 30'' (~ 1.25 pc) and derive dust and Lyman continuum (Lyc) luminosities of 7.5 * 107 Lo and 1.2 * 1050 s-1, respectively.
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