On a method to resolve the nuclear activity in galaxies 3 Fig. 2.— Spectrum of the nucleus (top) and 1 . $^{′′}$5 West off-nucleus region of NGC 1358. The best fitted linear combination of single stellar populations shows the stellar features fitted simultaneously as Gaussian fits to the emission lines have been applied. The residuals (data - total fit) show that the fits have been successful ([OI] λ 6300, 6364 have been excluded in this fit). Here we ask ourselves, what additional information on the structure and ionization distribution in the nuclear region do these differences in velocity behaviour give us? An explanation for the differences in velocity behaviour between the various emission lines would be that the whole central region consists of different gaseous com- ponents with different velocity behaviour, different exci- tation, and different dust absorption, and thus different sets of line ratios. When this region is smoothed by the PSF, different distortions of the velocity field for the dif- ferent lines will result. Assume that the region considered covers N pixels and contains K separate structural components, each with its own line ratios and set of velocities. If J ( λ ) i and V ( λ ) i are the observed line intensities and velocities at pixel number i of a line with wavelength λ , we have where I ( λ ) kj is the intrinsic intensity, in the line λ , of the component k = (1 , . . . , K ) at the pixel j , and v$_{kj}$ the cor- responding velocity, which is supposed independent of λ , and p ( x ) the PSF. If L is the number of lines measured, we have in the general case KN ( L + 1) unknown and 2 NL equations. If the line ratios are all the same in the different components, equation (2) is independent of λ . In the simplest case, where we only consider one compo- nent ( K = 1), we have enough equations to solve for the unknown intensities and velocities for each λ , which can be made by CLEAN-like algorithms (H¨ ogbom 1974). 3. THEORY J ( λ ) i = ∑ j ∑ k I ( λ ) kj p ( i − j ) , (1) V ( λ ) i = 1 J ( λ ) i ∑ j ∑ k v$_{kj}$ I ( λ ) kj p ( i − j ) , (2) In the general case, we do not have a sufficient num- ber of equations to solve for the unknown intensities I ( λ ) kj and velocities v$_{kj}$ . If, however, we increase the num- ber of lines measured and at the same time impose some conditions on the unknowns in terms of models, we may in principle be able to reach a solution. For instance, we see from equations (1) and (2) that if the line ratios I ( λ$_{1}$ ) kj /I ( λ$_{2}$ ) kj depend on k , i.e., do vary from one compo- nent to another, then we have V ( λ$_{1}$ ) i − V ( λ$_{2}$ ) i = 0. Thus, the differences between velocities from different spectral lines give us information about differing line ratios be-