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The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density distribution functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-(
) superposition, the solution is mathematically not unique. Rather it contains a range of possible solutions (kernel) which is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two parts 
(g) and 
(g) expressed by even- and odd-order terms of the series expansions. For the calculation of the even part (g), the positivity condition for all pole figures contributes essentially to an `economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time.
) superposition, the solution is mathematically not unique. Rather it contains a range of possible solutions (kernel) which is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two parts 
(g) and 
(g) expressed by even- and odd-order terms of the series expansions. For the calculation of the even part (g), the positivity condition for all pole figures contributes essentially to an `economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time.
