for polynomial interpolation to the polynomial formulation in the complex plane yields that the solution can be written in the form where The factor in this formula compensates for the fact that the complex plane formulation contains also negative powers of and is therefore not a polynomial expression in . The correctness of this expression can easily be verified by observing that and that is a linear combination of the right powers of . Upon using the identity the coefficient can be written in the form Even number of points If the number of points N is even, say N=2K, applying the Lagrange formula for polynomial interpolation to the polynomial formulation in the complex plane yields that the solution can be written in the form where Here, the constants can be chosen freely. This is caused by the fact that the interpolating function () contains an odd number of unknown constants. A common choice is to require that the highest frequency is of the form a constant times , i.e. the term vanishes, but in general the phase of the highest frequency can be chosen to be . To get an expression for , we obtain by using () that () can be written on the form This yields and Note that care must be taken in order to avoid infinities caused by zeros in the denominators. Equidistant nodes Further simplification of the problem is possible if nodes are equidistant, i.e. see Zygmund for more details. Odd number of points Further simplification by using () would be an obvious approach, but is obviously involved. A much simpler approach is to consider the Dirichlet kernel where is odd. It can easily be seen that is a linear combination of the right powers of and satisfies Since these two properties uniquely define the coefficients in (), it follows that Here, the sinc-function prevents any singularities and is defined by Even number of points For even, we define the Dirichlet kernel as Again, it can easily be seen that is a linear combination of the right powers of , does not contain the term and satisfies Using these properties, it follows that the coefficients in () are given by Note that does not contain the as well. Finally, note that the function vanishes at all the points . Multiples of this term can, therefore, always be added, but it is commonly left out.