\input zb-basic \input zb-ioport \iteman{io-port 05069083} \itemau{Averbuch, A.; Coifman, R.R.; Donoho, D.L.; Elad, M.; Israeli, M.} \itemti{Fast and accurate polar Fourier transform.} \itemso{Appl. Comput. Harmon. Anal. 21, No. 2, 145-167 (2006).} \itemab A fast high accuracy polar fast Fourier transform (FFT) is developed. Consider a polar grid of frequencies $\xi_{p,q} = \{\xi_x [p,q], \xi_y [p,q]\}$ in the circle inscribed in the fundamental region $\xi \in [-\pi, \pi)^2 ,$ $$\cases \xi_x [p,q]= {\pi p\over N}\cos (\pi q/2N) \\ & \text { for }-N \leq p \leq N-1,\quad 0 \leq q \leq 2N-1 \\ \xi_y [p,q]= {\pi p\over N} \sin (\pi q/2N) \endcases.$$ Given digital Cartesian data $f [i_1, i_2], \; i_1, i_2 = 0, \ldots , N-1,$ the polar FT is defined to be the collection of samples $\{F(\xi_{p,q})\}$, where $$F(\xi_{p,q}) = \sum^{N-1}_{i_1=0}\ \sum^{N-1}_{i_2 =0} f[i_1,i_2]\ \exp(-i(i_1 \xi_x [p,q]-i_2 \ \xi_y ]p,q])).$$ The proposed method for the fast evaluation of $\{F (\xi_{p,q} ) \}$ factors the problem into two steps. First, a pseudo-polar FFT is applied, in which a pseudo-polar sampling set is used, and second, a conversion from pseudo-polar to polar FT is performed by univariate polynomial interpolations. The pseudo-polar FFT is an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. For a given $N \times N$ signal the proposed algorithm has an arithmetical complexity of ${\cal O} (N^2 \log N)$. Further, the conversion process is described and an error analysis is given. \itemrv{Gerlind Plonka (Duisburg)} \itemcc{} \itemut{polar coordinates; Cartesian coordinates; pseudo-polar coordinates; fast Fourier transform; unequally-sampled FFT; interpolation; linogram; algorithm; error analysis} \itemli{doi:10.1016/j.acha.2005.11.003} \end