Geodesic
Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms
Diskretnaya Matematika, Tome 12 (2000) no. 3, pp. 124-153.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give a fast algorithm for multiplication of polynomials with real-valued coefficients without resort to complex numbers and the fast Fourier transformation. The efficiency of this algorithm is compared with the multiplication algorithm based on the discrete Hartley transformation. We demonstrate that the complexity of the Hartley transformation coincides, to within a linear term, with the complexity of the Fourier transformation, but the use of the Hartley transformation leads to a more efficient multiplication algorithm. We give analogues of these results for finite fields. In some cases, the multiplicative constants in bounds for the complexity of multiplication of polynomials and of the Fourier and Hartley transformations over finite fields are smaller than those in the case of the field of real numbers. This research was supported by the Russian Foundation for Basic Research, grant 99–01–01175, and the Federal Special Program ‘Integration’, grant 473. 
@article{DM_2000_12_3_a8,
     author = {S. B. Gashkov},
     title = {Remarks on the fast multiplication of polynomials, and {Fourier} and {Hartley} transforms},
     journal = {Diskretnaya Matematika},
     pages = {124--153},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2000_12_3_a8/}
}
TY  - JOUR
AU  - S. B. Gashkov
TI  - Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms
JO  - Diskretnaya Matematika
PY  - 2000
SP  - 124
EP  - 153
VL  - 12
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2000_12_3_a8/
LA  - ru
ID  - DM_2000_12_3_a8
ER  - 
%0 Journal Article
%A S. B. Gashkov
%T Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms
%J Diskretnaya Matematika
%D 2000
%P 124-153
%V 12
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2000_12_3_a8/
%G ru
%F DM_2000_12_3_a8
S. B. Gashkov. Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms. Diskretnaya Matematika, Tome 12 (2000) no. 3, pp. 124-153. http://geodesic.mathdoc.fr/item/DM_2000_12_3_a8/

[1] Noden P., Kitte K., Algebraicheskaya algoritmika, Mir, Moskva, 1999

[2] Bleikhut R., Bystrye algoritmy tsifrovoi obrabotki signalov, Mir, Moskva, 1989 | MR

[3] Akho A., Khopkroft E., Ulman D., Postroenie i analiz vychislitelnykh algoritmov, Mir, Moskva, 1979 | MR | Zbl

[4] Blasenko V. A., Lappa Yu. M., Yaroslavskii L. P., Metody sinteza bystrykh algoritmov svertki i spektralnogo analiza signalov, Nauka, Moskva, 1990 | MR

[5] Sorensen H. V., Burrus C. S., Fast DFT and convolution algoritms Handbook for Digital Signal Processing, eds. Mitra S. K., Wiley, New York, 1993, 491–610 | Zbl

[6] Breisuell R., Preobrazovanie Khartli, Mir, Moskva, 1990 | MR

[7] Makklellan Dzh., Reider Ch., Primenenie teorii chisel v tsifrovoi obrabotke signalov, Radio i svyaz, Moskva, 1983

[8] Devenport Dzh., Sire I., Turne E., Kompyuternaya algebra, Mir, Moskva, 1991 | MR

[9] Karatsuba A. A., Ofman Yu. P., “Umnozhenie mnogoznachnykh chisel na avtomatakh”, Dokl. AN SSSR, 145:2 (1962), 293–294

[10] Karatsuba A. A., “Slozhnost vychislenii”, Trudy Matem. in-ta im. V. A. Steklova, 211, 1995, 186–202 | MR | Zbl

[11] Shenkhage A., Shtrassen V., “Bystroe umnozhenie bolshikh chisel”, Kibern. sb., 10 (1973), 87–98

[12] Schönhage A., “Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2”, Acta Informatica, 7 (1977), 395–398 | DOI | MR | Zbl

[13] Gashkov S. B., “O slozhnosti integrirovaniya ratsionalnykh drobei”, Trudy Matem. in-ta im. V. A. Steklova, 218, 1997, 122–133 | MR | Zbl

[14] Suzuki Y., Sone T., Kido K., “A new FFT algorithm of radix 3, 6, 12”, IEEE Trans. A.S.S.P., 34, no. 2, 1986, 380–383 | MR