def dft(self, chi = lambda x: x):
"""
A discrete Fourier transform "over `\QQ`" using exact
`N`-th roots of unity.
EXAMPLES::
sage: J = range(6)
sage: A = [ZZ(1) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dft(lambda x:x^2)
Indexed sequence: [6, 0, 0, 6, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: s.dft()
Indexed sequence: [6, 0, 0, 0, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: G = SymmetricGroup(3)
sage: J = G.conjugacy_classes_representatives()
sage: s = IndexedSequence([1,2,3],J) # 1,2,3 are the values of a class fcn on G
sage: s.dft() # the "scalar-valued Fourier transform" of this class fcn
Indexed sequence: [8, 2, 2]
indexed by [(), (1,2), (1,2,3)]
sage: J = AbelianGroup(2,[2,3],names='ab')
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Multiplicative Abelian group isomorphic to C2 x C3
sage: J = CyclicPermutationGroup(6)
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Cyclic group of order 6 as a permutation group
sage: p = 7; J = range(p); A = [kronecker_symbol(j,p) for j in J]
sage: s = IndexedSequence(A,J)
sage: Fs = s.dft()
sage: c = Fs.list()[1]; [x/c for x in Fs.list()]; s.list()
[0, 1, 1, -1, 1, -1, -1]
[0, 1, 1, -1, 1, -1, -1]
The DFT of the values of the quadratic residue symbol is itself, up to
a constant factor (denoted c on the last line above).
.. TODO::
Read the parent of the elements of S; if `\QQ` or `\CC` leave as
is; if AbelianGroup, use abelian_group_dual; if some other
implemented Group (permutation, matrix), call .characters()
and test if the index list is the set of conjugacy classes.
"""
J = self.index_object() ## index set of length N
N = len(J)
S = self.list()
F = self.base_ring() ## elements must be coercible into QQ(zeta_N)
if not(J[0] in ZZ):
G = J[0].parent() ## if J is not a range it is a group G
if J[0] in ZZ and F.base_ring().fraction_field()==QQ:
## assumes J is range(N)
zeta = CyclotomicField(N).gen()
FT = [sum([S[i]*chi(zeta**(i*j)) for i in J]) for j in J]
elif not(J[0] in ZZ) and G.is_abelian() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
if is_PermutationGroupElement(J[0]):
## J is a CyclicPermGp
n = G.order()
a = list(factor(n))
invs = [x[0]**x[1] for x in a]
G = AbelianGroup(len(a),invs)
## assumes J is AbelianGroup(...)
Gd = G.dual_group()
FT = [sum([S[i]*chid(G.list()[i]) for i in range(N)])
for chid in Gd]
elif not(J[0] in ZZ) and G.is_finite() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
## assumes J is the list of conj class representatives of a
## PermuationGroup(...) or Matrixgroup(...)
chi = G.character_table()
FT = [sum([S[i]*chi[i,j] for i in range(N)]) for j in range(N)]
else:
raise ValueError("list elements must be in QQ(zeta_"+str(N)+")")
return IndexedSequence(FT,J)
def dft(self, chi = lambda x: x):
"""
A discrete Fourier transform "over `\QQ`" using exact
`N`-th roots of unity.
EXAMPLES::
sage: J = range(6)
sage: A = [ZZ(1) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dft(lambda x:x^2)
Indexed sequence: [6, 0, 0, 6, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: s.dft()
Indexed sequence: [6, 0, 0, 0, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: G = SymmetricGroup(3)
sage: J = G.conjugacy_classes_representatives()
sage: s = IndexedSequence([1,2,3],J) # 1,2,3 are the values of a class fcn on G
sage: s.dft() # the "scalar-valued Fourier transform" of this class fcn
Indexed sequence: [8, 2, 2]
indexed by [(), (1,2), (1,2,3)]
sage: J = AbelianGroup(2,[2,3],names='ab')
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Multiplicative Abelian group isomorphic to C2 x C3
sage: J = CyclicPermutationGroup(6)
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Cyclic group of order 6 as a permutation group
sage: p = 7; J = range(p); A = [kronecker_symbol(j,p) for j in J]
sage: s = IndexedSequence(A,J)
sage: Fs = s.dft()
sage: c = Fs.list()[1]; [x/c for x in Fs.list()]; s.list()
[0, 1, 1, -1, 1, -1, -1]
[0, 1, 1, -1, 1, -1, -1]
The DFT of the values of the quadratic residue symbol is itself, up to
a constant factor (denoted c on the last line above).
.. TODO::
Read the parent of the elements of S; if `\QQ` or `\CC` leave as
is; if AbelianGroup, use abelian_group_dual; if some other
implemented Group (permutation, matrix), call .characters()
and test if the index list is the set of conjugacy classes.
"""
J = self.index_object() ## index set of length N
N = len(J)
S = self.list()
F = self.base_ring() ## elements must be coercible into QQ(zeta_N)
if not(J[0] in ZZ):
G = J[0].parent() ## if J is not a range it is a group G
if J[0] in ZZ and F.base_ring().fraction_field()==QQ:
## assumes J is range(N)
zeta = CyclotomicField(N).gen()
FT = [sum([S[i]*chi(zeta**(i*j)) for i in J]) for j in J]
elif not(J[0] in ZZ) and G.is_abelian() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
if is_PermutationGroupElement(J[0]):
## J is a CyclicPermGp
n = G.order()
a = list(factor(n))
invs = [x[0]**x[1] for x in a]
G = AbelianGroup(len(a),invs)
## assumes J is AbelianGroup(...)
Gd = G.dual_group()
FT = [sum([S[i]*chid(G.list()[i]) for i in range(N)])
for chid in Gd]
elif not(J[0] in ZZ) and G.is_finite() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
## assumes J is the list of conj class representatives of a
## PermuationGroup(...) or Matrixgroup(...)
chi = G.character_table()
FT = [sum([S[i]*chi[i,j] for i in range(N)]) for j in range(N)]
else:
raise ValueError("list elements must be in QQ(zeta_"+str(N)+")")
return IndexedSequence(FT,J)
