Fourier transform of a Gaussian in finite fields
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From: magya_bloom on 29 Mar 2010 17:22 On Mar 22, 7:07 am, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > On Mar 20, 5:03 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo.com> > wrote: > > > > > On Mar 20, 11:06 am, dvsarwate <dvsarw...(a)gmail.com> wrote: > > > > On Mar 20, 12:01 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo.com> > > > wrote: > > > > > is theFouriertransform of a Gaussian function another Gaussian in > > > > finite fields? Any relevant books containing that? thanks. > > > > If you know (and are willing to share with us) the > > > definition of a Gaussian function in a finite field, > > > the answer will be immediately obvious, and will > > > be Yes or No, though I can never remember which > > > it is. I don't know of any books containing this > > > information specifically, though it looks like a great > > > homework problem that could be included in some: > > > > "Prove or disprove: TheFouriertransform....." > > > > --Dilip Sarwate > > > assume the field is Zp (Z mod p, where p is a prime). Gaussian > > definition: > > f(x) = e^(i * (2 * pi)/p * k * (x-j)^2) , where k and j are in Zp..- Hide quoted text - > > > - Show quoted text - > > Please define e for us. What does *it* mean in Z/pZ? it's not in Zp, it is in C (complex nos.), hence the fourier transform.
From: magya_bloom on 29 Mar 2010 17:23 On Mar 21, 2:49 pm, Axel Vogt <&nore...(a)axelvogt.de> wrote: > magya_bl...(a)yahoo.com wrote: > > is theFouriertransform of a Gaussian function another Gaussian in > > finite fields? Any relevant books containing that? thanks. > > There is such stuff in arithmetic geometry (due to Deligne, I think) looking for Deligne. Perchance you know of the work?
From: Chip Eastham on 29 Mar 2010 23:04 On Mar 29, 5:22 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo.com> wrote: > On Mar 22, 7:07 am, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > > > > > On Mar 20, 5:03 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo.com> > > wrote: > > > > On Mar 20, 11:06 am, dvsarwate <dvsarw...(a)gmail.com> wrote: > > > > > On Mar 20, 12:01 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo.com> > > > > wrote: > > > > > > is theFouriertransform of a Gaussian function another Gaussian in > > > > > finite fields? Any relevant books containing that? thanks. > > > > > If you know (and are willing to share with us) the > > > > definition of a Gaussian function in a finite field, > > > > the answer will be immediately obvious, and will > > > > be Yes or No, though I can never remember which > > > > it is. I don't know of any books containing this > > > > information specifically, though it looks like a great > > > > homework problem that could be included in some: > > > > > "Prove or disprove: TheFouriertransform....." > > > > > --Dilip Sarwate > > > > assume the field is Zp (Z mod p, where p is a prime). Gaussian > > > definition: > > > f(x) = e^(i * (2 * pi)/p * k * (x-j)^2) , where k and j are in Zp.- Hide quoted text - > > > > - Show quoted text - > > > Please define e for us. What does *it* mean in Z/pZ? > > it's not in Zp, it is in C (complex nos.), hence the fourier transform. There is a notion of a Fourier transform of a complex-valued function on a cyclic group (or more generally, a finite abelian group): [Fourier transform on finite groups -- Wikipedia] http://en.wikipedia.org/wiki/Fourier_transform_on_finite_groups So, assuming you have a function f:Z/pZ -> C (complex numbers), you can define a Fourier transform in a standard way. What I'm missing is the thing you were asked about early on in the thread, namely defining "Gaussian" function. Later on in the thread you offered this definition: > f(x) = e^(i * (2 * pi)/p * k * (x-j)^2), > where k and j are in Zp. The question then comes down to how one can evaluate this expression. Since x in Z/pZ, let's dispense for the moment with k and j, and focus on the case k = 1, j = 0: f(x) = e^(i * (2*pi)/p * x) What complex value is this? Okay, letting x be any integer in a residue class mod p will give the same value (due to the imaginary period 2pi*i of the exponential function), so the function f(x) is well-defined if we take that approach. Note however that this is a matter of taking Z/pZ not to be a finite field, but a cyclic group. The resulting construction is referred to as a discrete Fourier transform over the complex numbers. One can _also_ contemplate a theory of discrete Fourier transforms over _any_ field (including finite fields). Inquiring minds want to know: what kind of representation of a cyclic(?) group are you trying to work with? A group action (matrices) over the complex numbers C? Or something more general, as the original subject line might suggest? regards, chip
From: magya_bloom on 30 Mar 2010 00:47 On Mar 29, 8:04 pm, Chip Eastham <hardm...(a)gmail.com> wrote: > On Mar 29, 5:22 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo.com> > wrote: > > > > > On Mar 22, 7:07 am, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > > > > On Mar 20, 5:03 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo.com> > > > wrote: > > > > > On Mar 20, 11:06 am, dvsarwate <dvsarw...(a)gmail.com> wrote: > > > > > > On Mar 20, 12:01 pm, "magya_bl...(a)yahoo.com" <magya_bl...(a)yahoo..com> > > > > > wrote: > > > > > > > is theFouriertransform of a Gaussian function another Gaussian in > > > > > > finite fields? Any relevant books containing that? thanks. > > > > > > If you know (and are willing to share with us) the > > > > > definition of a Gaussian function in a finite field, > > > > > the answer will be immediately obvious, and will > > > > > be Yes or No, though I can never remember which > > > > > it is. I don't know of any books containing this > > > > > information specifically, though it looks like a great > > > > > homework problem that could be included in some: > > > > > > "Prove or disprove: TheFouriertransform....." > > > > > > --Dilip Sarwate > > > > > assume the field is Zp (Z mod p, where p is a prime). Gaussian > > > > definition: > > > > f(x) = e^(i * (2 * pi)/p * k * (x-j)^2) , where k and j are in Zp.- Hide quoted text - > > > > > - Show quoted text - > > > > Please define e for us. What does *it* mean in Z/pZ? > > > it's not in Zp, it is in C (complex nos.), hence thefouriertransform. > > There is a notion of aFouriertransform of > a complex-valued function on a cyclic group > (or more generally, a finite abelian group): > > [Fouriertransform on finite groups -- Wikipedia]http://en.wikipedia.org/wiki/Fourier_transform_on_finite_groups > > So, assuming you have a function f:Z/pZ -> C > (complex numbers), you can define aFourier > transform in a standard way. > > What I'm missing is the thing you were asked > about early on in the thread, namely defining > "Gaussian" function. Later on in the thread > you offered this definition: > > > f(x) = e^(i * (2 * pi)/p * k * (x-j)^2), > > where k and j are in Zp. > > The question then comes down to how one can > evaluate this expression. Since x in Z/pZ, > let's dispense for the moment with k and j, > and focus on the case k = 1, j = 0: > > f(x) = e^(i * (2*pi)/p * x) > > What complex value is this? Okay, letting x > be any integer in a residue class mod p will > give the same value (due to the imaginary > period 2pi*i of the exponential function), > so the function f(x) is well-defined if we > take that approach. > > Note however that this is a matter of taking > Z/pZ not to be a finite field, but a cyclic > group. The resulting construction is referred > to as a discreteFouriertransform over the > complex numbers. One can _also_ contemplate > a theory of discreteFouriertransforms over > _any_ field (including finite fields). > > Inquiring minds want to know: what kind of > representation of a cyclic(?) group are you > trying to work with? A group action (matrices) > over the complex numbers C? Or something more > general, as the original subject line might > suggest? > > regards, chip Your interpretation is exactly right. Z/pZ has the additive structure, cyclic, so the Gaussian function is defined and the Fourier transform over L^2(Z/pz) as well. It can be generalized to any finite field. The problem comes from another one which is connected to Heisenberg group algebra.
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