From continuous to the discrete Fourier transform: classical and quantum aspects 1 Institut de Mathématiques de Bourgogne, Dijon, France 2 Sankt-Petersburg University of Aerospace Instrumentation (SUAI), St-Petersburg, Russia. emails: matveev@u-bourgogne.fr, vladimir.matveev9@gmail.com lecture delivered at the international conference "Recent Advances in quantum Integrable Systems" ( RAQIS 14 ) 1-5 September 2014, Dijon, France September 5-th, 2014
1 Some general features of FT and DFT 2 The connection of the Gauss sum with the trace of the DFT matrix 3 The complete systems of the eigenvectors of Φ(n) and the real orthogonal matrices O(n) diagonalizing DFT 4 5 Poisson summation formula and intertwining relation between FT and DFT 6 7 Generating eigenvectors of DFT from absolutely convergent series. 8
We explain and generalise certain results by Mehta (1987) relating the eigenfunctions of quantum harmonic oscillator with the eigenvectors of the discrete Fourier transform (DFT). Using far more general construction based on a kind of Poisson summation, we associate the eigenvectors of the DFT with the sums of any absolutely convergent series. This construction in particular provides the overcomplete bases of the DFT eigenvectors by means of Jacobian theta functions or ν-theta functions and leads in a quite natural way to some new identities and addition theorems in the theory of special and q-special functions.
References 1.M.L. Mehta "Eigenvalues and Eigenvectors of the Finite Fourier Transform" J.Math.Phys. 28 (4) p.781-785 (1987) 2.V.B. Matveev "Intertwininig relations between the Fourier transform and discrete Fourier transform, the related functional densities and beyond" Inverse Problems 17, 633-657 (2001) 3. M Ruzzi Jacobi theta functions and discretee Fourier transforms, J.Math.Phys. 47, 063507 (2006) 4. D.V. Chudnovsky, G.V.Chudnovsky T.Morgan "Variance of signals and Their Finite Fourier transforms" in vol. D.Chudnovsky and G. Chudnovsky eds. additive number theory : Festschrift in honour of M.B. Nathanson, Springer Science +Buisness Media,LCC 2010 p.53-75 5. R. Malekar and H.Bhate "Discrete Fourier Transform and Jacobi function identities" J.Math. Phys. 51 023511 (2010) 6. R.Malekar and H.Bhate " Discrete Fourier transform and Riemann identities for theta functions" Applied Mathematical Letters 25 1415-1419 (2012) 7. M.Atakishiyeva, N. Atakishiev and T. Koornwinder "On a q-extension of Mehta s eigenvectors of the finite Fourier transform for q a root of unity. " arxiv :0811.4100v2 [math.ca] 21 May 2009 8. M.Atakishieva and N.Atakishiev "On integral and finite Fourier transforms of q-hermite polynomials" (2009 ) 9. D. Mumford "Tata lectures on theta" v.2
Spectral theorem for the operator roots of unity Suppose U is a bounded operator in a Hilbert space, U = U n = I, q := e 2iπ n, n Z +. n 1 m=0 q m P m, P m = 1 n 1 q jm U j. n n 1 P j = I, Pj 2 = I, UP j = q j P j, P j P m = 0, m j. j=0 Obviously U k = j=1 q jk P j. j=0
Spectral multiplicities m j of the eigenvalue q j of U In a finite dimensional case we have: m j := Tr P j = 1 n n 1 k=0 n 1 f (U) := f (q j )P j. j=1 q kj Tr U k.
Fourier Transform (FT) 1 f (k) = F(f )(k) := 2π R f (x)e ikx dx Inversion formula: f (x) = 1 2π R f (k)e ikx dk. Since (F 2 f )(x) = f ( x) F 4 = I, (F 3 f )(x) = (F 1 f )(x) = f ( x), (1) where I is an identity operator.
Spectral decomposition of F F = P 0 + ip 1 P 2 ip 3, P j = 1 4 3 ( i) jk F k k=0 Therefore any continuous absolutely integrable function f (x) generates the 4 eigenfunctions f j of F: F(f j ) = i j f j, j = 0, 1, 2, 3 f j (x) = f (x) + ( i) j f (x) + ( i) 2j f ( x) + ( i) 3j f ( x). This allows to generate the eigenfunctions of F quite different from the most known ψ n = e x2 2 H n (x), which are also eigenfunctions of the quantum harmonic oscillator H.
Example: f := exp( αx 2 + itx) with α > 0, t C F(f )(x) = 1 e (x t)2 4α 2α, F 2 (f )(x) = f ( x), F 3 (f )(x) = e (x+t)2 4α. Hence we can generate the 2-parametric family of eigenfunctions f j (x, α, t) = P j f of the Fourier operator:
f 0 = e αx2 +itx + e αx 2 itx + 1 e (x t)2 4α 2α e (x+t)2 4α, ( = 2 e αx2 cos tx + 1 e x2 +t 2 ) 4α cosh xt, 2α 2α f 1 = e αx2 +itx e αx 2 itx i 1 e (x t)2 4α e (x+t)2 4α 2α ( = 2i e αx2 sin tx + 1 e x2 +t 2 ) 4α sinh xt, 2α 2α f 2 = e αx2 +itx + e αx 2 itx 1 e (x t)2 4α + e (x+t)2 4α 2α ( = 2 e αx2 cos tx 1 e x2 +t 2 ) 4α cosh xt, 2α 2α f 3 = e αx2 +itx e αx 2 itx + i 1 e (x t)2 4α ie (x+t)2 4α 2α ( = 2i e αx2 sin tx + 1 e x2 +t 2 ) 4α sinh xt. 2α 2α
Since F is linear operator, the linear combinations of the functions above, corresponding to the different values of parameters α and t, are also producing the eigenfunctions. We can even integrate them with respect to t and α with some densities ρ(α, t), with compact support, vanishing for α ɛ, ɛ > 0. In this way we can generate eigenfunctions of F depending on any number of the functional parameters ρ i (α, t).
Creation and annihilation operators a := d d x + x, a := d d x + x, F a = ia F, F a = ia For any eigenfunction f S of F, corresponding to the eigenvalue i k, we have : F(a j f ) = i k+j a j f, F((a ) j f ) = i k j (a ) j f.
Schur matrix and DFT Φ jk (n) = 1 n q jk, j, k = 0,..., n 1, q = e 2iπ n, Φjk = Φ kj, Φ 1 = Φ. The DFT (discrete Fourier transform ) f f describes the action of Φ on the finite sequences,(vectors), f = (f 0,..., f n 1 ) : fk = 1 n 1 f j e 2πijk n, fk±n = f k n j=0
Inversion formula for the DFT and Parceval identity f j = 1 n 1 fk e 2πijk n n k=0 or f = Φ 1 f. n 1 n 1 f j 2 = f j 2. j=0 j=0
Square of the DFT matrix 1 0... 0 0 0 0... 0 1 Φ 2 = 0 0... 1 0...... 0 0 0 1 0... 0 In other words, Φ 2 jm = δ n j,m = Φ 2 mj, (Φ2 f ) j = f n j, Φ 4 = I, Φ 3 = Φ 1 = Φ = Φ. Tr Φ 2 = 3 + ( 1)n 2
The quantity G = n Tr Φ = is the famous Gauss sum: 1 n G(n) = Tr F = 1 n 1 n s=0 q s2 = 1 + ( i)n. 1 i
Spectral decomposition of the DFT operator Φ = 3 i j p j, p j = 1 ( i) jk Φ k, 4 j=0 k 3 p j = I, p j p k = p j δ jk. j=0 The spectral projectors p j are real symmetric matrices!
Generating eigenvectors of DFT from any finite sequence It follows from the spectral decomposition above, that any finite sequence sequence f j, j = 0,..., n 1 generates the eigenvector of Φ : Φv = i k v via a formula v j (k) = f j + ( i) k fj + ( 1) k f n j + ( i) 3k f j. Of course for the sequences f j extended periodically we can replace f n j by f j.
Spectral multiplicities of Φ(n) n = 4m + 2 m(1) = m + 1 m(i) = m m( 1) = m + 1 m( i) = m, n = 4m m(1) = m + 1 m(i) = m m( 1) = m m( i) = m 1 n = 4m + 1 m(1) = m + 1 m(i) = m m( 1) = m m( i) = m n = 4m + 3 m(1) = m + 1 m(i) = m + 1 m( 1) = m + 1 m( i) = m
Characteristic polynomial P(z) and determinant of Φ k=3 P(z) = (z i k ) m k, det F = k=0 k=3 k=1 i km k = i n(n 1) 2 +(n 1) 2 = i 3n2 5n 2 +1,
The explicit formulas for spectral projectors p j given above provide the complete set of eigenvectors of Φ: we have to take m 0 first columns of p 0, m 1 first nontrivial, (since first columns of p 1 and p 3 are formed by zeros), columns of p 1, m 2 first columns of p 2 and m 3 first nontrivial columns of p 3 to get it. The completeness follows from the relation p j = I. The eigenvector v(k, m), equal to the m-th column of p k, has the components v j (m, k) (equal to the matrix elements p mj = p jm of p k ), : 4v j (k, m) = δ jm + ( 1) k δ n j,m + ( i) k qjm 3k q jm + ( i) n n
Φ(2) := 1 2 ( 1 1 1 1 ),, F 2 = I, p 1 = p 3 = 0. Orthogonal matrix O 2 reducing F to the form diag[1, 1] = O 1 FO is O 2 = 2+1 4+2 2 1 4+2 2 1 2 4 2 2 1 4 2 2.
n=4 The DFT matrix Φ is Φ = 1 2 1 1 1 1 1 i 1 i 1 1 1 1 1 i 1 i Here p 3 = 0, since m( i) = m 3 = 0. O 4 : O 1 4 Φ O 4 = diag[1, 1, 1, i] O 4 = 1 2 1 2 1 0 1 0 1 2 1 2 1 0 1 0 1 2.
Generalized Poisson summation formula Let E be the space of the continuous functions absolutely integrable together with their Fourier transform and f E. Poisson summation formula: ab r Z f (b(ar + x)) = 2π m Z f (2πm(ab) 1 )e 2πimx a.
In order to create the DFT structure in the RHS of the Poisson formula, we specify a, b and x and write m as follows : 2π a = n, x = j, b = j = 0,..., n 1; n r Z m = nr + k, r Z, k = 0,..., n 1. The Poisson formula now takes a form: ( ) [ 2π f (nr + j) = 1 n 1 ( )] 2π f (nr + k) q jk. n n n k=0 r Z
Consider a following mapping M : M : E C n, M(f ) := v, v j = r Z f ( ) 2π (nr + j) ; j = 0,..., n 1. n we can interpret Poisson formula as an intertwining relation between FT and DFT. Namely the following statement holds.
Theorem The mapping M is intertwining F and Φ : Φ M = M F. The mapping M is surjective, it maps any eigenvector of F to some eigenvector of Φ. Any continuous, absolutely integrable eigenfunction ψ of F generates the eigenvector M(ψ) of Φ, corresponding to the same eigenvalue : F(ψ) = i k ψ Φ M(ψ) = i k M(ψ), k = 0, 1, 2, 3. In particular case, when f is any eigenfunction of H, this connection was found by Mehta (1987)
Let m Z g m be any absolutely convergent series and g(x) := m Z g m q mx,, q := exp 2iπ n, f j := r Z g rn+j. Obviously f j = f j+n. n 1 j=0 m Z qjl g nm+j = k Z g k q kl = g(l), l = 0,..., n 1; m Z g nm+j = 1 n n 1 l=0 g(l) q lj, q = e 2πi n.
Of course the second formula is immediate consequence of the first obtained by applying the inversion formula for DFT to its LHS. To prove the first formula we set r = nm + j, which implies that j = r nm, q jl = q rl lnm = q rl. Now it is clear that a double sum in the LHS of can be written as a single sum which is exactly the same as the RHS which completes the proof.
Theorem Let p Z g p is any absolutely converging series. Then vector v(k) with the components v j (k), v j (k) = m Z (g nm+j +( 1) k g nm j )+ ( i)k n m Z (g m +( 1) k g m )e 2πimj n, is an eigenvector of the DFT : Φv = i k v.
We can compare now the explicit expressions for the eigenvectors of Φ coming from spectral theorem with those obtained from the formula above. Assuming that g n are the functions depending on some parameters, we can expect that for the functions represented by the RHS of we can get in this way some functional identities. We will discuss here only the identities for generalized theta functions (ν-theta functions) although there are many other applications.
ν-theta functions The function θ(x, τ, ν), which we will shortly call ν-theta function, is defined by the formula θ(x, τ, ν) = m= e πiτm2ν +2πimx, ν Z +, Im τ > 0
It is obvious that θ(x, τ, ν) is an entire function of x satisfying the relation θ(x + 1, τ, ν) = θ(x, τ, ν). θ(x, τ, ν) reduces to the usual theta function θ(x, τ) when ν = 1.
ν-theta function satisfies the following PDE 2 (2π) 2ν 1 ( 1) ν θ τ = i 2ν θ x 2ν.
ν-theta function θ a,b (x, τ, ν) with characterstics [a, b] is introduced by the formulas θ a,b (x, τ, ν) = n Z exp[πiτ(n + a) 2ν + 2πi(n + a)(x + b)]. This function also gives the solution to a same parabolic PDE.
In the case ν = 1, θ a,b (x, τ, ν) reduces to the usual theta function with characteristics θ a,b (x, τ), simply connected with θ(x, τ): θ(x, τ) = θ(x + aτ + b, τ) exp[πia 2 τ + 2πia(x + b)].
Suppose that ( πiτm 2ν g m = exp n 2ν + 2πimx ). n We obviously have the formula g mn+j = θ j,0(x, τ, ν) n m Z
Proposition For any τ with Imτ > 0 vector v(τ, x, k, ν) with the components v j, v j (x, τ, k) = θ j,0(x, τ, ν) + ( 1)k θ j,0(x, τ, ν) ( n ) n + 1 n [( i) k θ j+x n, τ, ν + ( i) 3k θ n 2ν is the eigenvector of the DFT : Φ(n)v(k) = i k v(k). ( x j n, τ n 2ν, ν )]. This result already implies interesting functional relations between the ν-theta functions.
Let e k are the eigenvectors of Φ(n) : Φ(n)e k = i k e k. We obviously have the relation: Φ(n)(e 0 + e 1 + e 2 + e 3 ) = e 0 + ie 1 e 2 ie 3 Applying this relation to the eigenvectors of Φ(n) given by the series above we get the following statement:
Proposition In the case ν = 1 the result formulated below is well known and describes the connection between two bases in the space of entire functions with prescribed quasi periodicity properties (Mumford, Tata lectures on Thetas). The ν-theta functions ( x + k θ n, τ n 2ν, ν ), θ j/n,0 (x, τ, ν), k, j = 0,..., n 1 are connected by the discrete Fourier transform : θ j/n,0 (x, τ, ν) = 1 n θ ( x+k n n 1, τ n 2ν, ν ) = n 1 k=0 q jk θ ( x+k n, τ, ν ), n 2ν j=0 qjk θ j/n,0 (x, τ, ν).
In particular, taking j = 0 k = 0 in the formulas above we we obtain that n 1 θ(x, τ, ν) = 1 n k=0 θ ( x+k n, τ, ν ), n 2ν θ ( x n, τ, ν ) = n 1 n 2ν j=0 θ j/n,0(x, τ, ν)
Unitarity of the discrete Fourier transform implies now the following relations between the ν-theta functions (Parseval identity): n 1 ( x + k θ n k=0 ), τ 2 n 2ν, ν n 1 = n k=0 θ k/n,0 (x, τ, ν) 2
In the simplest case ν = 1, n = 2 we have two eigenvalues of Φ(2),- λ 1 = 1 and λ 2 = 1, and two eigenvectors v1, v( 1) The generaal formula expressing the components of these eigenvectors by means of Jacobi theta functions reads, (j = 0, 1): v j (1) = θ j/2,0 (x, τ)+θ j/2,0 (x, τ)+ 1 2 [θ ( x + j 2, τ 4 v j ( 1) = θ j/2,0 (x, τ)+θ j/2,0 (x, τ) 1 ( x + j [θ 2 2, τ 4 ) ( x j + θ ) + θ 2, τ 4 ( x j 2, τ 4 )], )]
Since Φ(2)(v(1) + v(2)) = v(1) v(2) the formulas above imply two identities: ( x θ(x, τ)+θ 1/2,0 = θ 2 4), τ, θ(x, τ) θ 1/2,0 (x, τ) = θ ( x + 1 2, τ ). 4
Suppose that Φv(x, τ) = v(x, τ). v(x + τ/2, τ) and v(y + 1, τ) are again eigenvectors, corresponding to the same eigenvalue λ 1 = 1. Therefore det [v(x + τ/2, τ), v(y + 1, τ)] = 0. This gives a quadratic relation between theta functions allowing to prove extended and classical quartic Riemann identities.
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