Rational aproximation in L_{2}

                                                           arl2 [2]
J. Grimm (MIAOU, INRIA)

The system of equations is obtained in the following way

• g = ∑_{i=0}^{m} g_{i} z^{i},

• q = z^{n} + ∑_{i=0}^{n−1} q_{i} z^{i}

• q̃ = z^{n}\overline{q}(1/z)

• By Euclidean division, gq̃ = V q + R. ψ = ||V ||^{2},

• If q is complex, the parameters are a_{i} and b_{i}, the real and imaginary parts of q_{i}. Take

\begin{matrix}x_{i} = \frac{∂ψ}{∂q_{i}} \\ y_{i} = \frac{∂ψ}{∂a_{i}} \\ z_{i} = \frac{∂ψ}{∂b_{i}} \end{matrix}

for i = 0, …, n − 1.

The equations are

x_{0} = ⋯ = x_{n−1} = 0  (1)

or

y_{0} = ⋯ = y_{n−1} = z_{0} = ⋯ = z_{n−1} = 0  (2)

Its an approximation problem in the analytic space L_{2}(ℂ), (see [1]). It has applications in Signal Processing.

Example 1:

# complex case, real and imaginary parts of $g_i$.
-0.398808768921536+I*(-0.014562755402622),0.181660872059936+I*(0.0165755065464342),
-0.0263001643656528+I*(-0.00882979179591983),-0.0446385628766925+I*(-0.00280946428836964),
0.0665382103437165+I*(0.0110027342600263),-0.0582110239179303+I*(-0.0129857778487792),
0.029732011018205+I*(0.00927607754783639),0.00015411195559675+I*(-0.00221536041038356),
-0.0184382220445118+I*(-0.00475066776583862),0.0261407621185394+I*(0.00899137685465235),
-0.0258416616507867+I*(-0.00987449719197549),0.0168948477652733+I*(0.00781857567828262),
-0.00325222404009983+I*(-0.0035177449999695),-0.00747284988204079+I*(-0.00157755070130404),
0.012790951437277+I*(0.00547618172625035),-0.0146654001533705+I*(-0.00698272221377631),
0.0129080742396486+I*(0.00614034618689353),-0.00663105426207005+I*(-0.00345120699056984),
-0.00105121062492546+I*(-0.000414060645842936),0.00608257640330125+I*(0.00426926773807864),
...

An extended list of coefficients is accessible \underline{arl2001.mpl}here

« For m > n, in the first case, n=2 should be easy. Problem c has 1500 solutions (estimate) case n=15 should have something like 150000000 solutions. Second example, n=8, problem b. Something like 1000000000000000 solutions » (J. Grimm).

The problems are a) Solve x_{i} = 0 (real case), system (1).

b) Solve y_{i} = z_{i} = 0 (complex case), system (1).

c) Same, but look only at solutions q such that q(z) = 0 implies |z| < 1.

References

[1] L. Baratchart. Sur l’approximation rationnelle L^{2} pour les systèmes dynamiques linéaires. Thèse de doctorat, Université de Nice, 1987.

[2] D. Bini and B. Mourrain. Polynomial test suite. 1996.