Rational aproximation in L₂

arl2 [2]

J. Grimm (MIAOU, INRIA)

The system of equations is obtained in the following way

g=S_i=0^m g_i zⁱ,
q=zⁿ+S_i=0^n-1 q_i zⁱ
q=zⁿq(1/z)
By Euclidean division, g q = V q+R. y=||V||²,
If q is complex, the parameters are a_i and b_i, the real and imaginary parts of q_i. Take

x_i=

І y

І q_i

y_i=

І y

І a_i

z_i=

І y

І b_i

for i=0,..., n-1.

The equations are

x₀=ЇЇЇ=x_n-1=0 (1)

y₀=ЇЇЇ=y_n-1=z₀=ЇЇЇ=z_n-1=0 (2)

Its an approximation problem in the analytic space L₂(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 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 L² pour les systХmes dynamiques linИaires. ThХse de doctorat, UniversitИ de Nice, 1987.
[2]: D. Bini and B. Mourrain. Polynomial test suite. 1996.

Rational aproximation in L2

References

Rational aproximation in L₂