A Takenaka-Malmquist system, also called a Malmquist-Takenaka system, is an orthonormal set of rational functions in the Hardy
space 
of the open unit disk 
. It provides generalized
Fourier series adapted to a chosen sequence of points
in 
(Takenaka 1925, Abdullayev and Savchuk 2024).
For a sequence 
, 
, ... in 
, with repetitions allowed, define
 |
(1)
|
where 
,
the empty product is 1, and the bar denotes complex
conjugate. Each factor in the product is a Blaschke
factor, so the product is a finite Blaschke
product up to a constant of absolute
value 1.
With the boundary inner product
 |
(2)
|
where boundary values are understood as radial limits almost everywhere, the functions satisfy
 |
(3)
|
where 
is the Kronecker delta. The system is complete
in the Hardy space 
iff
 |
(4)
|
Under this condition, every 
has the expansion
 |
(5)
|
with convergence in the 
norm (Abdullayev and Savchuk 2024).
In particular, choosing 
for every 
gives the monomials 
.
Xie and Mo (2026) combine these systems with the MUSIC algorithm to choose poles for approximating transfer functions.
