L/L'-Interpolation Problem: Theory & Applications

• The L/L'-interpolation problem is defined as a rational interpolation problem where one seeks a rational function R(z)=P(z)/Q(z) with P in L and Q in L' satisfying prescribed nodal conditions.
• It generalizes to a vector-polynomial module formulation, decomposing solutions via a finite set of generators and employing a height function for precise structural control.
• The framework has deep spectral origins, connecting rational and spline interpolation to inverse spectral analysis of finite band matrices and operator theory applications.

The $L/L'$-Interpolation Problem most classically denotes a rational interpolation problem in which one seeks a rational function $R(z)=P(z)/Q(z)$, with $P\in L$ and $Q\in L'$, satisfying prescribed conditions at finitely many nodes. In the module-theoretic formulation of vector polynomial interpolation, this becomes the $n=2$ case of the system

$\sum_{k=1}^n a_k(j)P_k(z_j)=0,$

and the resulting theory gives a complete structural description of all solutions in terms of generators of a polynomial module (Kudryavtsev et al., 2014). Other parts of the literature use the same label for spline interpolation attached to differential operators, limiting real interpolation spaces, and duality-based boundary-value solvability transfer (Kounchev et al., 2021, Doktorski et al., 2022, Dindoš et al., 18 Jan 2026). This suggests that the expression names a family of interpolation problems organized by paired spaces, operators, or dual regimes rather than a single universally fixed definition.

1. Classical rational formulation

In rational interpolation theory, the $L/L'$-interpolation problem is the problem of finding polynomials $P\in L$ and $Q\in L'$ such that

$\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$

or equivalently

$R(z)=P(z)/Q(z)$0

In the notation of the vector-polynomial problem, this is exactly the case $R(z)=P(z)/Q(z)$1 of

$R(z)=P(z)/Q(z)$2

with

$R(z)=P(z)/Q(z)$3

The ratio then satisfies

$R(z)=P(z)/Q(z)$4

The $R(z)=P(z)/Q(z)$5 theory is identified in (Kudryavtsev et al., 2014) with the rational interpolation problem, the Cauchy–Jacobi problem, and multipoint Padé approximants.

The importance of this identification is structural. Instead of studying only a numerator–denominator pair $R(z)=P(z)/Q(z)$6, the theory enlarges the setting to an $R(z)=P(z)/Q(z)$7-tuple of scalar polynomials. In that formulation, the pair $R(z)=P(z)/Q(z)$8 is replaced by a polynomial module and its solution submodule, so that rational interpolation appears as the first nontrivial member of a broader module-theoretic hierarchy (Kudryavtsev et al., 2014).

2. Vector-polynomial generalization

The general theory is developed in the space

$R(z)=P(z)/Q(z)$9

which is a complex vector space and also a module over the ring of scalar polynomials. For a scalar polynomial $P\in L$0,

$P\in L$1

A central device is the height function

$P\in L$2

defined by

$P\in L$3

It satisfies the fundamental covariance relation

$P\in L$4

This makes height the basic grading compatible with the module structure (Kudryavtsev et al., 2014).

The interpolation data consist of nodes $P\in L$5, not necessarily distinct, and coefficient vectors

$P\in L$6

satisfying

$P\in L$7

The interpolation problem is to find scalar polynomials $P\in L$8 such that

$P\in L$9

In vector form,

$Q\in L'$0

The same conditions are encoded by rank-one Hermitian nonnegative matrices $Q\in L'$1, built from $Q\in L'$2, so that

$Q\in L'$3

The solution set is

$Q\in L'$4

and it is a submodule of $Q\in L'$5 (Kudryavtsev et al., 2014).

This formulation contains the classical $Q\in L'$6 problem as the two-component case, but it also admits genuinely higher-dimensional constraints. A plausible implication is that the distinction between interpolation of ratios and interpolation of linear relations disappears once the problem is expressed module-theoretically.

3. Generator theory and complete characterization

The main structural result is a complete characterization of $Q\in L'$7 in terms of generators. For any $Q\in L'$8,

$Q\in L'$9

is the cyclic submodule generated by $n=2$0. If $n=2$1, then every nonzero $n=2$2 has height

$n=2$3

for some $n=2$4, so the heights in $n=2$5 lie in a single congruence class modulo $n=2$6 (Kudryavtsev et al., 2014).

The first generator $n=2$7 is defined as a nonzero element of $n=2$8 with minimal height: $n=2$9 Recursively, for $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$0, the $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$1-th generator $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$2 is chosen to have minimal height in

$\sum_{k=1}^n a_k(j)P_k(z_j)=0,$3

Because distinct generators occupy distinct congruence classes modulo $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$4, the sum is direct: $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$5 Moreover,

$\sum_{k=1}^n a_k(j)P_k(z_j)=0,$6

The height theory is sharp. For every integer $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$7, there exists $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$8 with $\sum_{k=1}^n a_k(j)P_k(z_j)=0,$9. The first generator satisfies

$L/L'$0

For the full generating system,

$L/L'$1

and the inequality becomes an exact formula when $L/L'$2: $L/L'$3 The proof uses the determinant

$L/L'$4

its vanishing at the interpolation nodes, and the existence of infinitely many points where $L/L'$5 are linearly independent (Kudryavtsev et al., 2014).

The final description is basis-like: $L/L'$6 where the $L/L'$7 are scalar polynomials. Equivalently,

$L/L'$8

There is no simple finite-dimensional “dimension” in the usual vector-space sense because the module is infinite-dimensional, but the number of generators is exactly $L/L'$9, which is the rank of $P\in L$0 as a module (Kudryavtsev et al., 2014).

For $P\in L$1, this recovers the rational $P\in L$2 case. There are two generators $P\in L$3, every solution has the form

$P\in L$4

and the heights satisfy

$P\in L$5

A common misconception is to treat the solution set as an ordinary finite-dimensional space of interpolation data; the module description shows instead that it is an infinite-dimensional object with a finite generating set (Kudryavtsev et al., 2014).

4. Spectral origin and inverse problems

The motivation for the vector-polynomial theory is spectral analysis of finite band matrices. An $P\in L$6 band matrix with bandwidth $P\in L$7 has nonzero entries confined to the main diagonal and $P\in L$8 diagonals above and below it. For wider band matrices, the recurrence relations involve $P\in L$9 consecutive terms of polynomials, and these are encoded by an $Q\in L'$0-vector polynomial $Q\in L'$1 whose components represent consecutive polynomial values in the recurrence (Kudryavtsev et al., 2014).

In that setting, the spectral conditions impose relations of the form

$Q\in L'$2

The nodes $Q\in L'$3 are typically eigenvalues or spectral points, the coefficients $Q\in L'$4 encode spectral weights or boundary conditions, and the vector polynomial $Q\in L'$5 encodes the system of polynomials generated by the recurrence relation associated with the band matrix. Solving the interpolation problem therefore yields vector polynomials whose structure reflects the band matrix’s spectral characteristics.

For inverse problems, the data are reversed. Given spectral data, one constructs the generators $Q\in L'$6 of $Q\in L'$7, and from these generators one can recover the recurrence coefficients, hence the band matrix entries. The paper explicitly states that the interpolation theory was developed with applications to inverse spectral analysis of symmetric band matrices with $Q\in L'$8 diagonals (Kudryavtsev et al., 2014).

This spectral origin explains why the $Q\in L'$9-component formulation is not merely a formal generalization of rational interpolation. It is tuned to recurrence systems and to the reconstruction of matrix coefficients from spectral information.

5. Terminological range in the literature

The same expression is used in several mathematically distinct settings. The following comparison records the usages documented in the cited papers.

Literature	Meaning of “$\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$0-interpolation”	Characteristic relation
Rational/vector polynomial theory (Kudryavtsev et al., 2014)	Rational interpolation or its $\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$1-component module-theoretic generalization	$\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$2
$\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$3-splines for fourth-order operators (Kounchev et al., 2021)	Interpolation and smoothing by natural $\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$4-splines with $\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$5	$\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$6
Limiting real interpolation (Doktorski et al., 2022, Doktorski, 2020)	Reiteration with slowly varying functions and limiting $\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$7- or $\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$8-spaces	$\frac{P(z_j)}{Q(z_j)}=\beta_j,\qquad j=1,\dots,N,$9, $R(z)=P(z)/Q(z)$00, $R(z)=P(z)/Q(z)$01
Parabolic boundary problems (Dindoš et al., 18 Jan 2026)	Interpolation of solvability of the $R(z)=P(z)/Q(z)$02 Neumann problem using $R(z)=P(z)/Q(z)$03 Dirichlet solvability for the adjoint	$R(z)=P(z)/Q(z)$04
Ordered ideals between $R(z)=P(z)/Q(z)$05 and $R(z)=P(z)/Q(z)$06 (Mekler, 2018)	Characterization of interpolation spaces via conditional expectations and doubly stochastic projections	$R(z)=P(z)/Q(z)$07

In the terminology of (Kounchev et al., 2021), the $R(z)=P(z)/Q(z)$08-interpolation problem can be viewed as interpolation and smoothing splines associated with a fourth-order differential operator $R(z)=P(z)/Q(z)$09, together with an associated second-order operator $R(z)=P(z)/Q(z)$10 (there denoted $R(z)=P(z)/Q(z)$11). Natural boundary conditions are

$R(z)=P(z)/Q(z)$12

and the unknown transformed nodal values satisfy a tridiagonal system

$R(z)=P(z)/Q(z)$13

which yields fast $R(z)=P(z)/Q(z)$14 algorithms under diagonal dominance assumptions (Kounchev et al., 2021).

In the theory of limiting real interpolation, the same label is tied to endpoint reiteration. Standard spaces

$R(z)=P(z)/Q(z)$15

are supplemented by limiting spaces

$R(z)=P(z)/Q(z)$16

and the point of the theory is that reiteration at $R(z)=P(z)/Q(z)$17 or $R(z)=P(z)/Q(z)$18 remains inside this enlarged scale (Doktorski et al., 2022, Doktorski, 2020).

In parabolic PDE, the phrase refers to interpolation of solvability. Assuming solvability of the $R(z)=P(z)/Q(z)$19 Neumann problem for

$R(z)=P(z)/Q(z)$20

and solvability of the $R(z)=P(z)/Q(z)$21 Dirichlet problem for the adjoint, one first proves Hardy-space solvability and then obtains solvability in $R(z)=P(z)/Q(z)$22 for all $R(z)=P(z)/Q(z)$23 (Dindoš et al., 18 Jan 2026).

This suggests that “$R(z)=P(z)/Q(z)$24-interpolation problem” is not a single standardized term across analysis. The most stable use is the rational-interpolation meaning, but several adjacent literatures employ it for paired operators, paired endpoint scales, or paired dual solvability regimes.

6. Dual, module, and operator formulations

Several related papers recast interpolation constraints as linear functionals, module decompositions, or predual pairings.

In linear combination interpolation, one fixes a polynomial

$R(z)=P(z)/Q(z)$25

and studies conditions such as

$R(z)=P(z)/Q(z)$26

or, more generally,

$R(z)=P(z)/Q(z)$27

Every function analytic near the roots of $R(z)=P(z)/Q(z)$28 admits a unique representation

$R(z)=P(z)/Q(z)$29

with $R(z)=P(z)/Q(z)$30 analytic near $R(z)=P(z)/Q(z)$31. The module is generated by two operations: substitution of $R(z)=P(z)/Q(z)$32 and multiplication by monomials $R(z)=P(z)/Q(z)$33, $R(z)=P(z)/Q(z)$34 (Alpay et al., 2014). This provides a one-point vector-valued reduction of a multipoint interpolation problem.

A different finite-dimensional perspective is the linear-algebraic dual-space formulation. In that framework, interpolation conditions are linear functionals in $R(z)=P(z)/Q(z)$35, and a general existence-and-uniqueness theorem states that if the interpolation functionals form a basis of $R(z)=P(z)/Q(z)$36 and $R(z)=P(z)/Q(z)$37, then the interpolation problem has a unique solution in $R(z)=P(z)/Q(z)$38 (Arutyunov et al., 21 Jun 2026). Dual basis functions are obtained from the inverse interpolation matrix, so mixed value-and-derivative schemes, Hermite splines, and trigonometric interpolation are all handled by the same matrix mechanism.

In Lipschitz interpolation, the duality becomes predual geometry. For a pointed metric space $R(z)=P(z)/Q(z)$39, the interpolation operator

$R(z)=P(z)/Q(z)$40

is paired with

$R(z)=P(z)/Q(z)$41

on the Lipschitz-free space, where

$R(z)=P(z)/Q(z)$42

The surjectivity of $R(z)=P(z)/Q(z)$43 is equivalent to the statement that the molecules $R(z)=P(z)/Q(z)$44 form an $R(z)=P(z)/Q(z)$45-equivalent basic sequence in $R(z)=P(z)/Q(z)$46, and the existence of a Beurling set is equivalent to the existence of a bounded projection from $R(z)=P(z)/Q(z)$47 onto that $R(z)=P(z)/Q(z)$48-copy (Jiménez-Vargas et al., 23 Mar 2025).

Taken together, these formulations point to a common structural pattern. A plausible implication is that many problems called $R(z)=P(z)/Q(z)$49-interpolation become transparent once the interpolation conditions are transferred from function values to module generators, dual bases, or predual molecules. In that sense, the vector-polynomial theory of (Kudryavtsev et al., 2014) occupies a central position: it turns a classical rational interpolation problem into a module decomposition problem with exact arithmetic control of generators, heights, and parametrization.