Spectral Interpolation Theorem

• Spectral interpolation theorem is a framework that uses spectral data—such as eigenvalues, Jordan blocks, and kernel spectra—to formulate interpolation problems.
• It bridges local interpolation conditions with global solutions via techniques like Oka theory and holomorphic liftings in complex domains.
• The theorem also underpins numerical superconvergence and operator interpolation, extending classical methods to high-dimensional and nonlinear settings.

Searching arXiv for recent and foundational papers using the phrase and closely related formulations. arxiv_search("^^^^^^^^3all:\3 interpolation theorem\"^^^^^^^^3 OR ti:\3^^^^^^^^"spectral interpolation\"^^^^^^^^3 OR abs:\3^^^^^^^^"spectral interpolation theorem\"", ^^^^3 OR ti:\3all:\3^^^^) The expression spectral interpolation theorem does not denote a single universally fixed statement. In the cited arXiv literature, it names a family of results in which interpolation is controlled by spectral data: characteristic polynomials and Jordan blocks for the spectral ball (&&&^^^^3all:\3^^^^&&&), orthogonal-polynomial or Hermite structure in high-order approximation (&&&^^^^3 OR ti:\3^^^^&&&, &&&^^^^3 OR abs:\3^^^^&&&), resolvent and eigenvalue data for operator interpolation (Colombo et al., 2024, Gowda, 2018, Gowda et al., 2019), and Fourier or kernel spectra in discrete sampling and RKHS theory (Bitzer et al., 22 Aug 2025, Haldar, 2024, Siripuram et al., 2014). The most literal theorem under that name in matrix-valued complex analysis is the spectral Nevanlinna–Pick lifting theorem, which reduces a global interpolation problem in the spectral ball to local jet conditions in the symmetrized polydisc (&&&^^^^3all:\3^^^^&&&).

^^^^3 OR ti:\3^^^^. Spectral ball, symmetrized polydisc, and the matrix-valued problem

For the spectral-ball formulation, the basic domain is

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^

where PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^ is the spectral radius. For PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3^^^^, $\Omega_1=\mathbb{D}$. If $\sigma_1,\dots,\sigma_n$ are the elementary symmetric polynomials and $\mathrm{EV}$ sends a matrix to the unordered list of its eigenvalues, then

$$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$

and

$$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$

The map $\pi=(\pi_1,\dots,\pi_n)$ is a holomorphic surjection

$\pi:\Omega_n\longrightarrow G_n,$

where

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^

is the symmetrized polydisc (&&&^^^^3all:\3^^^^&&&).

The spectral Nevanlinna–Pick problem asks for a holomorphic map

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^

with prescribed values PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3^^^^ at distinct points PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^. The weaker lifting problem prescribes only a holomorphic map PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4 satisfying PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5, and asks whether there exists PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6 such that

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7

The paper states that solving this lifting problem reduces the spectral Nevanlinna–Pick problem to an interpolation problem in the taut domain PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8, which is better behaved from the point of view of hyperbolic geometry (&&&^^^^3all:\3^^^^&&&).

A generic fibre of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9, over a point whose eigenvalues are all distinct, consists of a single similarity class of matrices and is an PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3all:\3^^^^-homogeneous manifold. This fibre geometry is one reason that interpolation in PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3 OR ti:\3^^^^ is naturally expressed through the spectral data encoded by PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3 OR abs:\3^^^^ rather than through matrix entries themselves (&&&^^^^3all:\3^^^^&&&).

^^^^3 OR abs:\3^^^^. The lifting theorem as a local jet interpolation criterion

The main statement is Theorem ^^^^3 OR ti:\3^^^^.4:

The spectral Nevanlinna-Pick lifting problem can be solved if and only if it can be solved locally around the interpolation points which means that PRESERVED_PLACEHOLDER_^^^^3 OR abs:\33^^^^ holds for each interpolation point.

Here PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3^^^^4 is the local jet condition from Proposition ^^^^3 OR ti:\3^^^^.^^^^3 OR abs:\3^^^^. If PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3^^^^5, PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3^^^^6, and PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3^^^^7, with Jordan decomposition

PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3^^^^8

then a local holomorphic lift exists near PRESERVED_PLACEHOLDER_^^^^3 OR abs:\3^^^^9 with prescribed value $\Omega_1=\mathbb{D}$^^^^3all:\3^^^^ if and only if

$\Omega_1=\mathbb{D}$^^^^3 OR ti:\3^^^^

Equivalently, there exists a holomorphic $\Omega_1=\mathbb{D}$^^^^3 OR abs:\3^^^^ with $\Omega_1=\mathbb{D}$3 and $\Omega_1=\mathbb{D}$4 for all $\Omega_1=\mathbb{D}$5 if and only if, for every $\Omega_1=\mathbb{D}$6, the local jet conditions $\Omega_1=\mathbb{D}$7 computed from the Jordan blocks of $\Omega_1=\mathbb{D}$8 are satisfied (&&&^^^^3all:\3^^^^&&&).

This is the precise sense in which the global matrix-valued interpolation problem becomes a jet interpolation problem into $\Omega_1=\mathbb{D}$9. The polynomial

$\sigma_1,\dots,\sigma_n$^^^^3all:\3^^^^

plays the role of the characteristic polynomial of any lift $\sigma_1,\dots,\sigma_n$^^^^3 OR ti:\3^^^^ with $\sigma_1,\dots,\sigma_n$^^^^3 OR abs:\3^^^^. The local vanishing-order condition forces $\sigma_1,\dots,\sigma_n$3 to match the eigenvalue multiplicities and nilpotent structure encoded by the Jordan form of the target matrix (&&&^^^^3all:\3^^^^&&&).

For $\sigma_1,\dots,\sigma_n$4, $\sigma_1,\dots,\sigma_n$5, $\sigma_1,\dots,\sigma_n$6, and $\sigma_1,\dots,\sigma_n$7 is the identity. The Jordan form is trivial, the condition $\sigma_1,\dots,\sigma_n$8 is vacuous, and the lifting problem becomes the classical scalar Nevanlinna–Pick problem. This identifies the spectral theorem as a genuine higher-dimensional generalization rather than a different problem of unrelated type (&&&^^^^3all:\3^^^^&&&).

3. Oka-theoretic mechanism and geometric consequences

The lifting problem admits a section-theoretic reformulation. For a holomorphic $\sigma_1,\dots,\sigma_n$9, define the pullback space

$\mathrm{EV}$^^^^3all:\3^^^^

with projection $\mathrm{EV}$^^^^3 OR ti:\3^^^^. A lifting $\mathrm{EV}$^^^^3 OR abs:\3^^^^ is equivalent to a holomorphic section

$\mathrm{EV}$3

The theorem therefore becomes a statement about the existence of global holomorphic sections from local ones (&&&^^^^3all:\3^^^^&&&).

The proof uses Forstnerič’s Oka principle for branched maps. Over the set of cyclic matrices, $\mathrm{EV}$4 is a holomorphic submersion, and one constructs a dominating spray through the conjugation action

$\mathrm{EV}$5

whose derivative at $\mathrm{EV}$6 is

$\mathrm{EV}$7

This surjects onto the vertical tangent space and shows that $\mathrm{EV}$8 is an elliptic submersion off the branching locus. Pullbacks preserve this elliptic structure off the corresponding branching set, and Proposition ^^^^3 OR ti:\3^^^^.^^^^3 OR abs:\3^^^^ supplies the required local holomorphic liftings at the interpolation nodes (&&&^^^^3all:\3^^^^&&&).

A second ingredient is topological triviality. The fibres of $\mathrm{EV}$9 are $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$^^^^3all:\3^^^^-connected: each fibre, each stratum, and each connected component of a stratum is $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$^^^^3 OR ti:\3^^^^-connected, hence path connected. This removes topological obstructions to producing a continuous section with prescribed values at the nodes. The Oka theorem then deforms that continuous section to a holomorphic section while fixing its values to given order at the interpolation set and avoiding the branching locus off the nodes (&&&^^^^3all:\3^^^^&&&).

The same geometric analysis yields Corollary 3.^^^^3 OR abs:\3^^^^: for $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$^^^^3 OR abs:\3^^^^, the spectral ball $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$3 is a union of immersed complex lines, and therefore there exists no bounded from above strictly plurisubharmonic function on $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$4. In interpolation language, this helps explain why the reduction to $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$5 is natural: the spectral ball is non-hyperbolic, whereas the symmetrized polydisc is the taut base domain carrying the relevant spectral information (&&&^^^^3all:\3^^^^&&&).

4. Numerical spectral interpolation and superconvergence

In numerical analysis, the phrase usually denotes exponential convergence of spectral interpolants together with the identification of points where derivatives or function values converge faster than the global norm suggests. On $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$6, the paper on superconvergence of spectral interpolation studies analytic $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$7 extended to a Bernstein ellipse $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$8 and uses contour integral remainder formulas for Chebyshev-based interpolation. The global error behaves like $$\pi_j := \sigma_j\circ \mathrm{EV},\qquad j=1,\dots,n,$$9 with polynomial prefactors in $$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$^^^^3all:\3^^^^, while derivative superconvergence occurs at zeros of derivatives of the nodal polynomial $$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$^^^^3 OR ti:\3^^^^ (&&&^^^^3 OR ti:\3^^^^&&&).

For interpolation at the zeros of $$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$^^^^3 OR abs:\3^^^^, one has $$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$3. First-derivative superconvergence occurs at the zeros of

$$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$4

namely

$$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$5

Second-derivative superconvergence occurs at the zeros of $$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$6, equivalently at solutions of

$$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$7

The same paper treats Chebyshev–Lobatto, Chebyshev–Radau, Legendre, and derivative-interpolation variants, and shows that when one interpolates first derivatives, function values superconverge at extremals or closely related nodes of the same orthogonal polynomial family (&&&^^^^3 OR ti:\3^^^^&&&).

On $$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$8, Hermite spectral interpolation uses the zeros of $$\chi_A(\lambda)=\det(\lambda I-A)=\lambda^n+\sum_{j=1}^n(-1)^j\,\pi_j(A)\lambda^{n-j}.$$9, equivalently of $\pi=(\pi_1,\dots,\pi_n)$^^^^3all:\3^^^^, as interpolation nodes in

$\pi=(\pi_1,\dots,\pi_n)$^^^^3 OR ti:\3^^^^

For analytic $\pi=(\pi_1,\dots,\pi_n)$^^^^3 OR abs:\3^^^^ in a strip $\pi=(\pi_1,\dots,\pi_n)$3, the Hermite interpolant $\pi=(\pi_1,\dots,\pi_n)$4 satisfies

$\pi=(\pi_1,\dots,\pi_n)$5

First-derivative superconvergence occurs at the zeros $\pi=(\pi_1,\dots,\pi_n)$6 of $\pi=(\pi_1,\dots,\pi_n)$7, and second-derivative superconvergence at the zeros $\pi=(\pi_1,\dots,\pi_n)$8 of $\pi=(\pi_1,\dots,\pi_n)$9. The sharp gain is a factor $\pi:\Omega_n\longrightarrow G_n,$^^^^3all:\3^^^^: $\pi:\Omega_n\longrightarrow G_n,$^^^^3 OR ti:\3^^^^ The same superconvergence nodes reappear for Hermite spectral collocation under the assumption $\pi:\Omega_n\longrightarrow G_n,$^^^^3 OR abs:\3^^^^ (&&&^^^^3 OR abs:\3^^^^&&&).

Setting	Interpolation nodes	Superconvergence points
Chebyshev value interpolation	zeros of $\pi:\Omega_n\longrightarrow G_n,$3, $\pi:\Omega_n\longrightarrow G_n,$4, $\pi:\Omega_n\longrightarrow G_n,$5	zeros of $\pi:\Omega_n\longrightarrow G_n,$6 and $\pi:\Omega_n\longrightarrow G_n,$7
Derivative interpolation on $\pi:\Omega_n\longrightarrow G_n,$8	Gauss, Lobatto, or Radau nodes	extremals or related nodes of the same family
Hermite interpolation on $\pi:\Omega_n\longrightarrow G_n,$9	zeros of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3all:\3^^^^ or PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3 OR ti:\3^^^^	zeros of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3 OR abs:\3^^^^ and PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\33^^^^

A frequent misconception is that superconvergence merely means “faster global convergence.” The cited papers state a stricter phenomenon: the global spectral rate is unchanged, but at distinguished points determined by the orthogonal basis, the leading contribution to the error cancels, and the local error gains an additional algebraic factor (&&&^^^^3 OR ti:\3^^^^&&&, &&&^^^^3 OR abs:\3^^^^&&&).

5. Operator, Jordan-algebra, and nonlinear forms of spectral interpolation

In operator theory on quaternionic Banach spaces, the spectral object is the quadratic polynomial

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^4

and interpolation is phrased through the pseudo PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^5-resolvent PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^6. Under the structural assumptions of Definition 3.^^^^3 OR abs:\3^^^^, Theorem 3.5 gives a resolvent-based characterization of

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^7

while Theorems 3.6 and 3.7 state that, for every PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^8,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^9

This is a quaternionic analogue of the classical fact that domains of powers interpolate between one another (Colombo et al., 2024).

In Euclidean Jordan algebras, spectral interpolation is expressed through eigenvalue vectors and the spectral PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3all:\3^^^^-norm

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3 OR ti:\3^^^^

The interpolation theorem of Theorem 6.^^^^3 OR ti:\3^^^^ states that if

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3 OR abs:\3^^^^

then for every linear map PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\33^^^^,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^4

A Riesz–Thorin type refinement interpolates between two different spectral norms: PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^5 with PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^6 (Gowda, 2018, Gowda et al., 2019).

A distinct spectral functional appears in the PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^7-trace program, where

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^8

is the PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^9-th elementary symmetric polynomial of the eigenvalues of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3all:\3^^^^, and

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3 OR ti:\3^^^^

The key interpolation input is a PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3 OR abs:\3^^^^-trace Stein–Hirschman inequality for holomorphic matrix families PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\33^^^^, obtained by lifting to the exterior algebra. This yields joint concavity of

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3^^^^4

for the Lieb-allowed range of parameters, and also the concavity of

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3^^^^5

on PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3^^^^6 (&&&^^^^3 OR abs:\3^^^^8&&&).

The nonlinear interpolation theorem of a different paper extends the Riesz–Thorin paradigm from linear operators to analytic maps on Sobolev balls: PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3^^^^7 For PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3^^^^8,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR abs:\3^^^^9

The paper applies this to spectral quantities of one-dimensional Schrödinger operators, obtaining fractional-order asymptotics for the Floquet exponents PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33all:\3^^^^ and for periodic eigenvalue combinations from integer-order estimates (&&&^^^^3 OR abs:\3^^^^9&&&).

6. RKHS interpolation spaces and weighted spectral priors

For reproducing kernel Hilbert spaces, the interpolation couple is

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33 OR ti:\3^^^^

where PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33 OR abs:\3^^^^ is an RKHS with bounded measurable kernel PRESERVED_PLACEHOLDER_^^^^3 OR ti:\333^^^^, compactly embedded into PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^4, and PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^5 has eigenvalues PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^6 and eigenfunctions PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^7. The spectral representation theorem states that for PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^8 and PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^9,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4^^^^3all:\3^^^^

with equivalence of norms. Writing

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4^^^^3 OR ti:\3^^^^

the norm of the interpolation space is equivalent to a dyadic PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4^^^^3 OR abs:\3^^^^-sum of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^43-blocks built from the weighted coefficients PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^44. For PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^45, this reduces to the familiar Hilbertian formula

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^46

(Bitzer et al., 22 Aug 2025).

The same paper gives an exact criterion for the embedding

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^47

namely uniform boundedness of the dual sequence norm of

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^48

for almost every PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^49. When this embedding holds, the interpolation space admits a Banach-space-of-functions realization with continuous point evaluations (Bitzer et al., 22 Aug 2025).

A different spectral-interpolation line generalizes Shannon reconstruction through weighted Hilbert spaces. For PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5^^^^3all:\3^^^^-bandlimited signals and a weight PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5^^^^3 OR ti:\3^^^^ satisfying

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5^^^^3 OR abs:\3^^^^

the norm is

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^53

The minimum-PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^54-norm interpolant from samples PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^55, PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^56, has the form

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^57

with coefficients determined by the Gram matrix

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^58

When PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^59 on PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6^^^^3all:\3^^^^ and PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6^^^^3 OR ti:\3^^^^, PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6^^^^3 OR abs:\3^^^^ becomes the sinc kernel and the formula reduces to truncated Shannon interpolation. When PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^63, it coincides with the LMMSE or GP-regression interpolator for a stationary process with PSD PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^64. The paper emphasizes that this weighted framework is particularly useful for interpolating sub-Nyquist data (Haldar, 2024).

7. Discrete Fourier interpolation, vector polynomials, and band-matrix spectral data

On the finite cyclic group PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^65, the bandlimited space

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^66

admits interpolation from a sampling set PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^67 exactly when the submatrix

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^68

is invertible. Orthogonal interpolation occurs when this submatrix is unitary up to scaling. Writing PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^69, an index set PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7^^^^3all:\3^^^^ is an orthogonal sampling set for PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7^^^^3 OR ti:\3^^^^ if and only if PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7^^^^3 OR abs:\3^^^^ and

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^73

This condition is encoded by the difference graph PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^74, whose vertices are PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^75 and whose edges join PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^76 whenever PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^77. Then PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^78 is an orthogonal sampling set if and only if PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^79 is a maximum clique in PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8^^^^3all:\3^^^^. For PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8^^^^3 OR ti:\3^^^^, the paper proves that

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8^^^^3 OR abs:\3^^^^

that the maximum clique size of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^83 is PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^84, and that PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^85 has an orthogonal sampling set if and only if PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^86 tiles PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^87 (Siripuram et al., 2014).

For PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^88-dimensional vector polynomials,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^89

the paper on vector-polynomial interpolation studies the condition

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9^^^^3all:\3^^^^

equivalently

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9^^^^3 OR ti:\3^^^^

with PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9^^^^3 OR abs:\3^^^^ nonnegative rank-one Hermitian matrices. The solution set PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^93 is organized by the height

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^94

The paper constructs PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^95 generators PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^96 such that

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^97

and proves the full structure theorem

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^98

where PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^99. The paper states that these results generalize rational interpolation and have applications to direct and inverse spectral analysis of band matrices (Kudryavtsev et al., 2014).

Taken together, these discrete and algebraic results show that the phrase spectral interpolation theorem often signals the same structural pattern in very different settings: interpolation is possible precisely when a spectral object—Fourier support, an idempotent zero set, a kernel eigenexpansion, a Jordan decomposition, or a rank-one spectral datum—satisfies a compatibility condition strong enough to produce a canonical basis of interpolants.