---
title: Spectral Interpolation Theorem
url: https://www.emergentmind.com/topics/spectral-interpolation-theorem
type: topic
---

# Spectral Interpolation Theorem

Searching arXiv for recent and foundational papers using the phrase and closely related formulations.
arxiv_search("all:\"spectral interpolation theorem\" OR ti:\"spectral interpolation\" OR abs:\"spectral interpolation theorem\"", 10)
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 [1501.07145], orthogonal-polynomial or Hermite structure in high-order approximation [1204.5813], [2507.15350], resolvent and eigenvalue data for operator interpolation [2402.10383], [1809.05417], [1905.02572], and Fourier or kernel spectra in discrete sampling and RKHS theory [2508.16492], [2410.21184], [1411.7086]. 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 [1501.07145].

## 1. Spectral ball, symmetrized polydisc, and the matrix-valued problem

For the spectral-ball formulation, the basic domain is
\[
\Omega_n := \{ A \in \mathrm{Mat}(n\times n;\mathbb{C}) : \rho(A) < 1\},
\]
where \(\rho(A)\) is the spectral radius. For \(n=1\), \(\Omega_1=\mathbb{D}\). 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
\[
G_n:=(\sigma_1,\dots,\sigma_n)(\mathbb{D}^n)\subset\mathbb{C}^n
\]
is the symmetrized polydisc [1501.07145].

The spectral Nevanlinna–Pick problem asks for a holomorphic map
\[
F:\mathbb{D}\longrightarrow \Omega_n
\]
with prescribed values \(F(a_j)=A_j\) at distinct points \(a_1,\dots,a_m\in\mathbb{D}\). The weaker lifting problem prescribes only a holomorphic map \(f:\mathbb{D}\to G_n\) satisfying \(f(a_j)=\pi(A_j)\), and asks whether there exists \(F\) such that
\[
\pi\circ F=f
\quad\text{and}\quad
F(a_j)=A_j,\qquad j=1,\dots,m.
\]
The paper states that solving this lifting problem reduces the spectral Nevanlinna–Pick problem to an interpolation problem in the taut domain \(G_n\), which is better behaved from the point of view of hyperbolic geometry [1501.07145].

A generic fibre of \(\pi\), over a point whose eigenvalues are all distinct, consists of a single similarity class of matrices and is an \(\mathrm{SL}_n(\mathbb{C})\)-homogeneous manifold. This fibre geometry is one reason that interpolation in \(\Omega_n\) is naturally expressed through the spectral data encoded by \(\pi\) rather than through matrix entries themselves [1501.07145].

## 2. The lifting theorem as a local jet interpolation criterion

The main statement is Theorem 1.4:

> *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 \((*)\) holds for each interpolation point.*

Here \((*)\) is the local jet condition from Proposition 1.2. If \(f\in\mathcal{O}(U,G_n)\), \(p\in\mathbb{D}\), and \(M\in\Omega_n\), with Jordan decomposition
\[
M\sim B_1\oplus\cdots\oplus B_s,
\qquad
m_j:=\operatorname{size}(B_j),
\]
then a local holomorphic lift exists near \(p\) with prescribed value \(M\) if and only if
\[
\frac{d^k}{dv^k}\chi_{f(v)}(v)\Big|_{v=p}
= O\bigl( (v-p)^{\,m_j-k}\bigr),
\qquad
0\le k\le m_j-1,\ \ 1\le j\le s.
\tag{*}
\]
Equivalently, there exists a holomorphic \(F:\mathbb{D}\to\Omega_n\) with \(\pi\circ F=f\) and \(F(a_j)=A_j\) for all \(j\) if and only if, for every \(j\), the local jet conditions \((*)\) computed from the Jordan blocks of \(A_j\) are satisfied [1501.07145].

This is the precise sense in which the global matrix-valued interpolation problem becomes a jet interpolation problem into \(G_n\). The polynomial
\[
\chi_{f(z)}(\lambda):=\lambda^n+\sum_{j=1}^n(-1)^j\,f_j(z)\lambda^{n-j}
\]
plays the role of the characteristic polynomial of any lift \(F(z)\) with \(\pi(F(z))=f(z)\). The local vanishing-order condition forces \(\chi_{f(z)}\) to match the eigenvalue multiplicities and nilpotent structure encoded by the Jordan form of the target matrix [1501.07145].

For \(n=1\), \(\Omega_1=\mathbb{D}\), \(G_1=\mathbb{D}\), and \(\pi\) is the identity. The Jordan form is trivial, the condition \((*)\) 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 [1501.07145].

## 3. Oka-theoretic mechanism and geometric consequences

The lifting problem admits a section-theoretic reformulation. For a holomorphic \(f:\mathbb{D}\to G_n\), define the pullback space
\[
f^*(\Omega_n):=\{(z,A)\in \mathbb{D}\times\Omega_n : f(z)=\pi(A)\},
\]
with projection \(f^*(\pi):f^*(\Omega_n)\to\mathbb{D}\). A lifting \(F\) is equivalent to a holomorphic section
\[
s:\mathbb{D}\to f^*(\Omega_n),\qquad s(z)=(z,F(z)).
\]
The theorem therefore becomes a statement about the existence of global holomorphic sections from local ones [1501.07145].

The proof uses Forstnerič’s Oka principle for branched maps. Over the set of cyclic matrices, \(\pi\) is a holomorphic submersion, and one constructs a dominating spray through the conjugation action
\[
E:=\mathfrak{sl}_n(\mathbb{C})\times \pi^{-1}(U)\to \pi^{-1}(U),
\qquad
s(B,A):=\exp(B)\,A\,\exp(-B),
\]
whose derivative at \((0,A)\) is
\[
ds_{(0,A)}(C)=[A,C].
\]
This surjects onto the vertical tangent space and shows that \(\pi\) is an elliptic submersion off the branching locus. Pullbacks preserve this elliptic structure off the corresponding branching set, and Proposition 1.2 supplies the required local holomorphic liftings at the interpolation nodes [1501.07145].

A second ingredient is topological triviality. The fibres of \(\pi\) are \(\mathbb{C}\)-connected: each fibre, each stratum, and each connected component of a stratum is \(\mathbb{C}\)-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 [1501.07145].

The same geometric analysis yields Corollary 3.2: for \(n\ge 2\), the spectral ball \(\Omega_n\) is a union of immersed complex lines, and therefore there exists no bounded from above strictly plurisubharmonic function on \(\Omega_n\). In interpolation language, this helps explain why the reduction to \(G_n\) is natural: the spectral ball is non-hyperbolic, whereas the symmetrized polydisc is the taut base domain carrying the relevant spectral information [1501.07145].

## 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 \([-1,1]\), the paper on superconvergence of spectral interpolation studies analytic \(u\) extended to a Bernstein ellipse \(E_\rho\) and uses contour integral remainder formulas for Chebyshev-based interpolation. The global error behaves like \(\rho^{-N}\) with polynomial prefactors in \(N\), while derivative superconvergence occurs at zeros of derivatives of the nodal polynomial \(\omega_{N+1}\) [1204.5813].

For interpolation at the zeros of \(T_{N+1}\), one has \(\omega_{N+1}(x)=T_{N+1}(x)\). First-derivative superconvergence occurs at the zeros of
\[
T_{N+1}'(x)=(N+1)U_N(x),
\]
namely
\[
x^*=y_k=\cos\frac{k\pi}{N+1},\qquad k=1,\dots,N.
\]
Second-derivative superconvergence occurs at the zeros of \(T_{N+1}''(x)\), equivalently at solutions of
\[
(N+1)\cos((N+1)\theta)\sin\theta
=
\sin((N+1)\theta)\cos\theta,
\qquad
x^*=\cos\theta.
\]
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 [1204.5813].

On \(\mathbb{R}\), Hermite spectral interpolation uses the zeros of \(H_{n+1}\), equivalently of \(\psi_{n+1}\), as interpolation nodes in
\[
\mathbb{H}_n=\mathrm{span}\{\psi_0,\dots,\psi_n\}.
\]
For analytic \(f\) in a strip \(\mathcal{S}_\rho\), the Hermite interpolant \(h_n\) satisfies
\[
\|(f-h_n)^{(m)}\|_{L^\infty(\mathbb{R})}
\le
\mathcal{K}_m
\begin{cases}
n^{1/6} e^{-\rho\sqrt{2n}}, & m=0,\\[0.5ex]
n^{m/2} e^{-\rho\sqrt{2n}}, & m\ge 1.
\end{cases}
\]
First-derivative superconvergence occurs at the zeros \(\{\tau_j\}\) of \(\psi_{n+1}'\), and second-derivative superconvergence at the zeros \(\{\eta_j\}\) of \(\psi_{n+1}''\). The sharp gain is a factor \(n^{1/2}\):
\[
|(f-h_n)'(\tau_j)|
\le
C\, n^{-1/2}\|(f-h_n)'\|_\infty,
\qquad
|(f-h_n)''(\eta_j)|
\le
C\, n^{-1/2}\|(f-h_n)''\|_\infty.
\]
The same superconvergence nodes reappear for Hermite spectral collocation under the assumption \(u\in\mathbb{H}_{n+1}\) [2507.15350].

| Setting | Interpolation nodes | Superconvergence points |
|---|---|---|
| Chebyshev value interpolation | zeros of \(T_{N+1}\), \(T_{N+1}-T_{N-1}\), \(T_{N+1}\pm T_N\) | zeros of \(\omega'_{N+1}\) and \(\omega''_{N+1}\) |
| Derivative interpolation on \([-1,1]\) | Gauss, Lobatto, or Radau nodes | extremals or related nodes of the same family |
| Hermite interpolation on \(\mathbb{R}\) | zeros of \(H_{n+1}\) or \(\psi_{n+1}\) | zeros of \(\psi'_{n+1}\) and \(\psi''_{n+1}\) |

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 [1204.5813], [2507.15350].

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

In operator theory on quaternionic Banach spaces, the spectral object is the quadratic polynomial
\[
Q_s(T):=T^2-2\operatorname{Re}(s)\,T+|s|^2 I,
\]
and interpolation is phrased through the pseudo \(S\)-resolvent \(Q_s^{-1}(T)\). Under the structural assumptions of Definition 3.2, Theorem 3.5 gives a resolvent-based characterization of
\[
(X,D(T^n))_{\theta,p},
\]
while Theorems 3.6 and 3.7 state that, for every \(n<k<m\in\mathbb N\),
\[
D(T^k)\in J_{\frac{k-n}{m-n}}(D(T^n),D(T^m))
\quad\text{and}\quad
D(T^k)\in K_{\frac{k-n}{m-n}}(D(T^n),D(T^m)).
\]
This is a quaternionic analogue of the classical fact that domains of powers interpolate between one another [2402.10383].

In Euclidean Jordan algebras, spectral interpolation is expressed through eigenvalue vectors and the spectral \(p\)-norm
\[
\|x\|_p:=\|\lambda(x)\|_p.
\]
The interpolation theorem of Theorem 6.1 states that if
\[
\frac{1}{p}=\frac{1-\theta}{r}+\frac{\theta}{s},
\]
then for every linear map \(T:V\to V\),
\[
\|T\|_{p\to p}\le \|T\|_{r\to r}^{\,1-\theta}\,\|T\|_{s\to s}^{\,\theta}.
\]
A Riesz–Thorin type refinement interpolates between two different spectral norms:
\[
\|T\|_{r_\theta\to s_\theta}
\le
C\,\|T\|_{r_0\to s_0}^{\,1-\theta}\,\|T\|_{r_1\to s_1}^{\,\theta},
\]
with \(1\le C\le 4\) [1809.05417], [1905.02572].

A distinct spectral functional appears in the \(k\)-trace program, where
\[
\operatorname{Tr}_k[A]
\]
is the \(k\)-th elementary symmetric polynomial of the eigenvalues of \(A\), and
\[
\phi(A):=(\operatorname{Tr}_k[A])^{1/k}.
\]
The key interpolation input is a \(k\)-trace Stein–Hirschman inequality for holomorphic matrix families \(G(z)\), obtained by lifting to the exterior algebra. This yields joint concavity of
\[
(A,B)\longmapsto \phi\Big((B^{\frac{q}{2}}K^*A^pKB^{\frac{q}{2}})^s\Big)
\]
for the Lieb-allowed range of parameters, and also the concavity of
\[
A\longmapsto \phi\big(\exp(H+\log A)\big)
\]
on \(\mathbf{H}_n^{++}\) [1904.03304].

The nonlinear interpolation theorem of a different paper extends the Riesz–Thorin paradigm from linear operators to analytic maps on Sobolev balls:
\[
F:B^a_{\mathbb C}(R)\to \ell^{p,\alpha+\beta a},
\qquad
F:B^b_{\mathbb C}(R)\to \ell^{p,\alpha+\beta b}.
\]
For \(s=(1-\theta)a+\theta b\),
\[
\sup_{q\in B^s_{\mathbb C}(R)}
\|F(q)\|_{p,\alpha+\beta s}
\le
M_a^{1-\theta}M_b^\theta.
\]
The paper applies this to spectral quantities of one-dimensional Schrödinger operators, obtaining fractional-order asymptotics for the Floquet exponents \(\kappa_n\) and for periodic eigenvalue combinations from integer-order estimates [1306.5721].

## 6. RKHS interpolation spaces and weighted spectral priors

For reproducing kernel Hilbert spaces, the interpolation couple is
\[
\bigl(L^2(\nu),[H]_{\sim}\bigr),
\]
where \(H\) is an RKHS with bounded measurable kernel \(k\), compactly embedded into \(L^2(\nu)\), and \(T_k=I_k I_k^*\) has eigenvalues \(\mu_i\) and eigenfunctions \(e_i\). The spectral representation theorem states that for \(0<\theta<1\) and \(1\le r\le\infty\),
\[
H_\nu^{(\theta,r)}=[L^2(\nu),[H]_{\sim}]_{\theta,r},
\]
with equivalence of norms. Writing
\[
f=\sum_{i\in I} b_i [e_i]_{\sim}\quad\text{in }L^2(\nu),
\]
the norm of the interpolation space is equivalent to a dyadic \(\ell^r\)-sum of \(\ell^2\)-blocks built from the weighted coefficients \(b_i\mu_i^{-\theta/2}\). For \(r=2\), this reduces to the familiar Hilbertian formula
\[
\|f\|_{[L^2,[H]_{\sim}]_{\theta,2}}^2
\asymp
\sum_{i\in I} |\langle f,[e_i]_{\sim}\rangle_{L^2}|^2 \mu_i^{-\theta}
\]
[2508.16492].

The same paper gives an exact criterion for the embedding
\[
H_\nu^{(\theta,r)}\hookrightarrow L^\infty(\nu),
\]
namely uniform boundedness of the dual sequence norm of
\[
\bigl(e_i(x)\mu_i^\theta\bigr)_{i\in I}
\]
for almost every \(x\). When this embedding holds, the interpolation space admits a Banach-space-of-functions realization with continuous point evaluations [2508.16492].

A different spectral-interpolation line generalizes Shannon reconstruction through weighted Hilbert spaces. For \(B\)-bandlimited signals and a weight \(W(\Omega)\) satisfying
\[
0<L<W(\Omega)<U<\infty
\quad\text{a.e. on }[-2\pi B,2\pi B],
\]
the norm is
\[
\|x\|_{\mathcal W}
=
\sqrt{\frac{1}{2\pi}\int_{-2\pi B}^{2\pi B}|X(\Omega)|^2W(\Omega)\,d\Omega}.
\]
The minimum-\(\mathcal W\)-norm interpolant from samples \(x[n]=x(nT)\), \(-N\le n\le N\), has the form
\[
\hat{x}_w(t)=\sum_{n=-N}^N c_n\,\psi(t-nT),
\qquad
\psi(t)=\frac{1}{2\pi}\int_{-2\pi B}^{2\pi B}\frac{e^{j\Omega t}}{W(\Omega)}\,d\Omega,
\]
with coefficients determined by the Gram matrix
\[
[\mathbf R]_{mn}=\psi(mT-nT).
\]
When \(W(\Omega)=1\) on \([-\pi/T,\pi/T]\) and \(B=\frac{1}{2T}\), \(\psi\) becomes the sinc kernel and the formula reduces to truncated Shannon interpolation. When \(W(\Omega)=1/S(\Omega)\), it coincides with the LMMSE or GP-regression interpolator for a stationary process with PSD \(S(\Omega)\). The paper emphasizes that this weighted framework is particularly useful for interpolating sub-Nyquist data [2410.21184].

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

On the finite cyclic group \(\mathbb{Z}_N\), the bandlimited space
\[
B^J=\{f:\mathbb{Z}_N\to\mathbb{C}:(Ff)(n)=0\text{ for }n\notin J\}
\]
admits interpolation from a sampling set \(I\) exactly when the submatrix
\[
A_{I,J}:=E_I^\top F^{-1}E_J
\]
is invertible. Orthogonal interpolation occurs when this submatrix is unitary up to scaling. Writing \(h_J:=F^{-1}1_J\), an index set \(I\) is an orthogonal sampling set for \(B^J\) if and only if \(|I|=|J|\) and
\[
h_J(i_1-i_2)=0
\quad\text{for all distinct }i_1,i_2\in I.
\]
This condition is encoded by the difference graph \(G(h_J)\), whose vertices are \(\mathbb{Z}_N\) and whose edges join \(i_1\neq i_2\) whenever \(h_J(i_1-i_2)=0\). Then \(I\) is an orthogonal sampling set if and only if \(I\) is a maximum clique in \(G(h_J)\). For \(N=p^M\), the paper proves that
\[
B^J \text{ has an orthogonal sampling set } \iff |J|=p^{|D(h_J)|},
\]
that the maximum clique size of \(G(h_J)\) is \(p^{|D(h_J)|}\), and that \(B^J\) has an orthogonal sampling set if and only if \(J\) tiles \(\mathbb{Z}_N\) [1411.7086].

For \(n\)-dimensional vector polynomials,
\[
\mathbb P=\{\mathbf p(z)=(P_1(z),\dots,P_n(z))^T\},
\]
the paper on vector-polynomial interpolation studies the condition
\[
\sum_{k=1}^n a_k(j)P_k(z_j)=0,\qquad j=1,\dots,N,
\]
equivalently
\[
(\mathbf p(z_j),\sigma_j\mathbf p(z_j))=0,
\]
with \(\sigma_j\) nonnegative rank-one Hermitian matrices. The solution set \(\mathbb S(n,N)\) is organized by the height
\[
h(\mathbf p)=
\max_{j\in\{1,\dots,n\}}
\bigl\{n\,\deg P_j(z)+j-1\bigr\}.
\]
The paper constructs \(n\) generators \(\mathbf r_1,\dots,\mathbf r_n\) such that
\[
\sum_{j=1}^n h(\mathbf r_j)=Nn+\frac{n(n-1)}{2},
\]
and proves the full structure theorem
\[
\mathbb S(n,N)=\mathbb M(\mathbf r_1)\oplus\cdots\oplus\mathbb M(\mathbf r_n),
\]
where \(\mathbb M(\mathbf r_j)=\{S\mathbf r_j:S\text{ scalar polynomial}\}\). The paper states that these results generalize rational interpolation and have applications to direct and inverse spectral analysis of band matrices [1401.5384].

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.

Source: https://www.emergentmind.com/topics/spectral-interpolation-theorem