---
title: Radial Fourier Interpolation Formulas
url: https://www.emergentmind.com/topics/radial-fourier-interpolation-formulas
type: topic
---

# Radial Fourier Interpolation Formulas

Searching arXiv for recent and foundational papers on radial/Fourier interpolation formulas to ground the article.
Radial Fourier interpolation formulas are exact reconstruction identities in Euclidean Fourier analysis that recover a radial function, or the radial component of a more general function, from discrete data in the radial variable. Depending on the setting, the sampled data consist of values of the function, values of its Fourier transform, and in derivative-sensitive formulations also radial derivatives of both at distinguished radii such as zeros of Bessel functions, the radii \(\sqrt{n}\), or the radii \(\sqrt{2n}\). The modern literature presents several analytically distinct versions of this theme: de Branges–Hankel interpolation for entire functions of exponential type, modular and Poincaré-series constructions at quadratic radii, sphere-restriction formulas for non-radial Schwartz functions, and dimension-specific formulas in dimensions \(8\) and \(24\) tied to the \(E_8\) and Leech lattices [1503.05178][1701.00265][1902.05438][2002.11627][2108.11828].

## 1. Radial reduction and the one-dimensional analytic model

For a radial function \(f(x)=f_0(|x|)\) on \(\mathbb{R}^d\), the Fourier transform remains radial and reduces to a one-dimensional Hankel transform of order \(\nu=\frac d2-1\). In the normalization used in the de Branges/Bessel treatment, the radial Fourier transform can be written as
\[
\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,
\]
while the radial \(L^2\) measure becomes
\[
\int_{\mathbb R^d} |f(x)|^2\,dx = |S^{d-1}|\int_0^\infty |f_0(r)|^2 r^{d-1}\,dr.
\]
Thus the natural one-dimensional weight is \(r^{d-1}\,dr\), equivalently \(|t|^{2\nu+1}dt\) with \(2\nu+1=d-1\) [1503.05178].

This reduction admits an exact dimension-walk identity. If \(f(|\cdot|)\) is radial in dimensions \(n\) and \(n+2\), then
\[
\mathcal{F}_{n+2}(f)(r) = -\frac{1}{2\pi r}\frac{d}{dr}\,\mathcal{F}_n(f)(r),
\]
and, iterating,
\[
\mathcal{F}_{2k+1}(f)=\left(-\frac{1}{2\pi r}\frac{d}{dr}\right)^k\mathcal{F}_1(f),\qquad
\mathcal{F}_{2k+2}(f)=\left(-\frac{1}{2\pi r}\frac{d}{dr}\right)^k\mathcal{F}_2(f).
\]
In this sense, all odd-dimensional radial Fourier transforms are determined by dimension \(1\), and all even-dimensional transforms by dimension \(2\) [1112.5469].

A closely related multiradial formulation expresses higher-dimensional transforms through differential operators applied to lower-dimensional transforms, ultimately reducing each block variable to dimension \(1\) or \(2\) according to parity. In the ordinary radial case this yields the same recursion
\[
\mathcal{F}_{n+2}(\Phi)(r)= -\frac{1}{2\pi r}\frac{d}{dr}\mathcal{F}_n(\Phi)(r),
\]
together with Abel-type formulas for compactly supported radial Fourier data, such as
\[
\mathcal{F}_2(\Phi)(r)= 2\int_r^A \frac{\widehat{\phi}(w)}{\sqrt{w^2-r^2}}\,dw\,\chi_{(0,A)}(r).
\]
These formulas are not discrete sampling theorems, but they identify the one-dimensional analytic structure from which discrete radial interpolation formulas are built [1307.8408].

## 2. De Branges spaces, Bessel zeros, and Hermite-type reconstruction

One major approach treats radial Fourier interpolation as interpolation in de Branges spaces. The general theorem concerns a Hermite–Biehler function \(E\), its companion functions \(A_\alpha,B_\alpha\), and the zero set
\[
\mathcal T(\alpha)=\{x\in\mathbb R:\varphi(x)=\alpha \pmod{\pi}\},
\]
equivalently the real zeros of \(B_\alpha\). Under the hypotheses that \(\mathcal H(E^2)\) is closed by differentiation, that \(A_\alpha B_\alpha\notin \mathcal H(E^2)\), and that \(\varphi'(x)\) is bounded away from zero on \(\mathcal T(\alpha)\), every \(F\in \mathcal H^p(E^2)\) satisfies
\[
F(z)=B_\alpha(z)^2\sum_{t\in\mathcal T(\alpha)}
\left\{
\frac{F(t)}{B_\alpha'(t)^2(z-t)^2}
+\frac{F'(t)B_\alpha'(t)-F(t)B_\alpha''(t)}{B_\alpha'(t)^3(z-t)}
\right\},
\]
with uniform convergence on compact subsets of \(\mathbb C\) for \(p\in[1,2]\), and also for \(p\in(2,\infty)\) under the additional mean-type assumption \(v(E^*/E)<0\) [1503.05178].

This formula is the de Branges-space analogue of Vaaler’s cardinal interpolation identity
\[
F(z)=\frac{\sin^2(\pi z)}{\pi^2}\sum_{n\in\mathbb Z}
\left( \frac{F(n)}{(z-n)^2}+\frac{F'(n)}{z-n} \right),
\]
recovered by taking \(E(z)=e^{-i\pi z}\), so that \(B(z)=\sin\pi z\) and \(\mathcal T(0)=\mathbb Z\) [1503.05178].

The radial content enters through the homogeneous de Branges spaces generated by
\[
A_\nu(z)=\Gamma(\nu+1)\left(\frac z2\right)^{-\nu}J_\nu(z),\qquad
B_\nu(z)=\Gamma(\nu+1)\left(\frac z2\right)^{-\nu}J_{\nu+1}(z),
\]
with \(E_\nu=A_\nu-iB_\nu\), \(\nu>-1\). Here
\[
\int_{\mathbb R}|F(x)|^2 |E_\nu(x)|^{-2}\,dx
= c_\nu \int_{\mathbb R}|F(x)|^2 |x|^{2\nu+1}\,dx,
\]
so \(\mathcal H(E_\nu)\) identifies with entire functions of exponential type at most \(1\) in \(L^2(\mathbb R,|x|^{2\nu+1}dx)\), and \(\nu=\frac d2-1\) matches the radial Fourier weight in \(\mathbb R^d\) [1503.05178].

The explicit Bessel interpolation theorem then states that if \(p\in(0,\infty)\), \(\nu>-1\), and \(F\) is entire of exponential type at most \(2\) with
\[
\int_{\mathbb R}|F(t)|^p |t|^{2\nu+1}\,dt<\infty,
\]
then
\[
\frac{F(z)}{A_\nu(z)^2}
=
\sum_{A_\nu(s)=0}
\left\{
\frac{F(s)}{A_\nu'(s)^2(z-s)^2}
+\frac{F'(s)}{A_\nu'(s)^2(z-s)}
+\frac{(2\nu+1)F(s)}{s\,A_\nu'(s)^2(z-s)}
\right\},
\]
and analogously with \(B_\nu\) in place of \(A_\nu\), with uniform convergence on compact subsets away from the singularities. The interpolation nodes are exactly the zeros of \(J_\nu\) and \(J_{\nu+1}\). In radial Fourier language, these are distinguished radii determined by Bessel zeros rather than by an arithmetic progression [1503.05178].

The same paper uses this derivative-sensitive interpolation formula to prove uniqueness of extremal one-sided band-limited majorants and minorants for radial functions. The uniqueness argument is explicitly based on the fact that equality of a radial function and its derivative at the interpolation radii forces the difference to vanish identically in \(\mathcal H^1(E^2)\) [1503.05178].

## 3. The \(\sqrt{n}\) and \(\sqrt{2n/\lambda}\) node families

A second major lineage centers on quadratic radii. In one dimension, Radchenko and Viazovska constructed even Schwartz functions \(a_n\) such that every even Schwartz function satisfies
\[
f(x)=\sum_{n=0}^{\infty} a_n(x)f(\sqrt n)+\sum_{n=0}^{\infty} \widehat a_n(x)\widehat f(\sqrt n),
\]
with absolute convergence. The sampling set is therefore \(\{0,\pm\sqrt1,\pm\sqrt2,\pm\sqrt3,\dots\}\), and the basis is constructed from weakly holomorphic modular forms for the theta group [1701.00265].

In higher dimensions, this quadratic-radius structure persists in the radial setting. For \(p\ge5\), there exist even entire functions \(b_{p,n},\widetilde b_{p,n}\) such that for every radial Schwartz function on \(\mathbb R^p\),
\[
f(x)=\sum_{n=1}^\infty f(\sqrt n)\,b_{p,n}(x)+\sum_{n=1}^\infty \widehat f(\sqrt n)\,\widetilde b_{p,n}(x),
\]
with absolute convergence. For non-radial functions in dimension \(d\ge5\), the corresponding data are restrictions of \(f\) and \(\widehat f\) to the spheres \(\sqrt n\,S^{d-1}\), and the reconstruction formula is
\[
f(x)=\sum_{n=1}^{\infty}\int_{S^{d-1}} A_n(x,\zeta)f(\sqrt n\,\zeta)\,d\zeta
+\sum_{n=1}^{\infty}\int_{S^{d-1}} \widetilde A_n(x,\zeta)\widehat f(\sqrt n\,\zeta)\,d\zeta,
\]
with absolute convergence [2002.11627].

The non-radial formula is obtained from a tower of radial formulas in dimensions \(d,d+2,d+4,\dots\) via spherical harmonics. The transfer mechanism is encoded in the radialization operators \(L_u^{d+2m}\) and the Fourier compatibility
\[
\mathcal F\big(L_u^{d+2m}f\big)=i^m\,L_u^{d+2m}(\widehat f),
\]
so the degree-\(m\) spherical harmonic component in dimension \(d\) becomes a radial function in dimension \(d+2m\) [2002.11627].

The same quadratic-radius pattern extends to all dimensions in the radial case. If \(d\ge1\), \(k=\frac d2\), and
\[
v_-(k)=\Bigl[\frac{k+2}{4}\Bigr], \qquad v_+(k)=\Bigl[\frac{k+4}{4}\Bigr],
\]
then there exist even Schwartz functions \(b_{k,n}^{\varepsilon}\), \(n\ge v_\varepsilon(k)\), with \(\mathcal F(b_{k,n}^{\varepsilon})=\varepsilon\,b_{k,n}^{\varepsilon}\), and hence functions
\[
a_{k,n}= b_{k,n}^{+}+b_{k,n}^{-}, \qquad
\widetilde a_{k,n}= b_{k,n}^{+}-b_{k,n}^{-},
\]
such that every radial Schwartz function on \(\mathbb R^d\) satisfies
\[
f(x)=\sum_{n=v_-(k)}^\infty \Bigl( a_{k,n}(|x|)\, f(\sqrt n) +\widetilde a_{k,n}(|x|)\,\widehat f(\sqrt n)\Bigr),
\]
with convergence of the partial sums in the Schwartz topology [2103.12015].

A further extension replaces \(\sqrt n\) by \(\sqrt{2n/\lambda}\). For every real \(\lambda\ge2\) and every integer \(d\ge5\), with \(k=d/2\), there exist entire even functions \(a_{k,\lambda,n}\) and \(\tilde a_{k,\lambda,n}\) such that for all radial Schwartz functions on \(\mathbb R^d\),
\[
f(x)=\sum_{n=1}^\infty a_{k,\lambda,n}(|x|)\,f\!\left(\sqrt{\frac{2n}{\lambda}}\right)
+\sum_{n=1}^\infty \tilde a_{k,\lambda,n}(|x|)\,\widehat f\!\left(\sqrt{\frac{2n}{\lambda}}\right),
\]
with both series converging absolutely and uniformly on \(\mathbb R^d\). When \(\lambda>2\), the construction can be modified to work after deleting any finite set of initial nodes, so that interpolation is available on any cofinite subset of the corresponding radii [2108.11828].

These formulas are built from modular or Hecke-group generating functions attached to Gaussians \(e^{\pi i\tau |x|^2}\). In this framework the appearance of \(r=\sqrt n\), or more generally \(r=\sqrt{2n/\lambda}\), reflects the fact that the Gaussian restriction to a sphere of radius \(r\) is \(e^{\pi i\tau r^2}\), so the natural spectral variable is \(r^2\) rather than \(r\) [2002.11627][2103.12015][2108.11828].

## 4. Dimensions \(8\) and \(24\): interpolation with radial derivatives

In dimensions \(8\) and \(24\), radial Fourier interpolation takes a stronger Hermite-type form. If \((d,n_0)\) is \((8,1)\) or \((24,2)\), then every radial Schwartz function on \(\mathbb R^d\) is uniquely determined by
\[
f(\sqrt{2n}),\quad f'(\sqrt{2n}),\quad \widehat f(\sqrt{2n}),\quad \widehat f\,'(\sqrt{2n})
\qquad (n\ge n_0),
\]
and there exist radial Schwartz basis functions \(a_n,b_n,\widetilde a_n,\widetilde b_n\) such that
\[
\begin{aligned}
f(x) &= \sum_{n=n_0}^\infty f\big(\sqrt{2n}\big)\,a_n(x)
+\sum_{n=n_0}^\infty f'\big(\sqrt{2n}\big)\,b_n(x) \\
&\quad+\sum_{n=n_0}^\infty \widehat{f}\big(\sqrt{2n}\big)\,\widetilde{a}_n(x)
+\sum_{n=n_0}^\infty \widehat{f}\,'\big(\sqrt{2n}\big)\,\widetilde{b}_n(x),
\end{aligned}
\]
with absolute convergence [1902.05438].

The theorem is an isomorphism statement: the map from \(f\) to the four rapidly decreasing sample sequences is an isomorphism from \(S_{\mathrm{rad}}(\mathbb R^d)\) onto \(S(\mathbb N)^4\). The basis functions satisfy the corresponding Kronecker interpolation conditions, and in fact
\[
\widetilde a_n=\widehat{a_n},\qquad \widetilde b_n=\widehat{b_n}.
\]
The derivative data are radial derivatives, meaning that if \(f(x)=F(|x|)\), then \(f'(r)=F'(r)\), and similarly for \(\widehat f\) [1902.05438].

The basis is encoded by generating functions
\[
F(\tau,x)=\sum_{n \ge n_0} a_n(x)e^{2\pi i n \tau}
+2\pi i \tau\sum_{n \ge n_0}\sqrt{2n}\, b_n(x)e^{2\pi i n \tau},
\]
\[
\widetilde F(\tau,x)=\sum_{n \ge n_0} \widetilde a_n(x)e^{2\pi i n \tau}
+2\pi i \tau\sum_{n \ge n_0}\sqrt{2n}\, \widetilde b_n(x)e^{2\pi i n \tau},
\]
which satisfy
\[
F(\tau+2,x)-2F(\tau+1,x)+F(\tau,x)=0,
\]
\[
\widetilde F(\tau+2,x)-2\widetilde F(\tau+1,x)+\widetilde F(\tau,x)=0,
\]
and the fundamental Gaussian identity
\[
F(\tau,x)+(i/\tau)^{d/2}\widetilde F(-1/\tau,x)=e^{\pi i \tau |x|^2}.
\]
The proof constructs \(F\) and \(\widetilde F\) from integral transforms of modular and quasimodular forms, extending Viazovska’s sphere-packing method [1902.05438].

This interpolation theorem is used in the proof of universal optimality of the \(E_8\) and Leech lattices. The sample radii \(\sqrt{2n}\) are exactly the shell radii of those lattices, and the theorem implies that the auxiliary radial Schwartz function required by linear programming is uniquely forced by its prescribed values and derivatives on lattice shells and dual lattice shells [1902.05438].

## 5. Perturbation theory, stability, and regularity of the bases

A substantial later development shows that these formulas are stable under perturbation of the nodes. In the all-dimensional radial \(\sqrt n\) theory, if \(d\ge1\), \(s>1\), \(\eta>0\), and
\[
|\varepsilon_n|+|\widetilde\varepsilon_n| < \delta (1+n)^{-d-(s/2)-2-\eta},
\]
then there exist radial functions \(h_{d,n},\widetilde h_{d,n}\) such that for all \(f\in V^s_{\mathrm{rad}}(\mathbb R^d)\),
\[
f= f(\varepsilon_0)h_{d,0}+\sum_{n=1}^\infty f(\sqrt n+\varepsilon_n)h_{d,n}
+\widehat f(\widetilde \varepsilon_0)\widetilde h_{d,0}
+\sum_{n=1}^\infty \widehat f(\sqrt n+\widetilde\varepsilon_n)\widetilde h_{d,n},
\]
with absolute convergence in a higher regularity space \(V^{s'}_{\mathrm{rad}}(\mathbb R^d)\) [2103.12015].

For the one-dimensional Radchenko–Viazovska formula, there is a sharper perturbative result: if
\[
\sup_{k\ge 0} |\varepsilon_k|(1+k)^{5/4}<\delta,
\]
with \(\varepsilon_0=0\), then there exist functions \(\varphi_j,\psi_j\) satisfying
\[
|\varphi_j(x)|+|\psi_j(x)|+|\widehat{\varphi_j}(x)|+|\widehat{\psi_j}(x)|
\ll (1+j)^{-10}(1+|x|)^{-10},
\]
such that every real-valued even Schwartz function satisfies
\[
f(x)=\sum_{j\ge 0}\Bigl(f(\sqrt j+\varepsilon_j)\varphi_j(x)+\widehat f(\sqrt j+\varepsilon_j)\psi_j(x)\Bigr).
\]
The same paper proves an analogue for the derivative interpolation formulas in dimensions \(8\) and \(24\): if \(|\varepsilon_k|<\delta\,k^{-C_0}\), then any radial Schwartz function is uniquely determined by the perturbed values
\[
\bigl(f(\sqrt{2n}+\varepsilon_n),\, f'(\sqrt{2n}+\varepsilon_n),\, \widehat f(\sqrt{2n}+\varepsilon_n),\, \widehat f'(\sqrt{2n}+\varepsilon_n)\bigr)_{n\ge n_0}.
\]
These perturbation results are obtained by viewing interpolation as invertibility of an operator that is a perturbation of the identity on weighted sequence spaces [2005.10337].

Regularity of the perturbed bases can be upgraded further. If
\[
|\varepsilon_n|< c(1+n)^{-\frac54-\delta},
\]
then the perturbed one-dimensional basis may be chosen in the Schwartz space, and the interpolation series converges absolutely in the Schwartz topology. If the perturbation is exponentially small,
\[
|\varepsilon_n|\le c e^{-Cn},
\]
then the perturbed basis functions remain entire of order \(2\) and finite type \(\pi\). In particular, vanishing on all but finitely many perturbed nodes forces analyticity of order \(2\), a property already known in the exact Radchenko–Viazovska case and now shown to persist under sufficiently small perturbations [2503.15733].

These stability results indicate that the interpolation formulas are not rigid artifacts of exact modular symmetry alone. A plausible implication is that the analytic structure of the underlying basis is preserved by perturbations well beyond bare uniqueness, although the available theorems are still formulated in specific radial or even-function settings [2103.12015][2005.10337][2503.15733].

## 6. Density, nonuniqueness, and structural limitations

Several results delineate what radial Fourier interpolation formulas do not provide. First, the sphere-restriction theorem in dimensions \(d\ge5\) is a sampling formula, not a free interpolation theorem: it reconstructs a Schwartz function from the restrictions of \(f\) and \(\widehat f\) to the spheres \(\sqrt n\,S^{d-1}\), but it does not characterize which sequences of sphere data arise from a Schwartz function. In fact, for \(d\ge5\), the map sending \(f\) to these restrictions has infinite-dimensional cokernel [2002.11627].

Second, general interpolation formulas involving both \(f\) and \(\widehat f\) are constrained by time-frequency density. If
\[
f(x)=\sum_{\lambda\in\Lambda} f^{(r(\lambda))}(\lambda)a_\lambda(x)
+\sum_{\mu\in M}\widehat f^{(k(\mu))}(\mu)b_\mu(x)
\]
holds for all compactly supported smooth functions under mild growth assumptions, then
\[
n_\Lambda(R_1)+n_M(R_2)\ge 4R_1R_2-C\log^{2+\varepsilon}(4R_1R_2).
\]
For even or odd functions the factor \(4\) becomes \(2\). This is a necessary density bound rather than a constructive interpolation theorem, but it explains why known formulas at quadratic radii are close to the phase-space barrier set by the Slepian \(4WT\) principle [2005.12836].

Third, the basis functions of the Radchenko–Viazovska formula exhibit substantial extra zero structure. For the Fourier-invariant basis \(b_n^+\), writing \(f_n(x)=b_n^+(\sqrt x)\), one has \(f_n(m)=\delta_{n,m}\) for \(m\in\mathbb Z_{\ge0}\), but \(f_n\) also possesses many “extraneous zeros” near the positive real axis. This leads to Fourier nonuniqueness pairs obtained by deleting one canonical sample point and inserting many extra real zeros of a basis function. The same analysis shows that the normalized basis functions do not form a Riesz basis in the Hilbert space
\[
\mathcal H=\left\{f:\ \int_{-\infty}^{\infty}\big(|f(x)|^2+|\widehat f(x)|^2\big)(1+x^2)\,dx<\infty\right\},
\]
so canonical interpolation at critical density does not imply frame-theoretic stability [2512.18677].

Fourth, higher-rank analogues of the square-root-lattice picture are obstructed. For lattices in \(\mathbb R^n\) with \(n\ge2\), the group-theoretic conditions needed for a multivariable \(\sqrt{\Lambda}\)-type interpolation formula generally fail if the associated subgroup of \(\mathrm{PSL}_2(\mathbb R)^n\) is discrete. The one-radial-variable case avoids this obstruction because the relevant Hecke-group subgroup \(\Gamma(\lambda)=\langle T^\lambda,V^\lambda\rangle\) is discrete and free for \(\lambda\ge2\) [2108.11828].

These limitations clarify several common misconceptions. Radial Fourier interpolation is not a single theorem but a family of exact reconstruction mechanisms with different node geometries and different admissible function spaces. Some formulas are free interpolation theorems, some are sampling identities, some are perturbative stability results, and some are dimension-reduction identities rather than discrete sampling formulas. The subject is unified not by a common kernel, but by a common principle: radial Fourier symmetry reduces multidimensional analysis to a one-dimensional analytic structure—Bessel, de Branges, modular, or Hecke-theoretic—from which exact reconstruction becomes possible in selected settings [1503.05178][2002.11627][2108.11828][2512.18677].

Source: https://www.emergentmind.com/topics/radial-fourier-interpolation-formulas