Radial Fourier Interpolation Formulas

• Radial Fourier interpolation formulas are exact reconstruction identities that recover radial function components from discrete samples of both the function and its Fourier transform.
• They employ methods such as de Branges-Hankel interpolation, modular and Poincaré-series constructions, and dimension-reduction via differential operators to simplify high-dimensional Fourier analysis.
• These formulas exhibit stability under perturbations and are instrumental in applications like sphere packing problems and proving uniqueness in extremal band-limited function scenarios in dimensions 8 and 24.

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 (Gonçalves, 2015, Radchenko et al., 2017, Cohn et al., 2019, Stoller, 2020, Radchenko et al., 2021).

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

$\sqrt{2n}$0

Thus the natural one-dimensional weight is $\sqrt{2n}$1, equivalently $\sqrt{2n}$2 with $\sqrt{2n}$3 (Gonçalves, 2015).

This reduction admits an exact dimension-walk identity. If $\sqrt{2n}$4 is radial in dimensions $\sqrt{2n}$5 and $\sqrt{2n}$6, then

$\sqrt{2n}$7

and, iterating,

$\sqrt{2n}$8

In this sense, all odd-dimensional radial Fourier transforms are determined by dimension $\sqrt{2n}$9, and all even-dimensional transforms by dimension $8$0 (Grafakos et al., 2011).

A closely related multiradial formulation expresses higher-dimensional transforms through differential operators applied to lower-dimensional transforms, ultimately reducing each block variable to dimension $8$1 or $8$2 according to parity. In the ordinary radial case this yields the same recursion

$8$3

together with Abel-type formulas for compactly supported radial Fourier data, such as

$8$4

These formulas are not discrete sampling theorems, but they identify the one-dimensional analytic structure from which discrete radial interpolation formulas are built (Bernicot et al., 2013).

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 $8$5, its companion functions $8$6, and the zero set

$8$7

equivalently the real zeros of $8$8. Under the hypotheses that $8$9 is closed by differentiation, that $24$0, and that $24$1 is bounded away from zero on $24$2, every $24$3 satisfies

$24$4

with uniform convergence on compact subsets of $24$5 for $24$6, and also for $24$7 under the additional mean-type assumption $24$8 (Gonçalves, 2015).

This formula is the de Branges-space analogue of Vaaler’s cardinal interpolation identity

$24$9

recovered by taking $E_8$0, so that $E_8$1 and $E_8$2 (Gonçalves, 2015).

The radial content enters through the homogeneous de Branges spaces generated by

$E_8$3

with $E_8$4, $E_8$5. Here

$E_8$6

so $E_8$7 identifies with entire functions of exponential type at most $E_8$8 in $E_8$9, and $f(x)=f_0(|x|)$0 matches the radial Fourier weight in $f(x)=f_0(|x|)$1 (Gonçalves, 2015).

The explicit Bessel interpolation theorem then states that if $f(x)=f_0(|x|)$2, $f(x)=f_0(|x|)$3, and $f(x)=f_0(|x|)$4 is entire of exponential type at most $f(x)=f_0(|x|)$5 with

$f(x)=f_0(|x|)$6

then

$f(x)=f_0(|x|)$7

and analogously with $f(x)=f_0(|x|)$8 in place of $f(x)=f_0(|x|)$9, with uniform convergence on compact subsets away from the singularities. The interpolation nodes are exactly the zeros of $\mathbb{R}^d$0 and $\mathbb{R}^d$1. In radial Fourier language, these are distinguished radii determined by Bessel zeros rather than by an arithmetic progression (Gonçalves, 2015).

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 $\mathbb{R}^d$2 (Gonçalves, 2015).

3. The $\mathbb{R}^d$3 and $\mathbb{R}^d$4 node families

A second major lineage centers on quadratic radii. In one dimension, Radchenko and Viazovska constructed even Schwartz functions $\mathbb{R}^d$5 such that every even Schwartz function satisfies

$\mathbb{R}^d$6

with absolute convergence. The sampling set is therefore $\mathbb{R}^d$7, and the basis is constructed from weakly holomorphic modular forms for the theta group (Radchenko et al., 2017).

In higher dimensions, this quadratic-radius structure persists in the radial setting. For $\mathbb{R}^d$8, there exist even entire functions $\mathbb{R}^d$9 such that for every radial Schwartz function on $\nu=\frac d2-1$0,

$\nu=\frac d2-1$1

with absolute convergence. For non-radial functions in dimension $\nu=\frac d2-1$2, the corresponding data are restrictions of $\nu=\frac d2-1$3 and $\nu=\frac d2-1$4 to the spheres $\nu=\frac d2-1$5, and the reconstruction formula is

$\nu=\frac d2-1$6

with absolute convergence (Stoller, 2020).

The non-radial formula is obtained from a tower of radial formulas in dimensions $\nu=\frac d2-1$7 via spherical harmonics. The transfer mechanism is encoded in the radialization operators $\nu=\frac d2-1$8 and the Fourier compatibility

$\nu=\frac d2-1$9

so the degree-$$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$0 spherical harmonic component in dimension $$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$1 becomes a radial function in dimension $$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$2 (Stoller, 2020).

The same quadratic-radius pattern extends to all dimensions in the radial case. If $$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$3, $$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$4, and

$$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$5

then there exist even Schwartz functions $$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$6, $$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$7, with $$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$8, and hence functions

$$\widehat f_0(\rho) = (2\pi)^{d/2}\rho^{-(d/2-1)} \int_0^\infty f_0(r)\,J_{d/2-1}(2\pi r\rho)\, r^{d/2}\,dr,$$9

such that every radial Schwartz function on $L^2$0 satisfies

$L^2$1

with convergence of the partial sums in the Schwartz topology (Ramos et al., 2021).

A further extension replaces $L^2$2 by $L^2$3. For every real $L^2$4 and every integer $L^2$5, with $L^2$6, there exist entire even functions $L^2$7 and $L^2$8 such that for all radial Schwartz functions on $L^2$9,

$\sqrt{2n}$00

with both series converging absolutely and uniformly on $\sqrt{2n}$01. When $\sqrt{2n}$02, 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 (Radchenko et al., 2021).

These formulas are built from modular or Hecke-group generating functions attached to Gaussians $\sqrt{2n}$03. In this framework the appearance of $\sqrt{2n}$04, or more generally $\sqrt{2n}$05, reflects the fact that the Gaussian restriction to a sphere of radius $\sqrt{2n}$06 is $\sqrt{2n}$07, so the natural spectral variable is $\sqrt{2n}$08 rather than $\sqrt{2n}$09 (Stoller, 2020, Ramos et al., 2021, Radchenko et al., 2021).

4. Dimensions $\sqrt{2n}$10 and $\sqrt{2n}$11: interpolation with radial derivatives

In dimensions $\sqrt{2n}$12 and $\sqrt{2n}$13, radial Fourier interpolation takes a stronger Hermite-type form. If $\sqrt{2n}$14 is $\sqrt{2n}$15 or $\sqrt{2n}$16, then every radial Schwartz function on $\sqrt{2n}$17 is uniquely determined by

$\sqrt{2n}$18

and there exist radial Schwartz basis functions $\sqrt{2n}$19 such that

$\sqrt{2n}$20

with absolute convergence (Cohn et al., 2019).

The theorem is an isomorphism statement: the map from $\sqrt{2n}$21 to the four rapidly decreasing sample sequences is an isomorphism from $\sqrt{2n}$22 onto $\sqrt{2n}$23. The basis functions satisfy the corresponding Kronecker interpolation conditions, and in fact

$\sqrt{2n}$24

The derivative data are radial derivatives, meaning that if $\sqrt{2n}$25, then $\sqrt{2n}$26, and similarly for $\sqrt{2n}$27 (Cohn et al., 2019).

The basis is encoded by generating functions

$\sqrt{2n}$28

$\sqrt{2n}$29

which satisfy

$\sqrt{2n}$30

$\sqrt{2n}$31

and the fundamental Gaussian identity

$\sqrt{2n}$32

The proof constructs $\sqrt{2n}$33 and $\sqrt{2n}$34 from integral transforms of modular and quasimodular forms, extending Viazovska’s sphere-packing method (Cohn et al., 2019).

This interpolation theorem is used in the proof of universal optimality of the $\sqrt{2n}$35 and Leech lattices. The sample radii $\sqrt{2n}$36 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 (Cohn et al., 2019).

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{2n}$37 theory, if $\sqrt{2n}$38, $\sqrt{2n}$39, $\sqrt{2n}$40, and

$\sqrt{2n}$41

then there exist radial functions $\sqrt{2n}$42 such that for all $\sqrt{2n}$43,

$\sqrt{2n}$44

with absolute convergence in a higher regularity space $\sqrt{2n}$45 (Ramos et al., 2021).

For the one-dimensional Radchenko–Viazovska formula, there is a sharper perturbative result: if

$\sqrt{2n}$46

with $\sqrt{2n}$47, then there exist functions $\sqrt{2n}$48 satisfying

$\sqrt{2n}$49

such that every real-valued even Schwartz function satisfies

$\sqrt{2n}$50

The same paper proves an analogue for the derivative interpolation formulas in dimensions $\sqrt{2n}$51 and $\sqrt{2n}$52: if $\sqrt{2n}$53, then any radial Schwartz function is uniquely determined by the perturbed values

$\sqrt{2n}$54

These perturbation results are obtained by viewing interpolation as invertibility of an operator that is a perturbation of the identity on weighted sequence spaces (Ramos et al., 2020).

Regularity of the perturbed bases can be upgraded further. If

$\sqrt{2n}$55

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,

$\sqrt{2n}$56

then the perturbed basis functions remain entire of order $\sqrt{2n}$57 and finite type $\sqrt{2n}$58. In particular, vanishing on all but finitely many perturbed nodes forces analyticity of order $\sqrt{2n}$59, a property already known in the exact Radchenko–Viazovska case and now shown to persist under sufficiently small perturbations (Cassese et al., 19 Mar 2025).

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 (Ramos et al., 2021, Ramos et al., 2020, Cassese et al., 19 Mar 2025).

6. Density, nonuniqueness, and structural limitations

Several results delineate what radial Fourier interpolation formulas do not provide. First, the sphere-restriction theorem in dimensions $\sqrt{2n}$60 is a sampling formula, not a free interpolation theorem: it reconstructs a Schwartz function from the restrictions of $\sqrt{2n}$61 and $\sqrt{2n}$62 to the spheres $\sqrt{2n}$63, but it does not characterize which sequences of sphere data arise from a Schwartz function. In fact, for $\sqrt{2n}$64, the map sending $\sqrt{2n}$65 to these restrictions has infinite-dimensional cokernel (Stoller, 2020).

Second, general interpolation formulas involving both $\sqrt{2n}$66 and $\sqrt{2n}$67 are constrained by time-frequency density. If

$\sqrt{2n}$68

holds for all compactly supported smooth functions under mild growth assumptions, then

$\sqrt{2n}$69

For even or odd functions the factor $\sqrt{2n}$70 becomes $\sqrt{2n}$71. 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 $\sqrt{2n}$72 principle (Kulikov, 2020).

Third, the basis functions of the Radchenko–Viazovska formula exhibit substantial extra zero structure. For the Fourier-invariant basis $\sqrt{2n}$73, writing $\sqrt{2n}$74, one has $\sqrt{2n}$75 for $\sqrt{2n}$76, but $\sqrt{2n}$77 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

$\sqrt{2n}$78

so canonical interpolation at critical density does not imply frame-theoretic stability (Berghaus et al., 21 Dec 2025).

Fourth, higher-rank analogues of the square-root-lattice picture are obstructed. For lattices in $\sqrt{2n}$79 with $\sqrt{2n}$80, the group-theoretic conditions needed for a multivariable $\sqrt{2n}$81-type interpolation formula generally fail if the associated subgroup of $\sqrt{2n}$82 is discrete. The one-radial-variable case avoids this obstruction because the relevant Hecke-group subgroup $\sqrt{2n}$83 is discrete and free for $\sqrt{2n}$84 (Radchenko et al., 2021).

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 (Gonçalves, 2015, Stoller, 2020, Radchenko et al., 2021, Berghaus et al., 21 Dec 2025).