Ball Harmonics Expansion Overview

• Ball harmonics expansion is an operator-adapted angular–radial decomposition that separates a function’s spherical modes from its radial components using tailored bases.
• It underpins applications in spectral analysis, imaging, inverse problems, and PDE discretizations by adapting the radial basis to the underlying operator or boundary condition.
• Fast computational techniques and orthogonal polynomial bases enhance its efficiency, accuracy, and applicability in high-dimensional simulations and data reconstructions.

Searching arXiv for recent and foundational papers on ball harmonics expansion. Ball harmonics expansion denotes a family of separable representations for functions, kernels, or fields on a ball in which the angular dependence is expanded in spherical harmonics or related spherical modes, while the radial dependence is carried by basis functions determined by the underlying operator, boundary condition, or reproducing structure. In the Dirichlet-Laplacian setting on the real unit ball, the canonical basis consists of eigenfunctions $$\psi_{l,m,n}(r,\theta,\phi)=C_{l,n}j_l(\alpha_{l,n}r)Y_l^m(\theta,\phi)$$ with eigenvalues $\lambda_{l,n}=\alpha_{l,n}^2$ (Kileel et al., 2024). In broader usage, the same separation principle appears in shellwise spherical-harmonic reconstructions, in vector spherical harmonics, in Sobolev-orthogonal polynomial bases on the ball, and in reproducing-kernel expansions for $M$-harmonic and $H$-harmonic functions (Flamant et al., 2016).

1. Core separation principle

At its most basic, ball harmonics expansion is the statement that a function on a ball can be decomposed into angular modes on the sphere and radial modes on the interval. In the single-particle imaging formulation, this is written as

$$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$

with admissible radial choices such as spherical Bessel $j_\ell$, Zernike radial polynomials, or Fourier–Bessel modes; the same source emphasizes that a shellwise discretization in radius yields a practical ball expansion sampled in the radial coordinate (Flamant et al., 2016). On the unit ball in $\mathbb{R}^3$, the Dirichlet Laplacian eigenbasis gives a distinguished version of the expansion that is orthonormal, rotation-steerable, and frequency-ordered (Kileel et al., 2024). On the unit disk, the analogous basis is the Fourier–Bessel basis, also referred to as the harmonics on the disk (Marshall et al., 2022).

The literature does not impose a single universal radial factor. Instead, the radial part is adapted to the operator under study: Dirichlet eigenfunctions for the Laplacian, shellwise samples in diffraction reconstruction, Sobolev orthogonal polynomials for variational PDE discretizations, or hypergeometric radial factors in reproducing-kernel problems. This suggests that “ball harmonics expansion” is best understood as an operator-adapted angular–radial decomposition rather than a single fixed basis.

Setting	Angular component	Radial or kernel component
Dirichlet Laplacian on $B\subset\mathbb{R}^3$	$Y_l^m$	$j_l(\alpha_{l,n}r)$
Shellwise diffraction intensity	$\lambda_{l,n}=\alpha_{l,n}^2$0 on each shell	discrete shells $\lambda_{l,n}=\alpha_{l,n}^2$1
Disk harmonics on $\lambda_{l,n}=\alpha_{l,n}^2$2	$\lambda_{l,n}=\alpha_{l,n}^2$3	$\lambda_{l,n}=\alpha_{l,n}^2$4
Complex-ball $\lambda_{l,n}=\alpha_{l,n}^2$5-harmonic kernels	unitary spherical harmonics $\lambda_{l,n}=\alpha_{l,n}^2$6	$\lambda_{l,n}=\alpha_{l,n}^2$7

2. Spectral bases, orthogonality, and rotation

For the Dirichlet Laplacian on the unit ball, separation of variables yields the radial equation

$\lambda_{l,n}=\alpha_{l,n}^2$8

whose regular solutions are spherical Bessel functions $\lambda_{l,n}=\alpha_{l,n}^2$9; the Dirichlet boundary condition forces $M$0 to be a positive zero $M$1 of $M$2 (Kileel et al., 2024). With

$M$3

the basis is orthonormal in $M$4, and the corresponding coefficients are obtained by projection against $M$5 (Kileel et al., 2024). In the more general radius-$M$6 formulation, the orthonormal basis of $M$7 is

$M$8

where $M$9 is the $H$0-th positive root of $H$1 and $H$2 (Kruse et al., 1 Sep 2025).

A central structural property is steerability. Under a rotation $H$3, spherical harmonics transform by Wigner $H$4-matrices,

$H$5

so the ball-harmonic coefficients transform block-diagonally in $H$6 and the radial index (Flamant et al., 2016). In the Dirichlet-Laplacian basis used for subtomogram alignment, this becomes

$H$7

which is the basis of exact rotational matching in coefficient space (Kruse et al., 1 Sep 2025). The disk analogue is even simpler: rotating an image by angle $H$8 multiplies each Fourier–Bessel coefficient by $H$9 (Marshall et al., 2022).

Vector-valued variants preserve the same separation principle. For divergence-free fields on the unit ball in $$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$0, the expansion is organized by vector spherical harmonics, and the toroidal mode is singled out by the identity

$$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$1

The resulting divergence-free vectorial ball PSWFs are

$$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$2

so the scalar basis and the divergence-free vector basis are linked by the solenoidal operator $$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$3 (Zhang et al., 2020).

3. Kernel-theoretic expansions on real and complex balls

Ball harmonics expansion also appears in reproducing-kernel theory, where the object being expanded is a kernel rather than a generic $$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$4 function. On the unit ball

$$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$5

the invariant Laplacian

$$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$6

defines the class of $$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$7-harmonic functions, and the restriction representation of $$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$8 on $$f(r,\theta,\phi)=\sum_{n,\ell,m} a_{n\ell m}R_{n\ell}(r)Y_{\ell m}(\theta,\phi),$$9 decomposes multiplicity-free as

$j_\ell$0

where $j_\ell$1 consists of restrictions of harmonic polynomials homogeneous of degree $j_\ell$2 in $j_\ell$3 and degree $j_\ell$4 in $j_\ell$5 (Englis et al., 2022). The associated Poisson–Szegő kernel admits the expansion

$j_\ell$6

and the $j_\ell$7-harmonic Szegő kernel satisfies

$j_\ell$8

with an explicit closed form in terms of the four-variable hypergeometric function $j_\ell$9 (Englis et al., 2022). The same work gives a general series for weighted $\mathbb{R}^3$0-harmonic Bergman kernels and argues against a simple closed form in higher dimensions by exhibiting coefficients involving $\mathbb{R}^3$1 already in the case $\mathbb{R}^3$2 (Englis et al., 2022).

A related but distinct real-ball theory concerns $\mathbb{R}^3$3-harmonic functions, i.e. functions annihilated by the hyperbolic Laplacian on $\mathbb{R}^3$4. There the Szegő kernel is

$\mathbb{R}^3$5

and it admits an explicit series expansion via Exton’s triple hypergeometric function $\mathbb{R}^3$6, as well as a finite sum representation in Gauss $\mathbb{R}^3$7 functions (Moravík, 13 Oct 2025). In the weighted Bergman setting,

$\mathbb{R}^3$8

which realizes the kernel as a spherical/radial decomposition into zonal harmonics and radial factors (Moravík, 13 Oct 2025).

These kernel-theoretic examples show that ball harmonics expansion is not restricted to eigenfunction expansions of the Laplacian. It also functions as a representation-theoretic device for decomposing reproducing kernels into spherical projectors and radial coefficients.

4. Fast transforms and PDE-adapted polynomial bases

A major recent development is the construction of fast transforms between Cartesian data and ball-harmonic coefficients. For voxelized data on an $\mathbb{R}^3$9 grid, fast expansion into the Dirichlet-Laplacian basis on the unit ball is achieved by combining a 3D NUFFT, spherical harmonic transforms on $B\subset\mathbb{R}^3$0, and radial interpolation from Chebyshev nodes to the Dirichlet spectrum $B\subset\mathbb{R}^3$1 (Kileel et al., 2024). The resulting forward and inverse transforms satisfy relative $B\subset\mathbb{R}^3$2–$B\subset\mathbb{R}^3$3 accuracy $B\subset\mathbb{R}^3$4 and run in

$B\subset\mathbb{R}^3$5

whereas naive dense application of the expansion operators costs $B\subset\mathbb{R}^3$6 (Kileel et al., 2024). The 2D analogue, the Fast Disk Harmonics Transform, computes Fourier–Bessel coefficients from an $B\subset\mathbb{R}^3$7 Cartesian image in $B\subset\mathbb{R}^3$8 operations and preserves the diagonal action of rotations and radial convolutions in coefficient space (Marshall et al., 2022).

Spectral methods on the ball also motivate polynomial ball-harmonic systems that are not Dirichlet eigenfunctions. For the stationary Schrödinger equation

$B\subset\mathbb{R}^3$9

a fully diagonalized method is obtained by working in the polynomial space

$Y_l^m$0

and constructing basis functions

$Y_l^m$1

where the radial factors are Sobolev orthogonal polynomials associated with a univariate Sobolev inner product (Piñar, 22 Jan 2026). Because the basis is orthogonal in the variational inner product, the Galerkin matrix becomes diagonal mode-by-mode (Piñar, 22 Jan 2026).

An earlier ball-polynomial construction uses the inner product

$Y_l^m$2

which adds a boundary term on the sphere (Martínez et al., 2015). The associated basis functions keep the spherical harmonics unchanged and modify only the radial Jacobi factor to a Jacobi-type polynomial

$Y_l^m$3

thereby producing orthogonal ball expansions adapted to the ball-plus-sphere geometry (Martínez et al., 2015). For $Y_l^m$4, these polynomials satisfy a fourth-order PDE on the ball (Martínez et al., 2015).

5. Imaging, alignment, and field simulation

In inverse problems, ball harmonics expansion is often used to decouple angular and radial degrees of freedom while preserving exact rotation laws. In single-particle imaging with X-ray lasers, the diffraction intensity on each shell is expanded as

$Y_l^m$5

with Friedel symmetry forcing odd $Y_l^m$6 to vanish, and the radial dependence handled by processing concentric shells $Y_l^m$7 independently (Flamant et al., 2016). The shell-by-shell EMC algorithm alternates expansion, maximization, and compression; for each shell, the dominant M-step cost scales as

$Y_l^m$8

and the paper reports, for example, a full reconstruction of the STNV virus with $Y_l^m$9 shells, $j_l(\alpha_{l,n}r)$0, $j_l(\alpha_{l,n}r)$1, $j_l(\alpha_{l,n}r)$2 patterns, and R-factors $j_l(\alpha_{l,n}r)$3, $j_l(\alpha_{l,n}r)$4, and $j_l(\alpha_{l,n}r)$5 for truncated theory at $j_l(\alpha_{l,n}r)$6 respectively (Flamant et al., 2016).

For subtomogram alignment in cryo-ET, the Dirichlet-Laplacian basis on the ball is used directly. Rotational correlation between two volumes is expressed as

$j_l(\alpha_{l,n}r)$7

so rotation evaluation reduces to small matrix-vector contractions in $j_l(\alpha_{l,n}r)$8 blocks (Kruse et al., 1 Sep 2025). The reported implementation uses $j_l(\alpha_{l,n}r)$9 and $\lambda_{l,n}=\alpha_{l,n}^2$00, combines frequency marching with Newton refinement, and achieves at least $\lambda_{l,n}=\alpha_{l,n}^2$01 speed-up compared to FRM-style exhaustive rotational matching while retaining identical accuracy to FRM and often sub-degree accuracy; a representative case achieves approximately $\lambda_{l,n}=\alpha_{l,n}^2$02 orientation error (Kruse et al., 1 Sep 2025).

In plasma simulation, the same angular–radial decomposition appears in vector form. The VSHPIC algorithm expands electric and magnetic fields as

$\lambda_{l,n}=\alpha_{l,n}^2$03

and similarly for $\lambda_{l,n}=\alpha_{l,n}^2$04 (Wang et al., 2024). Projection of Maxwell’s equations onto the vector spherical harmonics reduces the field solve to a set of 1D radial systems for each retained $\lambda_{l,n}=\alpha_{l,n}^2$05 mode, while a staggered deposition scheme enforces the discrete continuity equation and achieves deviations smaller than $\lambda_{l,n}=\alpha_{l,n}^2$06 in double precision over long runs (Wang et al., 2024). The code also reproduces the cold-plasma dispersion relation $\lambda_{l,n}=\alpha_{l,n}^2$07 from the transformed $\lambda_{l,n}=\alpha_{l,n}^2$08 modes (Wang et al., 2024).

6. Special cases, misconceptions, and related expansions

Several special cases clarify what ball harmonics expansion is, and what it is not. On the unit disk, the Dirichlet eigenbasis reduces to the Fourier–Bessel system

$\lambda_{l,n}=\alpha_{l,n}^2$09

which is orthogonal, frequency-ordered, and steerable; on the complex unit disc, the $\lambda_{l,n}=\alpha_{l,n}^2$10-harmonic Szegő kernel reduces to the classical harmonic formula

$\lambda_{l,n}=\alpha_{l,n}^2$11

when $\lambda_{l,n}=\alpha_{l,n}^2$12 (Marshall et al., 2022, Englis et al., 2022). These low-dimensional cases are much simpler than their higher-dimensional analogues, where multivariable hypergeometric structure and logarithmic-type boundary behavior appear (Englis et al., 2022).

A common misunderstanding is to identify ball harmonics expansion exclusively with regular solid harmonics $\lambda_{l,n}=\alpha_{l,n}^2$13. The cited literature supports a broader view. Regular and irregular solid spherical harmonics are one important family, but Dirichlet eigenfunctions use $\lambda_{l,n}=\alpha_{l,n}^2$14, vectorial PSWFs use toroidal vector spherical modes, Sobolev methods use univariate Sobolev orthogonal polynomials in $\lambda_{l,n}=\alpha_{l,n}^2$15, and reproducing-kernel theories use hypergeometric radial factors (Kileel et al., 2024, Zhang et al., 2020, Piñar, 22 Jan 2026, Englis et al., 2022). A plausible implication is that the term is most coherent when read functionally: it denotes a decomposition on the ball into spherical angular modes and a problem-specific radial sector.

Related coordinate systems further delimit the scope of the method. Relationships between solid spherical and toroidal harmonics allow one to convert toroidal-harmonic representations into regular or irregular spherical expansions, with coefficients determined by forward-stable recurrences for $\lambda_{l,n}=\alpha_{l,n}^2$16 and $\lambda_{l,n}=\alpha_{l,n}^2$17 (Majic et al., 2018). However, those expansions have restricted convergence domains, and the paper emphasizes that the relationships between toroidal and spherical harmonics are non-invertible in general because singularity geometry determines whether a convergent expansion exists (Majic et al., 2018). This is an important caution: ball harmonics expansion is powerful when its radial basis matches the operator and singularity structure, but it is not basis-independent, nor does it erase domain-of-convergence issues.

Taken together, these developments place ball harmonics expansion at the intersection of harmonic analysis, spectral theory, numerical analysis, and computational imaging. Its unifying mechanism is separation into spherical angular modes and radial structure on the interval, while its diversity comes from the fact that the radial sector may encode Dirichlet spectra, Sobolev orthogonality, reproducing kernels, divergence constraints, or shellwise discretizations, depending on the problem class (Kileel et al., 2024).