transform op• Data kinds: signal × signal → table
• Call: import fullseye as fs; fs.ledger.hankel_transform(r, f, *, r_max: 'float | None' = None, order: 'int' = 0, n: 'int' = 256) -> 'dict' (to call the implementation directly, import mathtransforms; mathtransforms.hankel_transform(r, f, *, r_max: 'float | None' = None, order: 'int' = 0, n: 'int' = 256) -> 'dict'; from the registry, opsmath.get("hankel_transform"))
Hankel transform of order p, F(ν) = 2π ∫_0^∞ f(r) J_p(2πνr) r dr, by the quasi-discrete method.
(r, f) are radial samples (r ascending). They are put onto the sample points r_k by linear interpolation; outside r_max (default = max of r) the value is 0. If a callable is passed as f, it is evaluated directly at the sample points (r is ignored; r_max must be passed). The sample points are r_k = j_k R / S (j_k is the k-th zero of J_p, S = j_{n+1}), frequencies ν_k = j_k / (2πR). Because the kernel T of this transform is an involution (T·T = I), the same function also does the inverse transform (pass F as f).
Gate: exp(−π r²) is self-dual (F(ν) = exp(−π ν²)). The order-0 Hankel transform of an axisymmetric function equals the radial cross-section of its 2-D Fourier transform.
Returns: {"r", "f", "nu", "F", "T_involution_error"} (the last is ‖T·T − I‖_max).
Every mathops op validates its input before computing (nothing slips through silently):
• complex input raises ValueError — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• masked arrays with masked elements raise ValueError — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• NaN/Inf raises ValueError on every input (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• transforms_tour — py -3.11 examples/transforms_tour.py
table as input)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
transform)abel_transform · abel_inverse · abel_inverse_image · abel_revolve · tf_poles_zeros · tf_freq_response · tf_impulse_response · tf_step_response
*Provenance: mathtransforms.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.