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Integral-Transform Methods in Medical Artificial Intelligence: Theorems and a Numerical Illustration for Hankel-Bessel Front-Ends

Author(s) Prof. Dr. Balasaheb B. Waphare
Country India
Abstract Artificial intelligence in medicine now operates a layered mathematical machinery, much of which originates in the classical theory of integral transforms. It is argued here that this classical theory, comprising the Hankel transform, Bessel-wavelet representations, the Meijer transform and the Prasad-Kumar pseudodifferential calculus, occupies a structurally privileged but practically under-exploited position in medical-image and biosignal analysis. The argument is substantiated with six analytical results, a connecting lemma, and an illustrative numerical experiment on simulated biosignals. The results are: a regularity-preservation theorem for Hankel-based reconstructions from non-Cartesian k-space sampling (Theorem 2.1); a pointwise stability bound for Bessel-wavelet representations of finite-energy biosignals (Theorem 2.7); a discrete frame inequality for the Bessel-wavelet family on a Paley-Wiener subspace, with explicit frame bounds (Theorem 2.9); a Hankel-domain convolution lemma (Proposition 2.4) yielding a closed-form denoising error bound at the Pinsker rate for radial Sobolev classes (Theorem 2.5); a Rademacher-based excess-risk bound for classifiers built on Bessel-wavelet features (Theorem 2.11); and a privacy-preservation result for differentially-private stochastic gradient descent trained on top of integral-transform front-ends (Proposition 3.1). Theorem 2.11 is illustrated by a controlled numerical experiment on simulated electrocardiogram data with twelve-fold replication: the observed generalisation gap decays at the predicted inverse-square-root rate in the sample size, with the bound itself loose by two to three orders of magnitude in the regime tested, consistent with the well-known looseness of Rademacher bounds for linear classifiers on natural data.
Keywords Hankel Transform, Bessel-Wavelet, Pseudo-differential Operators, Denoising, Generalisation Bound, Differential Privacy, Medical Artificial Intelligence
Field Mathematics
Published In Volume 7, Issue 4, July-August 2026
Published On 2026-08-05
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