Discrete Hermite Transform

• Discrete Hermite transform is a mathematical framework that constructs orthogonal signal representations using classical and q-Hermite polynomials, offering robust analysis and efficient algorithms.
• Matrix and spectral approaches mimic continuous Hermite functions through three-term recurrences and optimized Gaussian-type vectors, ensuring spectral concentration and uncertainty optimization.
• Applications span real-time compressive sensing, biomedical signal classification, and quantum computing, with practical algorithms enhancing sparsity and computational stability.

The discrete Hermite transform is a broad class of mathematical transforms constructed using discretized Hermite functions, polynomials, or their q-analogues, providing orthogonal bases for signal, data, and function representations. It encompasses developments ranging from the analytic theory of discrete Hermite (and q-Hermite) polynomials, matrix-based transforms that mimic the continuous Hermite transform, connections with discrete spectral methods, and practical algorithms for digital signal processing, compressive sensing, and quantum information tasks.

1. Foundational Theory: Discrete Hermite Polynomials and q-Analogues

Discrete Hermite polynomials—both classical and q-analogues—form the analytic backbone of the transform. In the classical setting, Hermite polynomials $H_n(x)$ are generated by the exponential formula

$$\exp(2xz - z^2 s) = \sum_{n=0}^{\infty} H_n(x,s) \frac{z^n}{n!}$$

with fundamental recurrence and differentiation properties ($d/dx\, H_n(x,s) = n H_{n-1}(x,s)$).

The q-analogues replace differentiation by the Jackson q-derivative $D_{(q)} f(x) = [f(x) - f(qx)] / [(1-q)x]$ and the classical integer $n$ with $[n] = (1-q^n)/(1-q)$, yielding the recurrence: $$H_{n+1}(x,s,q) = x H_n(x,s,q) - q^{n-1} s [n] H_{n-1}(x,s,q)$$ and operational formulas (Rodrigues-type) that elegantly generalize continuous Hermite representations: $$H_n(x, s, q) = \prod_{j=1}^{n} \left(x - q^{j-n} s D_{(q)}\right) 1$$ Orthogonality is maintained with respect to a linear functional $A$, satisfying $A(H_n(x,s,q)) = [n = 0]$. This ensures applicability in discrete transforms and their computational algorithms (Cigler, 2013).

Generalized discrete q-Hermite II polynomials further enable the explicit construction of q-deformed oscillator models, with first-order q-difference operators serving as ladder operators in the associated algebraic framework. These polynomials provide an orthonormal basis for weighted $L^2$ spaces, underpinning discrete Hermite transforms in nonuniform and “quantum-deformed” settings (Mezlini, 2015, Arjika, 2019).

2. Discrete Transform Construction: Matrix and Spectral Approaches

Several lines of work construct explicit, numerically tractable matrix or spectral transforms:

• Canonical eigenbases for the Discrete Fourier Transform (DFT), such as the minimal Hermite-type basis, are realized by symmetrized “Gaussian-type” and modified vectors whose entries, supports, and recurrence properties mimic, and converge to, Hermite functions under grid scaling. The construction is optimal under discrete uncertainty principles, mirroring Hardy’s theorem. Key features include three-term recurrence relations and spectral concentration properties (Kuznetsov et al., 2017, Kuznetsov, 2015).
• Expansions in Kravchuk functions, which discretize Hermite polynomials on uniform grids, provide an isospectral discretization of the quantum harmonic oscillator, with convergence to Hermite functions at a rate $O(h^{2-\delta})$ and efficient matrix exponentiation-based algorithms for practical transformation. Such schemes enable “Kravchuk transforms” for numerically stable Hermite-based processing, maintaining eigenvalues of the continuous operator and enabling robust time-propagation simulations (Chauleur et al., 2022).
• Rigged Hilbert space constructions and Gram–Schmidt orthogonalization produce full operator algebras—creation, annihilation, and number operators—on both $L^2$ (the unit circle) and the countable sequence space $\ell^2(\mathbb{Z})$, providing a framework for discrete Hermite harmonic analysis with guaranteed operator continuity and unitary mappings between the spaces (Celeghini et al., 2020).

3. Signal Processing Applications: Sparse Representations and Compressive Sensing

The discrete Hermite transform is particularly advantageous in compressive sensing and biomedical signal processing:

• Signals, such as ECG QRS complexes and ultra-wideband pulses, exhibit morphological similarity to Hermite basis functions and are therefore highly sparse in this domain. Discrete expansions, often via Gauss–Hermite quadrature (sampling at Hermite polynomial roots), yield efficient representations with only a few nonzero coefficients required for accurate reconstruction or classification (Brajovic et al., 2015, Vulaj et al., 2017, Brajovic, 2019).
• Statistical analysis of missing/time-domain samples or additive noise leads to explicit models for the Hermite coefficients: non-signal coefficients follow (half-)normal distributions while signal coefficients obey folded normal laws, enabling precise error/threshold analysis for support detection and necessitating hard-thresholding algorithms for noise removal. Optimal parameter tuning (scaling and alignment) is critical for maximizing sparsity, with automated procedures yielding 14% improvements in the number of significant coefficients retained for QRS compression (Brajovic et al., 2015, Brajovic, 2019).
• Discrete transforms have been utilized for robust polynomial approximation, fusing both value and derivative information via covariance-weighted orthogonalization and recurrence, yielding computationally stable fits for high-degree polynomials, particularly suitable for real-time geotechnical monitoring and similar high-noise environments (Ritt et al., 2019).

4. Advanced Topics: Quantum, q-Deformations, and Wavelet Extensions

Quantum information and analysis frameworks are rapidly developing discrete Hermite transform primitives:

• Efficient quantum algorithms implement the discrete Hermite transform in time logarithmic in dimension and inverse error, with basis states mapped to Hermite amplitudes—interpreted as Gaussian analogues of the Fourier transform. Fast-forward simulation of quantum harmonic oscillators is realized, yielding provable quantum query advantages in property testing and Gaussian Goldreich-Levin learning tasks. The Gaussian Poincaré inequality is foundational in sample complexity analysis, enforcing variance bounds in terms of expected squared gradient under the Gaussian measure (Jain et al., 6 Oct 2025).
• Discrete Hermite wavelet transforms extend classical theory, integrating wavelet-based multiresolution analysis with Hermite polynomial kernels. These methods decompose signals and images into approximate and detail coefficients using iterative filter banks, facilitating efficient color image compression. The orthogonality and energy-preserving properties of Hermite wavelets enable high compression ratios and quality (as measured by PSNR, CR, BPP, MSE), outperforming standard wavelet families (Muhi-Aldeen et al., 2023).

5. Connections, Generalizations, and Analytical Structures

The discrete Hermite transform subsumes broad algebraic, analytical, and computational frameworks:

• Summation and connection formulas for generalized discrete q-Hermite II polynomials link them to other families of q-orthogonal polynomials (q-Laguerre, Stieltjes–Wigert), facilitating inversion and kernel variants, and supporting flexible transform architectures tailored to specific signal features or algebraic deformation settings (Arjika, 2019).
• Rodrigues-type and operational formulas extend the set of computational methods for coefficient evaluation and expansion, essential for practical signal analysis, spectral estimation, and property testing tasks.
• Transform equivalence with continuous Hermite analysis is systematically maintained—discrete eigenvectors and kernels strongly converge to their continuous counterparts, preserving spectral, localization, and orthogonality properties (Kuznetsov et al., 2017, Chauleur et al., 2022).

6. Ongoing Challenges and Future Directions

Several open problems and research directions arise:

• Extending rigorous convergence results for discrete DFT eigenvectors to all Hermite functions (not only the first eight), with analytic proofs remaining challenging as combinatorics and asymptotics become involved for higher indices (Kuznetsov, 2015).
• The construction and utilization of discrete transforms based on q-Hermite polynomials for non-uniform, deformed, or quantum-lattice data, with further exploration of computational and spectral properties.
• Algorithmic integration for real-time, large-dimensional quantum and signal processing systems, leveraging fast matrix operations and leveraging uncertainty principles for optimal concentration and information-theoretic efficiency.

Table: Discrete Hermite Transform—Key Mathematical Objects

Transform Kernel Type	Defining Recurrence/Formula	Primary Domain/Use
Classical Hermite polynomials/funcs	$H_{n+1}(x) = x H_n(x) - n H_{n-1}(x)$	Continuous/Digital signals
Discrete q-Hermite polynomials	$$H_{n+1}(x,s,q) = x H_n(x,s,q) - q^{n-1} s [n] H_{n-1}(x,s,q)$$	q-deformed/Quantum models
Kravchuk functions	$$k_{n+1,h}(x) = 2x k_{n,h}(x) - 2n(1-h^2(n-1)/2)k_{n-1,h}(x)$$	Discretized oscillators
Hermite wavelet basis	Expansion via Hermite wavelets	Multiresolution/Image comp.
Minimal Hermite-type DFT eigenbasis	3-term recurrences, optimized width	Fourier spectral methods

The discrete Hermite transform thus serves as a unifying framework that connects orthogonal polynomial theory, discrete harmonic analysis, signal processing algorithms, quantum computation, and algebraic deformations. Its flexibility and depth enable broad applicability and ongoing research across computational mathematics, engineering, and physics.