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Chromatic derivatives, chromatic expansions and associated spaces II

Aleksandar Ignjatovic

TL;DR

<3-5 sentence high-level summary> The paper develops a unified framework of chromatic derivatives and chromatic expansions built on general families of orthogonal polynomials, enabling robust, local representations of band-limited signals. It constructs and analyzes spaces L^2_{M} and L^2_{M} with associated Fourier-type transforms, proving uniform convergence results and demonstrating operator compatibility. Through extensive classical-polynomial examples (Legendre, Chebyshev, Gegenbauer, Jacobi, Hermite, Laguerre, Herron), it shows how familiar special functions arise as basis elements and extends Bessel-function identities to a broad chromatic setting. The work also links these expansions to almost periodic signal spaces and provides a geometric interpretation via moving frames in l^2, highlighting practical implications for stable numerical differentiation and spectral-type analyses in signal processing.

Abstract

We study differential operators associated with families of polynomials orthonormal with respect to certain measures. These operators, when applied to the Fourier transforms of such measures, produce basis functions for expansions of functions analytic on some complex domains. For many classical families of orthogonal polynomials these basis functions are the familiar special functions, such as the Bessel and the spherical Bessel functions. Many familiar identities involving such special functions turn out to be just special cases of such expansions. We also use these differential operators to introduce some new spaces of almost periodic functions. The notions we study here have been successfully applied to signal processing, for example to recovery of band-limited signals from their non-uniform samples as well as from their zero crossings and the locations of their extremal points.

Chromatic derivatives, chromatic expansions and associated spaces II

TL;DR

<3-5 sentence high-level summary> The paper develops a unified framework of chromatic derivatives and chromatic expansions built on general families of orthogonal polynomials, enabling robust, local representations of band-limited signals. It constructs and analyzes spaces L^2_{M} and L^2_{M} with associated Fourier-type transforms, proving uniform convergence results and demonstrating operator compatibility. Through extensive classical-polynomial examples (Legendre, Chebyshev, Gegenbauer, Jacobi, Hermite, Laguerre, Herron), it shows how familiar special functions arise as basis elements and extends Bessel-function identities to a broad chromatic setting. The work also links these expansions to almost periodic signal spaces and provides a geometric interpretation via moving frames in l^2, highlighting practical implications for stable numerical differentiation and spectral-type analyses in signal processing.

Abstract

We study differential operators associated with families of polynomials orthonormal with respect to certain measures. These operators, when applied to the Fourier transforms of such measures, produce basis functions for expansions of functions analytic on some complex domains. For many classical families of orthogonal polynomials these basis functions are the familiar special functions, such as the Bessel and the spherical Bessel functions. Many familiar identities involving such special functions turn out to be just special cases of such expansions. We also use these differential operators to introduce some new spaces of almost periodic functions. The notions we study here have been successfully applied to signal processing, for example to recovery of band-limited signals from their non-uniform samples as well as from their zero crossings and the locations of their extremal points.

Paper Structure

This paper contains 32 sections, 52 theorems, 188 equations, 3 figures.

Table of Contents

  1. Motivation
  2. Numerical differentiation of band limited signals
  3. Chromatic derivatives
  4. Chromatic expansions
  5. A local definition of the scalar product in $\mathbf{BL}(\pi)$
  6. Basic Notions
  7. Families of orthogonal polynomials
  8. Space $L^2_{d\alpha}$.
  9. Chromatic derivatives
  10. Chromatic expansions
  11. Space ${\mathbf L}_{{\!{{\mathcal{M}}}}}^2$
  12. Space $L_{\!{{\mathcal{M}}}}^2$
  13. Relationship of $\mathcal{F}_{\!\!\!\!\!~_{\!\!\!{\mathcal{M}}}}[f](\omega)$ with the usual Fourier transform
  14. Uniform convergence of chromatic expansions
  15. Chromatic expansions of functions in $L_{\!{{\mathcal{M}}}}^2$
  16. ...and 17 more sections

Key Result

Proposition 1.1

Let ${\mathcal{K}}^{n}$ be the chromatic derivatives associated with the Legendre polynomials, let ${\mathrm j}_n(x)$ be the spherical Bessel function of the first kind of order $n$, and let $f$ be an arbitrary entire function; then for all $z,u\in\mathop{\mathds{C}}$, Moreover, if $f\in\mathbf{BL}(\pi)$, then the series converges uniformly on $\mathop{\mathds{R}}$ and also converges with respect

Figures (3)

  • Figure 1: Graphs of $\left(\frac{\omega}{\pi}\right)^n$ (left) and of $P^{{L}}_{{n}}(\omega)$ (right) for $n=15-18$.
  • Figure 2: Left: transfer function of the (normalised) standard derivative $\frac{1}{\pi^{32}}\frac{d^{32}}{dt^{32}}$ of order 32 (gray) and of its FIR filter approximation (black). Center: transfer function of the chromatic derivative ${\mathcal{K}}^{32}$ (gray) and of its FIR approximation (black). Right: the error of a 129 tap FIR filter approximation for ${\mathcal{K}}^{32}$ within its bandpass $[-.9\pi,.9\pi]$.
  • Figure 3: (left) plots of $\mathrm{sinc}\, (t)$ (black) and ${\mathcal{K}}^{15}[\mathrm{sinc}\, ](t)$ (grey); (right) A band-limited signal (black) and its chromatic approximation of order 15 (grey) and its Taylor approximation of the same order (dashed).

Theorems & Definitions (92)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • proof
  • ...and 82 more