Quadratic-Phase Fourier--Bessel Transform: definitions, properties and uncertainty principles

This manuscript introduces the quadratic-phase Fourier-Bessel transform, establishes its fundamental properties and associated convolution structures, and proves a Donoho-Stark-type uncertainty principle to extend classical results to this generalized framework.

The Gist

Imagine you are trying to listen to a song, but the music is constantly changing speed and pitch in a complex, swirling way. The standard tool for analyzing music, the Fourier Transform, is like a pair of glasses that works perfectly for steady, unchanging sounds. But when the music gets chaotic or "chirpy" (like a radar pulse or a bird's call that changes pitch), those glasses get blurry.

To fix this, mathematicians invented a new, more flexible pair of glasses called the Quadratic-Phase Fourier Transform. It can handle those swirling, changing sounds.

This paper takes that idea one step further. The author, Ahmed Saoudi, introduces a brand-new mathematical tool called the Quadratic-Phase Fourier–Bessel Transform. Think of this as a super-charged, multi-lens camera lens designed specifically for a very specific type of signal—one that behaves like ripples spreading out from a stone dropped in a pond (which is what "Bessel" functions describe).

Here is a breakdown of what the paper does, using simple analogies:

1. The New Tool: A Custom Lens

The author defines a new way to transform signals.

• The Old Way: Standard math tools treat signals as if they are static or change in simple ways.
• The New Way: This new transform uses a "kernel" (a mathematical recipe) that includes quadratic phases. Imagine the signal isn't just a flat line, but a curved surface. This tool can flatten that curve to analyze it properly.
• The "Bessel" Part: This adds a specific shape to the analysis, perfect for signals that radiate outward in circles or spheres (like sound waves in a room or light in an optical fiber).
• The "Knobs": The formula has five adjustable "knobs" (parameters $a, b, c, d, e$). By turning these knobs, this single new tool can actually mimic many other famous math tools (like the standard Fourier transform or the Fractional Fourier transform). It's a "Swiss Army Knife" of signal analysis.

2. Proving the Tool Works (Fundamental Properties)

Before using a new tool, you have to prove it doesn't break. The paper checks four main things:

• Continuity: If you make a tiny change to the input signal, the output doesn't suddenly explode or jump wildly. It changes smoothly.
• The "Fading" Rule (Riemann–Lebesgue): If you feed in a signal that is well-behaved, the result will eventually fade away to zero as you look further out. It won't stay loud forever.
• Reversibility: This is crucial. If you transform a signal, you must be able to transform it back to get the original signal exactly. The paper proves there is a specific "undo" button for this new transform.
• Energy Conservation (Parseval's Identity): Imagine the signal has a certain amount of "energy" (like the volume of a song). The paper proves that the total energy in the original signal is exactly the same as the total energy in the transformed signal. Nothing is lost or created; it's just rearranged.

3. Moving and Mixing Signals (Translation and Convolution)

To do real work with signals, you need to be able to move them around and mix them.

• Translation (Moving): In standard math, "moving" a signal is easy (just shift it left or right). In this new, curved world, "moving" is trickier. The author defines a special "Generalized Translation" operator. Think of it as a custom slider that moves the signal along the curved surface without distorting it.
• Convolution (Mixing): This is how you blend two signals together (like mixing two audio tracks). The paper defines a new way to mix signals that respects the rules of this new curved world. They prove that this mixing is fair: it doesn't matter which order you mix them in (commutative), and you can mix three signals in any grouping (associative).

4. The Uncertainty Principle (The "Fog" Rule)

This is the most famous part of signal analysis. There is a rule in physics and math called the Uncertainty Principle. It says: You cannot know exactly where a signal is (time) and exactly what its frequency is at the same time. It's like trying to take a photo of a fast-moving car: if you focus on the car's position, the background blurs; if you focus on the background, the car blurs.

The paper proves a Donoho–Stark-type uncertainty principle for this new tool.

• The Claim: If you try to squeeze a signal into a very small box (time-limited) AND try to squeeze its transformed version into a very small box (frequency-limited), you run into a hard limit.
• The Result: The paper calculates a mathematical "floor." It says the size of the time-box multiplied by the size of the frequency-box cannot be smaller than a specific number determined by the tool's settings. If you try to make both boxes too small, the math breaks. This confirms that even with this fancy new tool, nature still has a limit on how precisely we can pin down a signal.

Summary

Ahmed Saoudi has built a new mathematical microscope.

1. He defined the lens (The Transform).
2. He proved the lens is sharp and doesn't break (Continuity, Reversibility, Energy Conservation).
3. He figured out how to slide the lens and mix images (Translation and Convolution).
4. He measured the limits of the lens, proving that you can't see everything perfectly at once (Uncertainty Principle).

The paper is purely mathematical. It builds the foundation and the rules for this new tool, preparing the ground for future scientists to use it in fields like optics, radar, and signal processing, but the paper itself focuses strictly on establishing these mathematical rules.

Technical Summary

Technical Summary: Quadratic-Phase Fourier–Bessel Transform: Definitions, Properties, and Uncertainty Principles

Problem Statement and Context
The classical Fourier transform, while foundational, assumes signal stationarity, limiting its efficacy in analyzing non-stationary signals such as chirps, radar pulses, and optical wavefronts. To address this, the field has evolved through generalizations like the fractional Fourier transform and the linear canonical transform. More recently, the quadratic-phase Fourier transform (QPFT) was introduced as a unifying framework incorporating quadratic and linear phase terms via an exponential kernel.

However, a gap exists in extending this quadratic-phase framework to the Bessel domain, which is essential for problems involving radial symmetry and specific boundary conditions. This paper addresses the need to define and analyze a Quadratic-Phase Fourier–Bessel Transform (QPFBT) that integrates the flexibility of the QPFT with the structural properties of the Bessel transform. The work aims to establish a rigorous mathematical foundation for this new operator, including its fundamental properties, operational calculus (translation and convolution), and uncertainty principles.

Methodology
The authors define the QPFBT, denoted as $B^{a,b,c}_{d,e,\gamma}$, for an integrable function $h$ on the half-line $\mathbb{R}^+$, depending on five real parameters ($a, b, c, d, e$ with $b \neq 0$) and an index $\gamma > -1/2$. The transform is defined via an integral kernel involving a normalized Bessel function $j_\gamma$ and a quadratic-phase exponential term:
$$B^{a,b,c}_{d,e,\gamma}[h](t) = \frac{c_\gamma}{(ib)^{\gamma+1}} \int_0^\infty J^{a,b,c}_{d,e,\gamma}(t, s) h(s) s^{2\gamma+1} ds$$
where the kernel $J^{a,b,c}_{d,e,\gamma}$ combines phase modulation with the Bessel function.

The methodology proceeds through the following analytical steps:

1. Reduction Analysis: The authors demonstrate that by selecting specific parameter values, the QPFBT reduces to known transforms, including the linear canonical Fourier–Bessel transform, the fractional Fourier–Bessel transform, and the classical Fourier–Bessel transform.
2. Functional Analysis: The transform is studied within specific functional spaces, including $L^p_\gamma(\mathbb{R}^+)$ and the Schwartz space of even functions $S^*(\mathbb{R})$. Key properties such as continuity, the Riemann–Lebesgue lemma, and reversibility are proven using kernel bounds and Fubini's theorem.
3. Operational Calculus: To facilitate harmonic analysis, the authors define a generalized translation operator ($T^{a,b,d}_{t,\gamma}$) and a generalized convolution product ($\star$). These are constructed using the kernel of the translation operator associated with the standard Fourier–Bessel transform, modified by quadratic-phase factors.
4. Uncertainty Principles: The paper applies the developed framework to prove uncertainty principles, specifically adapting the Donoho–Stark theorem to the QPFBT setting. This involves defining time-limited and band-limited functions in the context of the new transform and analyzing the Hilbert-Schmidt norm of the associated projection operators.

Key Contributions and Results
The paper establishes the following core results:

• Foundational Properties:

  • Continuity and Boundedness: The transform maps $L^1_\gamma(\mathbb{R}^+)$ continuously into $C^*_{0}(\mathbb{R})$ (even continuous functions vanishing at infinity) and satisfies a specific norm inequality.
  • Riemann–Lebesgue Lemma: The transform of an $L^1_\gamma$ function vanishes at infinity.
  • Reversibility: An explicit inversion formula is derived, showing that the transform is invertible with an inverse defined by negating and permuting the parameters ($a, b, c, d, e \to -c, -b, -a, -e, -d$).
  • Parseval's Identity: The authors prove that the transform is an isometry on $L^2_\gamma(\mathbb{R}^+)$, preserving the $L^2$ norm (Plancherel theorem).

• Operational Tools:

  • Translation Operator: A generalized translation operator is defined and proven to be a contraction on $L^p_\gamma(\mathbb{R}^+)$ for $1 \le p \le \infty$.
  • Convolution Product: A convolution product is defined via the translation operator. The paper establishes its commutativity, associativity, and a Young's inequality for the QPFBT convolution, ensuring that the convolution of functions in $L^p$ and $L^q$ spaces remains in a corresponding $L^r$ space.

• Uncertainty Principles:

  • Donoho–Stark Type: The paper proves a Donoho–Stark-type uncertainty principle. It establishes a lower bound on the product of the measures of the time-support set ($M$) and the frequency-support set ($N$) for a function that is approximately time-limited and approximately band-limited. The bound depends on the parameters $b$ and $\gamma$, and the approximation errors $\epsilon_M$ and $\epsilon_N$.
  • Concentration Inequality: A second uncertainty principle is derived for functions in $L^1_\gamma \cap L^p_\gamma$, relating the norms of the function and its transform to the measures of their support sets.

Significance and Claims
The paper claims to extend the framework of quadratic-phase Fourier analysis by introducing the QPFBT, thereby unifying quadratic phase modulation with Bessel structures. The significance of this work lies in:

1. Generalization: It provides a broader, more flexible transform framework that encompasses several well-known integral transforms as special cases.
2. Analytical Tools: By establishing continuity, inversion, Parseval's identity, and a convolution theorem, the paper provides the necessary analytic tools for harmonic analysis in this generalized domain.
3. Theoretical Extension: The proof of the Donoho–Stark uncertainty principle extends classical results to this new setting, offering theoretical limits on the simultaneous localization of signals in the time and QPFBT frequency domains.

The authors note that these results lay the groundwork for future applications in signal processing and optics within generalized Bessel domains, though the current paper focuses strictly on the theoretical development of the transform and its properties.