This repository contains a Python implementation of Prolate Spheroidal Wave Functions (PSWFs) for both finite and infinite support. The project also verifies various mathematical properties of PSWFs and demonstrates their applications in signal processing, spectral analysis, and applied mathematics.
Prolate Spheroidal Wave Functions (PSWFs) are a special class of orthogonal functions that solve the time-bandwidth concentration problem. They arise in various disciplines and are closely related to Legendre polynomials and Fourier analysis.
PSWFs, denoted as ψ_n(c, x), are solutions to the prolate spheroidal wave equation:
d²ψ/dx² + (λ_n - c²x²)ψ = 0
where:
- λ_n are eigenvalues associated with the functions,
- c is the bandwidth parameter that controls spectral concentration,
- x represents the spatial or temporal domain.
These functions are optimal for representing band-limited signals in a given time interval and are extensively used in signal processing, information theory, and physics.
✅ Orthogonality: PSWFs form an orthogonal basis over both finite and infinite domains.
✅ Eigenfunction Representation: They serve as eigenfunctions of the time-limiting and band-limiting operators.
✅ Maximal Energy Concentration: Among all functions with a fixed bandwidth, PSWFs maximize energy concentration in a given time interval.
✅ Connection to Legendre Polynomials: PSWFs generalize Legendre polynomials and play a fundamental role in spectral analysis.
🔹 Signal Processing - Efficient representation of band-limited signals in finite time intervals.
🔹 Spectral Analysis - Used in eigenvalue problems related to Fourier transform operators.
🔹 Antenna Design - PSWFs are applied in beamforming and wave propagation modeling.
🔹 Quantum Mechanics - Solve certain Schrödinger equations with boundary constraints.
🔹 Optics & Imaging - Enhance image resolution and reconstruction techniques.
This implementation focuses on numerical computation of PSWFs, visualizing their properties, and verifying theoretical results.
🔹 Finite & Infinite Support Implementation
🔹 Numerical Computation of Eigenvalues & Eigenfunctions
🔹 Orthogonality Verification
🔹 Spectral Analysis Demonstrations
🔹 Efficient Computation using Python & NumPy
🚀 Future Enhancements: Additional PSWF applications in signal filtering, quantum mechanics, and advanced wave analysis.