A new approach to the interpolation of sampled data

Area of Science:

• Signal Processing
• Numerical Analysis
• Fourier Analysis

Background:

• Traditional interpolation methods often lack analytical tractability or computational simplicity.
• The need for efficient and mathematically flexible interpolation kernels in signal processing is well-established.

Purpose of the Study:

• To introduce a new class of interpolation kernels.
• To highlight their properties of local compactness and near band-limitation.
• To demonstrate their utility in analytical manipulations and numerical integration.

Main Methods:

• Construction of kernels as linear sums of Gaussian functions and their even derivatives.
• Derivation of a numerical Gaussian quadrature method for intractable integrals.
• Examination of mathematical properties including analytical manipulability.

Main Results:

• Kernels are locally compact in signal space and nearly band-limited in Fourier space.
• The proposed kernels facilitate analytical operations like convolutions and projection integrals.
• A numerical quadrature method is presented for integrals that resist analytical evaluation.

Conclusions:

• The new interpolation kernels offer a balance of computational simplicity and mathematical flexibility.
• These kernels are suitable for advanced signal processing applications requiring analytical manipulation.
• Further extensions to higher-order kernels are feasible and warrant further investigation.