Shannon–Gabor wavelet distributed approximating functional

The Shannon sampling theorem is critically reviewed from a physical point of view. An approximate sampling formula is proposed, combining Shannon sampling with a Gabor-distributed approximating functional (DAF) window function, which results in new Shannon–Gabor wavelet DAFs (SGWDs). They are extremely smooth, decay rapidly, have simultaneous time-frequency localization, and are also generalized delta sequences (reducing to the Dirac delta function under the limit of a zero window width). Shannonʹs sampling theorem is recovered exactly when the window is infinitely wide.Finally, SGWDs are well-behaved L2(R) kernels, and thus can be used for solving differential equations.