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Appendix: Properties of the First-Order Filter

Here we present simple proofs of the main properties of the first-order linear low-pass filter on the real line of the coordinate .

Subsections

Invariance of Linear Functions
Action on Powers and Polynomials
Differentiability of Filtered Functions
Continuity of Filtered Functions
Action on Dirac's Delta ``Function''
Reduction to the Identity
Points of Discontinuity
Points of Non-Differentiability
Invariance of Definite Integrals
Periodicity of Filtered Functions
Invariance of Averages Over the Period
Action on the Fourier Coefficients with
Completeness of the Set of Eigenfunctions