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Data Resonance

1.Bases induced by eigenmodes

The trigonometric functions ${1, \cos t, \sin t, \cos 2 t, \sin 2 t, \dots}$ have been introduced as a particularly appropriate basis for periodic functions. The functions $\cos (k t), \sin (k t)$ are also known as solution of the homogeneous differential equation

y^{''} + k^{2} y = 0 .

The diferential operator is a linear mapping

\frac{d^{q}}{d t^{q}} (α y + β z) = α \frac{d^{q} y}{d t^{q}} + β \frac{d^{q} z}{d t^{q}},

and hence has an associated linear mapping. An approximation of the second-order differentiation operation is given by the finite difference formulas

y_{i}^{''} = y^{''} (t_{i}) ≅ \frac{1}{h^{2}} (y_{i + 1} - 2 y_{i} + y_{i - 1})