Plain Tikhonov penalises \|m\|^2 — it prefers a small model. But "small" is rarely what we actually want; usually we want smooth. General Tikhonov swaps the penalty for the norm of a linear operator L applied to the model:

Choosing L is choosing what "simple" means — it encodes the prior belief about the solution.

• L = I: penalise size → a small solution (standard Tikhonov).
• L = D_1 (first difference, a discrete derivative): penalise slope → a flat solution.
• L = D_2 (second difference): penalise curvature → a smooth solution.

The penalty is a prior

The solution is again a modified normal equation, (G^{\mathsf T}G + \alpha^2 L^{\mathsf T}L)\,\hat m = G^{\mathsf T}d. The derivative penalty makes oscillatory, jagged solutions expensive, so the minimiser comes out smooth — exactly suppressing the high-frequency noise that ill-posedness amplifies. This is the same instinct as a smoothing prior, and it foreshadows the Bayesian reading: the penalty operator L is the inverse square root of a prior covariance — a precise statement of "I expect the answer to be smooth".

Smooth vs jagged

Two candidate solutions that fit the data about equally well: a smooth one and a jagged one carrying high-frequency wiggles. A first- or second-derivative penalty \|Lm\|^2 is small for the smooth curve and large for the jagged one, so general Tikhonov picks the smooth candidate — the wiggles are exactly the amplified noise we want gone.