# On One Property of Tikhonov Regularization Algorithm

## TL;DR

This paper demonstrates that Tikhonov regularization is minimax optimal for linear inverse problems with Gaussian noise, especially when the solution's Fourier coefficients satisfy certain a priori conditions, including membership in a Besov space.

## Contribution

It establishes the minimax optimality and asymptotic minimaxity of Tikhonov regularization under specific a priori Fourier coefficient conditions.

## Key findings

- Tikhonov regularization is minimax in the class of linear estimators.
- It is asymptotically minimax among all estimators.
- Results depend on a priori information about Fourier coefficients, especially in Besov spaces.

## Abstract

For linear inverse problem with Gaussian random noise we show that Tikhonov regularization algorithm is minimax in the class of linear estimators and is asymptotically minimax in the sense of sharp asymptotic in the class of all estimators. The results are valid if some a priori information on a Fourier coefficients of solution is provided. For trigonometric basis this a priori information implies that the solution belongs to a ball in Besov space $B^r_{2\infty}$.