# Order estimates of best orthogonal trigonometric approximations of classes of infinitely differentiable functions

## TL;DR

This paper derives precise order estimates for the best uniform orthogonal trigonometric approximations of classes of periodic functions with specific smoothness properties, focusing on cases where the derivative sequence decreases faster than any power but slower than geometric progression.

## Contribution

It provides exact order estimates for approximation errors of classes of infinitely differentiable functions with specific derivative decay rates, extending previous results to new decay regimes.

## Key findings

- Established exact order estimates in uniform norm for certain function classes.
- Derived similar estimates in L_s metrics for differentiable functions.
- Extended approximation theory to functions with derivatives decreasing faster than any power.

## Abstract

In this paper we establish exact order estimates for the best uniform orthogonal trigonometric approximations of the classes of $2\pi$-periodic functions, whose $(\psi,\beta)$-derivatives belong to unit balls of spaces $L_{p}$, $1\leq p<\infty$, in the case, when the sequence $\psi(k)$ tends to zero faster, than any power function, but slower than geometric progression. Similar estimates are also established in the $L_{s}$-metric, $1<s\leq\infty$ for the classes of differentiable functions, which $(\psi,\beta)$-derivatives belong to unit ball of space $L_{1}$.