In metric of spaces \(L_{s}, \ 1\leq s\leq\infty\), we obtain exact in order estimates of best \(m\)-term trigonometric approximations of classes of convolutions of periodic functions, that belong to unit all of space \(L_{p}, \ 1\leq p\leq\infty\), with generated kernel \(\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k)\cos(kt-\frac{\beta\pi}{2})\), \(\beta\in \mathbb{R}\), whose coefficients \(\psi(k)\) tend to zero not slower than geometric progression. Obtained estimates coincide in order with approximation by Fourier sums of the given classes of functions in \(L_{s}\)-metric. This fact allows to write down exact order estimates of best orthogonal trigonometric approximation and trigonometric widths of given classes.