Direct and inverse theorems of approximation

Remarks

Example: let $X$ be the continuous $2\pi$-periodic functions with the uniform norm, and $V_n$ the trigonometric polynomials of degree at most $n$. Jackson's inequality $E_n(f) \le C \, \omega(f, 1/n)$, with $\omega$ the modulus of continuity, is a direct theorem. Bernstein's theorem, that $E_n(f) \le M n^{-\alpha}$ for all $n \ge 1$ with $0 < \alpha < 1$ implies $\omega(f, \delta) \le M' \delta^{\alpha}$, is an inverse theorem. Combined, they say that $E_n(f) = O(n^{-\alpha})$ if and only if $f$ is Hölder continuous with exponent $\alpha$.

Non-example: the bound $E_n(f) \le \norm{f}$, from $0 \in V_n$, is not a direct theorem. It does not involve any smoothness of $f$ and gives no rate as $n \to \infty$.

Inverse theorems usually rest on inequalities of Bernstein type for the derivatives of elements of $V_n$. Direct theorems usually rest on good approximating operators, such as convolution with the Jackson kernel.