Peetre K-functional

Remarks

Example: take $X_0 = X_1 = \R$ with the absolute value. For a split $x = x_0 + x_1$ the cost is $\abs{x_0} + t \abs{x_1} \ge \min(1, t) (\abs{x_0} + \abs{x_1}) \ge \min(1, t) \abs{x}$, and putting all of $x$ in the cheaper space attains this, so $K(t, x) = \min(1, t) \abs{x}$.

Non-example: the K-functional is only defined on $X_0 + X_1$. For $X_0 = X_1 = \ell^1$ inside the space of all sequences, the sequence $(1, 1, 1, \dots)$ has no split at all, so $K(t, \cdot)$ is not defined there.

For each $x$, the function $t \mapsto K(t, x)$ is nondecreasing and concave, and $K(1, x)$ is the norm of $x$ in $X_0 + X_1$. For $0 < \theta < 1$ and $1 \le q \le \infty$, the real interpolation space $(X_0, X_1)_{\theta, q}$ consists of the $x$ for which $t \mapsto t^{-\theta} K(t, x)$ lies in $L^q((0, \infty), dt / t)$. In approximation theory one takes $X_0 = L^p[a, b]$ and $X_1$ the Sobolev space $W^{k, p}[a, b]$, measured by the seminorm $\norm{g^{(k)}}_p$. Then the quantity $K(t^k, f)$ is bounded above and below by constant multiples of the modulus of smoothness $\omega_k(f, t)_p$ for small $t$.