Functions of bounded or vanishing nonlinearity

A natural way to measure the nonlinearity of a function {f\colon I\to \mathbb R}, where {I\subset \mathbb R} is an interval, is the quantity {\displaystyle NL(f;I) = \frac{1}{|I|} \inf_{k, r}\sup_{x\in I}|f(x)-kx-r|} which expresses the deviation of {f} from a line, divided by the size of interval {I}. This quantity was considered in Measuring nonlinearity and reducing it.

Let us write {NL(f) = \sup_I NL(f; I)} where the supremum is taken over all intervals {I} in the domain of definition of {f}. What functions have finite {NL(f)}? Every Lipschitz function does, as was noted previously: {NL(f) \le \frac14 \mathrm{Lip}\,(f)}. But the converse is not true: for example, {NL(f)} is finite for the non-Lipschitz function {f(x)=x\log|x|}, where {f(0)=0}.

y  = x log|x|

The function looks nice, but {f(x)/x} is clearly unbounded. What makes {NL(f)} finite? Note the scale-invariant feature of NL: for any {t>0} the scaled function {f_t(x) = t^{-1}f(tx)} satisfies {NL(f_t)=NL(f)}, and more precisely {NL(f; tI) = NL(f_t; I)}. On the other hand, our function has a curious scaling property {f_t(x) = f(x) + x\log t} where the linear term {x\log t} does not affect NL at all. This means that it suffices to bound {NL(f; I)} for intervals {I} of unit length. The plot of {f} shows that not much deviation from the secant line happens on such intervals, so I will not bother with estimates.

The class of functions {f} with {NL(f)<\infty} is precisely the Zygmund class {\Lambda^*} defined by the property {|f(x-h)-2f(x)+f(x+h)| \le Mh} with {M} independent of {x, h}. Indeed, since the second-order difference {f(x-h)-2f(x)+f(x+h)} is unchanged by adding an affine function to {f}, we can replace {f} by {f(x)-kx-r} with suitable {k, r} and use the triangle inequality to obtain

{\displaystyle |f(x-h)-2f(x)+f(x+h)| \le 4 \sup_I |f(x)-kx-r| = 8h\; NL(f; I)}

where {I=[x-h, x+h]}. Conversely, suppose that {f\in \Lambda^*}. Given an interval {I=[a, b]}, subtract an affine function from {f} to ensure {f(a)=f(b)=0}. We may assume {|f|} attains its maximum on {I} at a point {\xi \le (a + b)/2}. Applying the definition of {\Lambda^*} with {x = \xi} and {h = \xi - a}, we get {|f(2\xi - a) - 2f(\xi )| \le M h}, hence {|f(\xi )| \le Mh}. This shows {NL(f; I)\le M/2}. The upshot is that {NL(f)} is equivalent to the Zygmund seminorm of {f} (i.e., the smallest possible M in the definition of {\Lambda^*}).

A function in {\Lambda^*} may be nowhere differentiable: it is not difficult to construct {f} so that {NL(f;I)} is bounded between two positive constants. The situation is different for the small Zygmund class {\lambda^*} whose definition requires that {NL(f; I)\to 0} as {|I|\to 0}. A function {f \in \lambda^*} is differentiable at any point of local extremum, since the condition {NL(f; I)\to 0} forces its graph to be tangent to the horizontal line through the point of extremum. Given any two points {a, b} we can subtract the secant line from {f} and thus create a point of local extremum between {a } and {b}. It follows that {f} is differentiable on a dense set of points.

The definitions of {\Lambda^* } and {\lambda^*} apply equally well to complex-valued functions, or vector-valued functions. But there is a notable difference in the differentiability properties: a complex-valued function of class {\lambda^*} may be nowhere differentiable [Ullrich, 1993]. Put another way, two real-valued functions in {\lambda^*} need not have a common point of differentiability. This sort of thing does not often happen in analysis, where the existence of points of “good” behavior is usually based on the prevalence of such points in some sense, and therefore a finite collection of functions is expected to have common points of good behavior.

The key lemma in Ullrich’s paper provides a real-valued VMO function that has infinite limit at every point of a given {F_\sigma} set {E} of measure zero. Although this is a result of real analysis, the proof is complex-analytic in nature and involves a conformal mapping. It would be interesting to see a “real” proof of this lemma. Since the antiderivative of a VMO function belongs to {\lambda^* }, the lemma yields a   function {v \in \lambda^*} that is not differentiable at any point of {E}. Consider the lacunary series {u(t) = \sum_{n=1}^\infty a_n 2^{-n} \cos (2^n t)}. One theorem of Zygmund shows that {u \in \lambda^*} when {a_n\to 0}, while another shows that {u} is almost nowhere differentiable when {\sum a_n^2 = \infty}. It remains to apply the lemma to get a function {v\in \lambda^*} that is not differentiable at any point where {u} is differentiable.

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