Besov Classes And Difference Moduli

Why Measure Smoothness at Many Scales?

A smoothness description needs to say more than how many derivatives a function has. It can also distinguish whether irregularity is concentrated in a few locations or spread across a domain, and how the size of a function’s changes varies from one scale to another. Besov classes make these features explicit by measuring finite differences at each scale and then aggregating those measurements.

This construction gives a family of smoothness classes indexed by three parameters. The parameter \(s\) specifies the smoothness order, \(p\) measures the size of each difference over space, and \(q\) measures how the differences behave across scales. The distinction between \(p\) and \(q\) is substantive: functions with similar pointwise regularity can have different scale-by-scale integrability.

Finite Differences and the Modulus of Smoothness

Let \(f\) be a measurable function on \(\mathbb{R}^d\). For a vector \(h\in\mathbb{R}^d\), its first difference is \(\Delta_h f(x)=f(x+h)-f(x)\). The order-\(m\) difference is obtained by applying this operation \(m\) times in the same direction.

Definition: For an integer \(m\geq 1\), the \(m\)th finite difference of \(f\) with step \(h\) is

\Delta_h^m f(x)=\sum_{j=0}^{m}(-1)^{m-j}\binom{m}{j}f(x+jh).

For \(1\leq p<\infty\), define the \(L^p\) modulus of smoothness by

\omega_m(f,t)_p=\sup_{|h|\leq t}\|\Delta_h^m f\|_{L^p(\mathbb{R}^d)},\qquad t>0.

For \(p=\infty\), use the essential supremum norm in this definition.

Finite differences are useful because they measure changes without requiring a classical derivative to exist. If \(f\) is a polynomial of degree less than \(m\), then \(\Delta_h^m f=0\). More generally, when derivatives exist, repeated applications of the fundamental theorem of calculus express an \(m\)th difference as an average of \(m\)th directional derivatives. The difference definition therefore applies both to differentiable functions and to functions with jumps or other irregularities.

The supremum over all steps with \(|h|\leq t\) records the largest \(L^p\) difference at scale at most \(t\). Increasing \(m\) beyond the desired smoothness order gives a useful way to detect smoothness of non-integer order without choosing a derivative of fractional order.

The Besov Definition

Fix \(s>0\) and choose any integer \(m>s\). For clarity, the definition below uses \(1\leq p,q\leq\infty\). The integral in the definition is taken over scales with measure \(dt/t\), which gives equal weight to equal multiplicative ranges of scale.

Definition: The inhomogeneous Besov norm \(B^s_{p,q}(\mathbb{R}^d)\) is

\|f\|_{B^s_{p,q}} = \|f\|_{L^p} + \left(\int_0^\infty \big[t^{-s}\omega_m(f,t)_p\big]^q\,\frac{dt}{t}\right)^{1/q}, \qquad q<\infty,

and

\|f\|_{B^s_{p,\infty}} = \|f\|_{L^p} + \sup_{t>0}t^{-s}\omega_m(f,t)_p.

The class \(B^s_{p,q}(\mathbb{R}^d)\) consists of the \(L^p\) functions for which the corresponding norm is finite. For \(p=\infty\), replace \(\|f\|_{L^p}\) with the essential supremum norm.

The first parameter, \(p\), governs the spatial norm of each difference. A small \(L^p\) difference can arise because changes are small in magnitude, because they occur on a small part of the domain, or both. The parameter \(q\) instead governs how the normalized differences \(t^{-s}\omega_m(f,t)_p\) aggregate over scales. When \(q=\infty\), only their largest value matters; when \(q<\infty\), their \(q\)th-power integral must also be finite.

The \(L^p\) term makes this an inhomogeneous norm: it controls the function itself as well as its small-scale differences. The restriction \(m>s\) is essential to the definition. For standard finite-difference characterizations, different integer choices above \(s\) yield equivalent norms; one can therefore choose a convenient such \(m\), rather than interpreting \(m\) as an additional smoothness parameter.

Worked Examples

Worked Example: A Jump and the Role of the Scale Parameter

Take \(f(x)=\mathbf{1}_{[0,1]}(x)\) on \(\mathbb{R}\), and fix a finite \(p\geq1\). We determine the order of its modulus for small \(t\), using any integer \(m\geq1\).

For \(0<h<1/(2m)\), \(\Delta_h^m f\) vanishes except within distance \(mh\) of one of the two endpoints of \([0,1]\). This is because, away from those endpoint neighborhoods, either all the points \(x,x+h,\ldots,x+mh\) lie in the interval or all lie outside it. In either case the finite difference is zero. Everywhere, its absolute value is at most \(\sum_{j=0}^m\binom{m}{j}=2^m\). The total length of the endpoint neighborhoods is at most \(4mh\). Thus

\|\Delta_h^m f\|_p\leq 2^m(4mh)^{1/p}.

There is also a matching lower bound. On \(-h\leq x<0\), the term \(f(x)\) is zero and \(f(x+jh)=1\) for \(1\leq j\leq m\), since \(mh<1\). The difference is constant on this interval, with absolute value \(\left|\sum_{j=1}^m(-1)^{m-j}\binom{m}{j}\right|=1\). Consequently \(\|\Delta_h^m f\|_p\geq h^{1/p}\). The same upper and lower orders hold for negative steps. Taking the supremum over steps up to \(t\), and using \(h=t\) for the lower bound, gives

\omega_m(f,t)_p\asymp t^{1/p}\qquad\text{for sufficiently small }t.

At small scales, the Besov integral therefore has the same convergence behavior as \(\int_0^c t^{(1/p-s)q}\,dt/t\), where \(c>0\) is small. For finite \(q\), this integral is finite exactly when \(s<1/p\): at \(s=1/p\), it becomes a constant times \(\int_0^c dt/t\), which diverges. For \(q=\infty\), the normalized modulus stays bounded exactly when \(s\leq1/p\). Thus a jump belongs to \(B^s_{p,q}\) below the threshold \(s=1/p\), and at the threshold only when \(q=\infty\).

Worked Example: A Smooth Function Has Small Differences

Let \(f\) be a compactly supported \(m\)-times continuously differentiable function on \(\mathbb{R}^d\), where \(m>s\) and \(1\leq p\leq\infty\). We derive a small-scale bound on its modulus.

Write \(h=tu\), where \(t=|h|\) and \(|u|=1\). Repeated use of the fundamental theorem of calculus gives

\Delta_{tu}^m f(x) = t^m\int_{[0,1]^m} D_u^m f\bigl(x+t(r_1+\cdots+r_m)u\bigr)\,dr_1\cdots dr_m,

where \(D_u^m\) is the \(m\)th derivative in direction \(u\). Translation does not change an \(L^p\) norm. By the integral form of the triangle inequality,

\|\Delta_{tu}^m f\|_p \leq t^m\|D_u^m f\|_p.

Expanding \(D_u^m\) in coordinate derivatives shows that \(\|D_u^m f\|_p\) is bounded uniformly over unit vectors \(u\): it is a finite sum of terms involving the \(L^p\) norms of the order-\(m\) partial derivatives, with coefficients bounded for \(|u|=1\). Hence \(\omega_m(f,t)_p\leq C t^m\) for \(0<t\leq1\), for a finite constant \(C\).

It follows that near zero the integrand defining the Besov norm is bounded by a constant times \(t^{(m-s)q}\) when \(q<\infty\), whose integral against \(dt/t\) is finite because \(m-s>0\). For scales \(t\geq1\), the triangle inequality gives \(\omega_m(f,t)_p\leq2^m\|f\|_p\), and \(\int_1^\infty t^{-sq}\,dt/t\) is finite. The supremum version is also finite for \(q=\infty\). Therefore \(f\in B^s_{p,q}(\mathbb{R}^d)\) for every \(s<m\) and every \(1\leq q\leq\infty\).

Worked Example: Exact Differences of a Periodic Sine Function

Consider \(f(x)=\sin(2\pi x)\) on the one-dimensional torus, with its \(L^p\) norm over one period. For periodic functions, use the same difference definition and integrate the scale norm over \(0<t\leq1\), including the \(L^p\) term.

Represent \(f\) as the imaginary part of \(e^{2\pi i x}\). Applying one difference multiplies this exponential by \(e^{2\pi i h}-1\); applying \(m\) differences multiplies it by \((e^{2\pi i h}-1)^m\). The resulting real function is a shifted sine multiplied by an amplitude, so its \(L^p\) norm is

\|\Delta_h^m f\|_p = \bigl(2|\sin(\pi h)|\bigr)^m\|\sin(2\pi\cdot)\|_p.

For \(0<t\leq1/4\), \(|\sin(\pi h)|\leq\pi |h|\) whenever \(|h|\leq t\), giving an upper bound proportional to \(t^m\). For a lower bound, choose \(h=t\); on this range, \(\sin(\pi t)\geq2t\). Therefore \(\omega_m(f,t)_p\asymp t^m\) for \(0<t\leq1/4\). The small-scale Besov integral converges whenever \(s<m\), for every finite \(q\), and the supremum is finite for \(q=\infty\). This is consistent with the function’s smoothness, but also shows explicitly how the finite-difference definition checks it.

What Besov Smoothness Adds

Besov classes provide a scale of regularity rather than a single derivative-based description. Their parameter \(s\) specifies the power against which differences are normalized, while \(p\) and \(q\) retain separate information about space and scales. The indicator example shows why the scale parameter matters: at the critical value \(s=1/p\), the same jump satisfies the \(B^s_{p,\infty}\) condition but fails the \(B^s_{p,q}\) condition for every finite \(q\). A statement about \(s\) alone would miss this distinction.

There are useful connections to familiar smoothness classes. For non-integer \(s>0\), the Hölder condition of order \(s\) can be expressed through a supremum bound on finite differences; in this setting it corresponds to the \(B^s_{\infty,\infty}\) scale. At integer smoothness, the related difference condition is often called a Zygmund condition, so it should not automatically be identified with classical \(C^s\) differentiability. In the \(L^2\) setting, the Fourier-defined Sobolev space \(H^s\) agrees with \(B^s_{2,2}\) under the usual equivalent-norm convention. For fractional orders between zero and one and \(1<p<\infty\), the Slobodeckij Sobolev space also has a standard \(B^s_{p,p}\) description. These relationships use particular parameter choices; the full Besov scale records more information than any one such choice.

Common Pitfall

Common mistake: Treating \(p\) and \(q\) as interchangeable because both occur as integrability exponents. The wrong reasoning is that two functions with the same \(s\) and \(p\) must have the same Besov membership, regardless of \(q\). The correction is that \(p\) measures each difference over the spatial domain, while \(q\) aggregates the normalized differences across scales. The jump example verifies that changing \(q\) alone can change membership at the critical smoothness \(s=1/p\).

A second caution is to use a difference order \(m\) strictly greater than \(s\). Choosing \(m\leq s\) does not provide the stated standard characterization. Also check the domain convention: the definition above is on \(\mathbb{R}^d\); periodic functions use periodic differences, while functions on a bounded region require an explicitly specified boundary or extension convention.

Check Your Understanding

Answer these questions using the definitions and derivations above.

In the Besov notation \(B^s_{p,q}\), what does each of \(s\), \(p\), and \(q\) measure?
For the indicator of an interval on \(\mathbb{R}\), what is the small-scale order of \(\omega_m(f,t)_p\) when \(p\) is finite?
At \(s=1/p\), why does the indicator belong to \(B^s_{p,\infty}\) but not \(B^s_{p,q}\) for finite \(q\)?
Why must the difference order \(m\) be chosen strictly larger than \(s\)?
What caution is needed when relating integer-order Besov smoothness to classical differentiability?