Why Smoothness Can Depend on Direction
A regression function can change rapidly in one predictor and gently in another. For example, a response surface might have a sharp transition with respect to a biomarker but vary gradually with age. Describing both directions by one smoothness index can hide this difference. Anisotropic Hölder classes instead assign a separate smoothness index to each coordinate.
The indices describe regularity in coordinate directions, not a claim that the function is constant or equally smooth in all directions. To make this precise, we define a class on the unit cube and require uniform control of the relevant derivatives and increments. Other equivalent or related conventions appear in the literature; here we use a tensor-product Hölder convention, with an explicit Lipschitz convention when an index is an integer.
Definition of an Anisotropic Hölder Ball
Let the domain be \([0,1]^d\), and let the coordinate-specific smoothness indices be \(\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_d)\), with each \(\alpha_j>0\). Write \(\alpha_j=k_j+\beta_j\), where \(k_j=\lceil\alpha_j\rceil-1\) is a nonnegative integer and \(0<\beta_j\leq 1\). Thus, if \(\alpha_j\) is an integer, \(\beta_j=1\) and \(k_j=\alpha_j-1\): the convention at integer indices is that the derivative of order \(\alpha_j-1\) is Lipschitz.
A multi-index is a vector \(\boldsymbol{r}=(r_1,\ldots,r_d)\) of nonnegative integers. The mixed partial derivative it indexes is denoted by \(D^{\boldsymbol{r}}f\), with \(r_j\) derivatives in coordinate \(j\). In this definition, we require these mixed partial derivatives for all \(0\leq r_j\leq k_j\). They must be continuous on the cube (with derivatives at the boundary understood as continuous extensions from the interior).
\(\displaystyle \max_{\boldsymbol{r}:\,0\leq r_j\leq k_j}\|D^{\boldsymbol{r}}f\|_\infty \leq L\),
such that, for every coordinate \(j\) and every multi-index in this range with \(r_j=k_j\),
\(\displaystyle |D^{\boldsymbol{r}}f(x+t e_j)-D^{\boldsymbol{r}}f(x)|\leq L|t|^{\beta_j}\)
whenever \(x\) and \(x+t e_j\) both lie in \([0,1]^d\). Here \(e_j\) is the \(j\)th coordinate vector and \(\|\cdot\|_\infty\) is the supremum norm. Equivalently, the maximum of the displayed derivative bounds and coordinatewise Hölder quotients is at most \(L\).
The derivative bounds include \(f\) itself, by taking \(\boldsymbol{r}=0\). The increment condition for coordinate \(j\) applies to derivatives at the highest required order \(k_j\) in that coordinate, including the allowed mixed derivatives in the other coordinates. When \(\alpha_j\leq1\), \(k_j=0\), so it applies directly to \(f\) and to its allowed derivatives in the other coordinates. The definition therefore controls more than just a collection of unrelated one-dimensional slices.
Reading the Smoothness Indices
The integer part \(k_j\) indicates how many derivatives are controlled along coordinate \(j\); the fractional part \(\beta_j\) specifies the Hölder continuity of the derivative at the highest required order. For example, an index \(3/2\) requires one derivative in that coordinate and a \(1/2\)-Hölder bound on that derivative. An index \(2\), under the convention above, requires one derivative with a Lipschitz bound. The indices can differ, so the number and regularity of derivatives required need not be the same from one coordinate to another.
The coordinatewise conditions also explain what the indices mean operationally: when all other coordinates are held fixed, changes along coordinate \(j\) are governed by \(\alpha_j\). The following examples check the conditions directly.
Worked Example: A Function with Different Fractional Smoothness
On \([0,1]^2\), consider \(f(x,y)=\sqrt{x}+y^2\) with indices \((\alpha_1,\alpha_2)=(1/2,2)\). We verify that it belongs to a ball with a finite radius.
For \(x,x+h\in[0,1]\), the square-root increment satisfies \[ |\sqrt{x+h}-\sqrt{x}|\leq\sqrt{|h|}. \] For \(h\geq0\), this follows by squaring the nonnegative sides: \((\sqrt{x+h}-\sqrt{x})^2\leq h\), since \(2\sqrt{x}\sqrt{x+h}\geq2x\). The case \(h<0\) follows by exchanging the two points. Thus the \(x\)-direction has the required exponent \(1/2\). Its increment is exactly \(1/2\) between \(x=0\) and \(x=1/4\), matching \(\sqrt{1/4}=1/2\).
For the \(y\)-direction, \(\alpha_2=2\) means \(k_2=1,\ \beta_2=1\). The derivative is \(\partial f/\partial y=2y\), and \[ |2(y+h)-2y|=2|h|. \] So this derivative is Lipschitz. It is bounded by \(2\); \(f\) is bounded by \(2\); and the mixed derivative \(\partial^2 f/(\partial x\,\partial y)\) is zero wherever required. The conditions are satisfied, for example, with a sufficiently large finite \(L\). The function has a fractional-index condition in \(x\) and a Lipschitz first-derivative condition in \(y\), rather than one common smoothness index.
Worked Example: Checking an Index Greater Than One
Let \(f(x,y)=x^2+|y-1/2|^{3/2}\) on \([0,1]^2\), with \((\alpha_1,\alpha_2)=(2,3/2)\). The coordinate orders are \(k_1=1,\beta_1=1\) and \(k_2=1,\beta_2=1/2\).
In \(x\), \(f_x=2x\), so \(|f_x(x+h,y)-f_x(x,y)|=2|h|\): the required Lipschitz bound holds. In \(y\), \[ f_y(x,y)=\tfrac32\,\operatorname{sgn}(y-\tfrac12)|y-\tfrac12|^{1/2}, \] where the derivative at \(y=1/2\) is \(0\). To check its \(1/2\)-Hölder condition, put \(u=y-1/2\) and \(v=y'-1/2\). If \(u,v\) have the same sign, \(|\sqrt{|u|}-\sqrt{|v|}|\leq\sqrt{|u-v|}\). If they have opposite signs, then \(|u-v|=|u|+|v|\), and \[ \sqrt{|u|}+\sqrt{|v|}\leq\sqrt{2(|u|+|v|)}. \] Consequently, \(|f_y(x,y)-f_y(x,y')|\leq(3\sqrt{2}/2)|y-y'|^{1/2}\). The mixed derivative \(f_{xy}\) is zero. All derivatives required by the definition are bounded on the cube, so \(f\) lies in an anisotropic Hölder ball of finite radius. The fractional behavior at \(y=1/2\) is compatible with one derivative and a \(1/2\)-Hölder condition on that derivative.
Worked Example: A Smooth Coordinate and a Rougher Coordinate
Consider \(g(x,y)=\sin(4\pi x)+\cos(\pi y)\) with indices \((1/2,2)\). In \(x\), the mean value theorem gives \[ |\sin(4\pi(x+h))-\sin(4\pi x)|\leq4\pi|h|\leq4\pi|h|^{1/2}, \] because \(|h|\leq1\) on the domain. The \(x\)-increment of \(g_y=-\pi\sin(\pi y)\) is zero, so the other derivative subject to the \(x\)-condition causes no difficulty.
In \(y\), \(g_y=-\pi\sin(\pi y)\), and \[ |g_y(x,y+h)-g_y(x,y)|\leq\pi^2|h|, \] again by the mean value theorem. This verifies the required Lipschitz condition for the first \(y\)-derivative. The function and all required derivatives are bounded on the cube, and the mixed derivative is zero. Hence \(g\) belongs to a finite-radius ball with these indices. The explicit bounds show that variation in \(x\) is controlled by a \(1/2\)-Hölder increment, while the \(y\)-derivative has a Lipschitz increment.
Two Consequences of the Definition
Coordinatewise conditions can be used without replacing the anisotropic description by an isotropic one. First, when all indices are at most one, they give a direct bound for a change in several coordinates. Second, for indices above one, the Hölder condition on a highest-order derivative controls the Taylor remainder along that coordinate.
\(\displaystyle |f(y)-f(x)|\leq L\sum_{j=1}^d |y_j-x_j|^{\alpha_j}.\)
Proof. Define \(z^{(0)}=x\), and for \(j=1,\ldots,d\), let \(z^{(j)}\) be the point obtained from \(x\) by replacing its first \(j\) coordinates with those of \(y\). Each consecutive pair differs only in coordinate \(j\), and both points are in the cube. Therefore, \[ |f(y)-f(x)| \leq\sum_{j=1}^d|f(z^{(j)})-f(z^{(j-1)})| \leq L\sum_{j=1}^d|y_j-x_j|^{\alpha_j}. \] This proves the claim. \(\square\)
For higher indices, apply a one-dimensional Taylor expansion to a coordinate section. The next result states the remainder bound, including its regularity assumptions.
\(\displaystyle \left|g(a+h)-\sum_{q=0}^{k}\frac{g^{(q)}(a)}{q!}h^q\right|\leq\frac{L}{(k-1)!(\beta+1)}|h|^{k+\beta}.\)
For \(k=0\), the corresponding statement is the defining increment bound \(|g(a+h)-g(a)|\leq L|h|^\beta\).
Proof. It is enough to consider \(h\geq0\); for \(h<0\), reverse the interval and use the same argument. Taylor's formula with integral remainder gives \[ g(a+h)=\sum_{q=0}^{k-1}\frac{g^{(q)}(a)}{q!}h^q+ \frac{1}{(k-1)!}\int_0^h(h-t)^{k-1}g^{(k)}(a+t)\,dt. \] Subtract the degree-\(k\) Taylor polynomial. Its order-\(k\) term is \(g^{(k)}(a)h^k/k!\), which equals \(\frac{g^{(k)}(a)}{(k-1)!}\int_0^h(h-t)^{k-1}\,dt\). Thus the difference is \[ \frac{1}{(k-1)!}\int_0^h(h-t)^{k-1}\bigl(g^{(k)}(a+t)-g^{(k)}(a)\bigr)\,dt. \] Its absolute value is at most \[ \frac{L}{(k-1)!}\int_0^h(h-t)^{k-1}t^\beta\,dt \leq\frac{Lh^{k-1}}{(k-1)!}\int_0^h t^\beta\,dt =\frac{Lh^{k+\beta}}{(k-1)!(\beta+1)}. \] This is the stated bound. \(\square\)
What the Indices Do—and Do Not—Say
Each \(\alpha_j\) records regularity in its own coordinate, and can therefore distinguish a rapidly varying direction from a slowly varying one. The coordinate Taylor bound makes this especially concrete: the error after a degree-\(k_j\) Taylor approximation in coordinate \(j\) is of order \(|h|^{\alpha_j}\). This is a local statement about that coordinate, under the class's derivative conditions. It is not a claim that every diagonal or oblique direction has exactly one of the listed indices.
Common pitfall: It is tempting to replace the vector \((\alpha_1,\ldots,\alpha_d)\) by its smallest entry and then say that this completely describes the function's smoothness. That discards the distinction the anisotropic class is designed to preserve. For instance, \((1/2,2)\) requires only a fractional increment bound in the first coordinate but controls a Lipschitz first derivative in the second. The minimum index alone does not express that second-coordinate condition. Conversely, assigning the larger index to every coordinate would impose regularity that has not been established.
The definition is also stronger than checking only the function's separate coordinate sections: it includes bounds for the required mixed derivatives and their coordinate increments. When verifying membership, check the relevant derivatives and conditions for each coordinate, rather than inferring the whole class from a plot or from a few one-dimensional slices.
Check Your Understanding
Use the definition and coordinatewise conditions to answer the following.
- For a coordinate index \(\alpha_j=5/2\), what are \(k_j\) and \(\beta_j\), and which derivative receives the fractional Hölder condition?
- Under the integer-index convention in this tutorial, what regularity does \(\alpha_j=3\) require?
- Why does the multi-coordinate increment proposition follow by changing one coordinate at a time?
- For \(f(x,y)=x^2+|y-1/2|^{3/2}\), which coordinate has the fractional-index condition, and what is its exponent?
- Why is it not enough to check only the function's coordinate sections when verifying the stated tensor-product class?