Smooth Bump Functions For Lower Bounds

Why Localized Smooth Perturbations Matter

A lower-bound construction often needs two regression functions that differ only on a small region, while both remain inside the same smoothness class. A bump function provides that local change: it is smooth everywhere, vanishes outside a compact set, and can be made narrower while its height is reduced in a controlled way. The key calculations are how this rescaling affects the Hölder norm, the integrated squared separation, and the conditional means of observed responses.

This tutorial develops those calculations directly. It stops at quantifying the perturbation and its effect on the observations; turning such a perturbation into a statistical lower bound is a separate argument.

A Concrete Smooth Bump

For \(t\in\mathbb{R}\), define

\psi_1(t)= \begin{cases} \exp\!\left(1-\dfrac{1}{1-t^2}\right), & |t|<1,\\[6pt] 0, & |t|\geq 1. \end{cases}

This function satisfies \(\psi_1(0)=1\), is positive for \(|t|<1\), and is zero outside \([-1,1]\). It is infinitely differentiable: at either endpoint, every derivative from within \((-1,1)\) is a finite sum of terms consisting of a rational function of \(1-t^2\) multiplied by \(\exp(-1/(1-t^2))\). As \(1-t^2\) decreases to zero, this exponential tends to zero faster than any power of \(1-t^2\), so all those derivatives tend to zero and match the derivatives of the zero function outside the interval.

For dimension \(d\), use the product bump

\psi(u)=\prod_{j=1}^{d}\psi_1(u_j), \qquad u=(u_1,\ldots,u_d)\in\mathbb{R}^d.

It is \(C^\infty\), supported on \([-1,1]^d\), and has \(\psi(0)=1\). Its squared integral \(C_2=\int_{\mathbb{R}^d}\psi(u)^2\,du\) is finite and strictly positive: finiteness follows from boundedness and compact support, and positivity follows because \(\psi\) is positive in a neighborhood of zero.

Definition: Given a center \(x_0\in(0,1)^d\), a width \(h>0\), and an amplitude \(A>0\), define the scaled bump \(g_h(x)=A h^\beta\psi((x-x_0)/h)\). Its support is contained in \(x_0+h[-1,1]^d\). When this cube lies inside \([0,1]^d\), the bump is a smooth perturbation supported entirely within the domain.

Worked Example: Locating and Scaling a Bump

Take \(d=1\), \(x_0=0.5\), \(h=0.2\), \(\beta=3/2\), and \(A=0.4\). Then \(g_h(x)=0.4(0.2)^{3/2}\psi_1((x-0.5)/0.2)\).

The support is where \(|(x-0.5)/0.2|\leq 1\), which is the interval \([0.3,0.7]\). This interval is contained in \([0,1]\). At the center, \(\psi_1(0)=1\), so the bump’s maximum is \(g_h(0.5)=0.4(0.2)^{3/2}=0.4(0.0894427\ldots)=0.0357771\ldots\).

The multiplier \(h^{3/2}\) makes the bump’s height decrease as its width decreases. The exact Hölder-norm calculation below explains why this particular power offsets the growth in derivatives.

Hölder-Norm Scaling

Let the domain be \([0,1]^d\). For a noninteger smoothness level \(\beta=m+\alpha\), where \(m\) is a nonnegative integer and \(0<\alpha<1\), use the Hölder norm that bounds all partial derivatives through order \(m\) and the \(\alpha\)-Hölder seminorms of the order-\(m\) derivatives. For integer \(\beta=m\geq1\), use the norm that bounds all partial derivatives through order \(m\). A Hölder ball of radius \(L\) consists of functions whose chosen norm is at most \(L\). These conventions suffice for the scaling calculation; the bump is smooth under either convention.

Result: For fixed \(\beta>0\) and the bump \(\psi\) above, there is a finite constant \(C\), depending on \(\psi,d,\beta\) but not on \(h,x_0,A\), such that, whenever \(0<h\leq1\) and the support is inside the domain, \(\|A h^\beta\psi((\cdot-x_0)/h)\|_{C^\beta}\leq CA\). Thus choosing \(A\leq L/C\) keeps every such bump in the Hölder ball of radius \(L\), uniformly over its width.

Proof. For a multi-index \(\nu\), the chain rule gives

D^\nu g_h(x) = A h^{\beta-|\nu|} (D^\nu\psi)((x-x_0)/h).

Every derivative of \(\psi\) is bounded. For \(|\nu|\leq m\), and \(h\leq1\), the factor \(h^{\beta-|\nu|}\) is at most one when \(\beta-|\nu|\geq0\). Therefore the required derivative suprema are bounded by a constant times \(A\).

In the noninteger case, take \(|\nu|=m\). The function \(D^\nu\psi\) is globally \(\alpha\)-Hölder with some finite constant \(C_\nu\). Indeed, its gradient is bounded, giving a Lipschitz bound for pairs at distance at most one; boundedness gives a constant bound for pairs at distance greater than one. Each bound implies an \(\alpha\)-Hölder bound in its respective range. Rescaling consequently gives

\frac{|D^\nu g_h(x)-D^\nu g_h(y)|}{\|x-y\|^\alpha} \leq A C_\nu h^{\beta-m-\alpha} = A C_\nu,

because \(\beta=m+\alpha\). In the integer case, the order-\(m\) derivative supremum is bounded by a constant times \(A h^{\beta-m}=A\). Combining the finitely many bounds proves the result. \(\square\)

Worked Example: Checking the Scaling for Smoothness \(3/2\)

For \(\beta=3/2\), write \(\beta=1+1/2\), so \(m=1\) and \(\alpha=1/2\). The bump’s first derivative is \(g_h'(x)=A h^{1/2}\psi_1'((x-x_0)/h)\), whose supremum is at most \(A h^{1/2}\|\psi_1'\|_\infty\leq A\|\psi_1'\|_\infty\) for \(h\leq1\).

For the required half-Hölder seminorm, rescaling the first derivative contributes \(A h^{1/2}\) from its amplitude and \(h^{-1/2}\) from its argument and distance rescaling. Their product is \(A h^0=A\). Thus neither the derivative bound nor the seminorm grows as the bump narrows. This verifies the scaling power for this smoothness level.

Integrated Squared Separation

The bump’s height is of order \(A h^\beta\), while its support has volume of order \(h^d\). The integral of its square therefore has a different power of \(h\) from either quantity alone.

Result: If the support of \(g_h\) lies inside \([0,1]^d\), then \(\|g_h\|_{L^2([0,1]^d)}^2=A^2h^{2\beta+d}C_2\), and hence \(\|g_h\|_{L^2([0,1]^d)}=A h^{\beta+d/2}\sqrt{C_2}\).

Proof. By definition, the squared norm is the integral of \(A^2h^{2\beta}\psi((x-x_0)/h)^2\) over the domain. The integrand vanishes outside the bump’s support, which lies inside the domain. Substitute \(u=(x-x_0)/h\); then \(dx=h^d\,du\), and the transformed domain includes the full support of \(\psi\). Therefore

\int_{[0,1]^d}g_h(x)^2\,dx = A^2h^{2\beta}h^d \int_{\mathbb{R}^d}\psi(u)^2\,du = A^2h^{2\beta+d}C_2.

Taking the nonnegative square root gives the stated \(L^2\) norm. \(\square\)

Worked Example: A Random-Design Mean-Shift Calculation

Suppose \(d=1\), \(\beta=1/2\), \(x_0=1/2\), \(h=1/10\), and \(A=1/5\). Let \(X_1,\ldots,X_{1000}\) be independent and uniform on \([0,1]\), and compare regression functions \(f_0(x)=0\) and \(f_1(x)=g_h(x)\). For one design point, the squared conditional-mean shift is \(g_h(X_i)^2\).

Using the uniform density and the substitution \(u=(x-x_0)/h\), its expectation is \(E[g_h(X_i)^2]=A^2h^{2\beta+1}C_2\). By linearity of expectation, the expected sum of squared conditional-mean shifts over all 1000 observations is \(1000A^2h^{2\beta+1}C_2\). Substituting the values gives

1000\left(\frac15\right)^2 \left(\frac1{10}\right)^{2(1/2)+1}C_2 = 1000\cdot\frac1{25}\cdot\left(\frac1{10}\right)^2C_2 = 1000\cdot\frac1{25}\cdot\frac1{100}C_2 = \frac25 C_2.

The exponent is \(2(1/2)+1=2\), so the width factor is \(1/100\). This is an expected sum for 1000 random design points, not an assertion that 1000 disjoint bumps fit into the domain.

How a Bump Changes Regression Observations

Consider the regression model \(Y_i=f(X_i)+\varepsilon_i\), where the errors have mean zero conditional on the design points. Replacing \(f_0\) by \(f_1=f_0+g_h\) changes the conditional mean of observation \(i\) by exactly \(g_h(X_i)\). If \(X_i\) falls outside the bump’s support, that change is zero.

Result: For fixed design points \(x_1,\ldots,x_n\), the vector of conditional-mean changes from adding \(g_h\) is \((g_h(x_1),\ldots,g_h(x_n))\). Its squared Euclidean length is \(\sum_{i=1}^n g_h(x_i)^2=A^2h^{2\beta}\sum_{i=1}^n\psi((x_i-x_0)/h)^2\). For independent random design points with density \(p\), its expected squared length is \(nA^2h^{2\beta+d}\int_{\mathbb{R}^d}\psi(u)^2p(x_0+hu)\,du\), provided the bump support lies in the domain.

Derivation. The fixed-design identity follows by substituting the definition of \(g_h\) into each squared mean change and summing. For random design, all \(n\) terms have the same expectation, so the expected sum is \(nE[g_h(X_1)^2]\). Integrating against the density and making the same change of variables as above gives

nE[g_h(X_1)^2] = n A^2h^{2\beta+d} \int_{\mathbb{R}^d}\psi(u)^2p(x_0+hu)\,du.

In particular, if \(p(x)\leq p_{\max}\) on the support, this expectation is at most \(nA^2h^{2\beta+d}p_{\max}C_2\). Under uniform design on \([0,1]^d\), it is exactly \(nA^2h^{2\beta+d}C_2\).

Worked Example: Evaluating Mean Shifts at Fixed Design Points

Take \(d=1\), \(\beta=1/2\), \(A=1/2\), \(h=1/4\), and \(x_0=1/2\). Use the regression model \(Y_i=f(X_i)+\varepsilon_i\) with \(\varepsilon_i\sim N(0,1)\), and compare \(f_0=0\) with \(f_1=g_h\). Consider fixed design points \(x_1=1/2\), \(x_2=5/8\), and \(x_3=3/4\).

The bump’s height multiplier is \(Ah^{1/2}=(1/2)(1/2)=1/4\). At \(x_1\), the scaled coordinate is zero, so the mean shift is \(1/4\). At \(x_2\), it is \(1/2\), and \(\psi_1(1/2)=\exp(1-1/(1-1/4))=\exp(-1/3)\); the shift is therefore \((1/4)e^{-1/3}\). At \(x_3\), the scaled coordinate is one, so the bump and mean shift are zero.

The squared length of the mean-shift vector is consequently \((1/4)^2+((1/4)e^{-1/3})^2+0^2 =(1/16)(1+e^{-2/3})\), approximately \((1/16)(1+0.5134)=0.0946\). The noise variance specifies the observation model, but the mean-shift calculation itself is determined by the bump values at the design points.

Common Mistakes and Practical Checks

Using the wrong power for height. A width-\(h\) bump with height independent of \(h\) generally has derivatives that grow as \(h\) shrinks. The factor \(h^\beta\) offsets this growth and keeps the Hölder norm bounded up to the fixed amplitude \(A\).
Confusing height with \(L^2\) size. The height is proportional to \(A h^\beta\), whereas the squared \(L^2\) norm is proportional to \(A^2h^{2\beta+d}\). The extra \(h^d\) comes from the shrinking support volume.
Dropping the dimension factor. A width-\(h\) cube in \(d\) dimensions has volume proportional to \(h^d\), not \(h\). Thus the squared separation has exponent \(2\beta+d\).
Forgetting to check support containment. The exact integral formula uses the full bump. If its support crosses the domain boundary, the domain integral can omit part of the bump, and the equality needs adjustment.
Replacing a random-design expectation by a fixed-design identity. For fixed design, sum the actual values \(g_h(x_i)^2\). For random design, integrate against the design density and then multiply by \(n\).

Key takeaway: A smooth localized perturbation of height \(A h^\beta\) and width \(h\) stays in a fixed \(C^\beta\) ball when \(A\) is chosen small enough. Its squared integrated size scales as \(A^2h^{2\beta+d}\), and its effect on regression means is obtained by evaluating the bump at the observed design points or averaging those squared values under the design distribution.

Check Your Understanding

Work through these questions by tracking the height, support, and design contribution separately.

For \(d=2\) and \(\beta=1\), what power of \(h\) appears in the squared \(L^2\) norm of a scaled bump?
Why does multiplying a bump of width \(h\) by \(h^\beta\) keep its \(C^\beta\) norm bounded as \(h\) decreases?
For a fixed design point outside the support, what is the change in the conditional regression mean?
If the random design density is bounded by \(p_{\max}\), give an upper bound for the expected sum of squared mean shifts.
In the worked random-design calculation, why is \((1/10)^{2(1/2)+1}\) equal to \(1/100\), rather than \(1/1000\)?