Lattice Operations on Functions

Maximum and Minimum as Operations on Functions

The previous tutorial studied addition and multiplication in \(C(K)\). Those algebraic operations combine function values by adding or multiplying them. Another useful way to combine two functions is to select, at each point, whichever value is larger or smaller. These pointwise maximum and minimum operations will be important in approximation theory because they allow us to build new continuous functions from existing ones.

Throughout this tutorial, \(K\) is a nonempty compact subset of \(\mathbb{R}\), and \(C(K)\) is equipped with the supremum norm. We compare functions by comparing their values at every point. This is an order on functions, rather than a comparison of their norms or of their values at just one selected point.

At each \(x\), the real numbers \(f(x)\) and \(g(x)\) have a maximum and a minimum. It remains to check that the resulting functions are continuous. One way to do this is to express these operations using addition and absolute value. Since the absolute value function on \(\mathbb{R}\) is continuous, \(|f-g|\) is continuous whenever \(f\) and \(g\) are continuous.

Proof. For real numbers \(a,b\), the identities

hold. To verify them, first suppose \(a\geq b\). Then \(|a-b|=a-b\), so the right sides are \(a\) and \(b\), respectively. If \(a<b\), then \(|a-b|=b-a\), so the right sides are \(b\) and \(a\), respectively. Applying these scalar identities with \(a=f(x)\) and \(b=g(x)\) proves both function identities at every \(x\in K\).

The functions \(f+g\) and \(|f-g|\) are continuous, so the two displayed formulas show that \(f\vee g\) and \(f\wedge g\) are continuous. At every point, their definitions give \(f\leq f\vee g\) and \(g\leq f\vee g\). If \(h\in C(K)\) also satisfies \(f\leq h\) and \(g\leq h\), then for each \(x\in K\),

Thus \(f\vee g\leq h\), proving it is the least such function. Similarly, \(f\wedge g\leq f\) and \(f\wedge g\leq g\). If \(h\leq f\) and \(h\leq g\), then \(h(x)\leq\min\{f(x),g(x)\}=(f\wedge g)(x)\) for every \(x\), so \(h\leq f\wedge g\). This proves the greatest-lower-bound property. \(\square\)

The least-upper-bound and greatest-lower-bound properties explain the word “lattice”: any pair of functions has a least upper bound and a greatest lower bound under the pointwise order. These are not the supremum and infimum of the set of all values of one function. They are functions constructed by comparing two functions at each point.

Identities and Positive Parts

The pointwise definitions immediately give familiar identities from real numbers. For example, \(f\vee g=g\vee f\), \(f\wedge g=g\wedge f\), and \(f\vee f=f\wedge f=f\). The lattice operations are also distributive: taking the maximum with a minimum gives the same result as taking the minimum of the two maxima. These statements hold because they hold for the real values at each point.

Proof. The first two identities follow by adding and subtracting the formulas in the Lattice Operations Theorem. For the distributive identity, fix \(x\in K\) and set \(a=f(x)\), \(b=g(x)\), and \(c=h(x)\). If \(b\leq c\), then \(\min\{b,c\}=b\), and \(\max\{a,b\}\leq\max\{a,c\}\). Hence

If \(c\leq b\), the same calculation with \(b\) and \(c\) interchanged gives the identity. It therefore holds at every \(x\), proving the function identity. For the other distributive identity, if \(b\leq c\), then \(\max\{b,c\}=c\) and \(\min\{a,b\}\leq\min\{a,c\}\), so

The case \(c\leq b\) follows by interchanging \(b\) and \(c\). This proves the second distributive identity pointwise as well. \(\square\)

A particularly useful construction is the positive part of a function: \(f^+=f\vee\mathbf{0}\). It retains the values of \(f\) where they are positive and replaces negative values by zero. The negative part is \(f^-=(-f)\vee\mathbf{0}\); it records the magnitude of the negative values. Both belong to \(C(K)\), since they are lattice operations on continuous functions.

The last identity follows by considering the two cases \(f(x)\geq0\) and \(f(x)<0\). In the first case \(f^+(x)=f(x)\) and \(f^-(x)=0\); in the second case \(f^+(x)=0\) and \(f^-(x)=-f(x)\). In either case their difference is \(f(x)\).

Stability in the Supremum Norm

Lattice operations do more than preserve continuity: small uniform changes in the inputs produce small uniform changes in their maximum or minimum. This stability is especially useful when the input functions are approximations. The estimate below is a pointwise fact about real numbers, transferred to functions by taking the supremum.

Proof. Set \(\delta=\max\{\|f-u\|_\infty,\|g-v\|_\infty\}\). By the definition of the supremum norm, for every \(x\in K\),

Taking maxima gives

Interchanging \((f,g)\) and \((u,v)\) gives \((u\vee v)(x)\leq(f\vee g)(x)+\delta\). Combining the two inequalities yields

Taking the supremum over \(x\in K\) proves the stated bound. For minima, the same argument applies with the inequalities reversed when taking the smaller value: from \(f(x)\leq u(x)+\delta\) and \(g(x)\leq v(x)+\delta\), we obtain \((f\wedge g)(x)\leq(u\wedge v)(x)+\delta\); interchanging the pairs gives the reverse bound. Thus the minimum estimate follows too. \(\square\)

Why Lattice Operations Matter in Approximation

A continuous function built with maxima and minima remains in \(C(K)\), even when its formula changes from place to place. This permits useful constructions such as clipping a function to a prescribed range or joining candidate approximants by selecting whichever is larger at each point. The supremum-norm stability theorem adds a quantitative guarantee: if each input is uniformly close to its target, then the resulting maximum and minimum are uniformly close as well.

This estimate also explains a practical way to use approximation results. Suppose \(u\) and \(v\) approximate \(f\) and \(g\) within \(\varepsilon\) in the supremum norm. Then their pointwise maximum approximates \(f\vee g\) within \(\varepsilon\), and their pointwise minimum approximates \(f\wedge g\) within \(\varepsilon\). No additional loss of accuracy is introduced by taking the lattice operation.

A common mistake is to treat the maximum of two continuous functions as though the choice between them must be fixed across the entire domain. It need not be: the function that is larger can change with \(x\). The pointwise maximum remains continuous, including at points where the two functions cross, because the formulas for the maximum agree at every equality point. The lattice identities and norm estimate give precise ways to work with this changing choice.