Question

Let \(X=C[-1,1]\) with norm \(\|x\|=\sup \\{|x(t)|: t \in[-1,1]\\}\), for \(x \in X\). Let \(\mathrm{A}\) be the set of constant functions in \(C[-1,1]\). Let \(x \in X\) be defined by \(x(t)=t^{2}\). Compute dist \((x, A)\). Does \(x\) have a best approximation in \(A\) ?

Short answer

Answer: The distance between the function \(x(t) = t^2\) and the set of constant functions in \(A\) is 1. The best approximation of \(x(t)\) in \(A\) exists and it is the constant function \(y(t) = 0\).

Step-by-step solution

Defining the distance between a function and a set of functions using the given norm

The distance between a function \(x(t)\) and a set of functions \(A\) in a normed space is defined by $$ \text{dist}(x, A) = \inf_{y \in A} \|x - y\| $$ where \(\inf\) refers to the infimum or greatest lower bound.

Computing the distance between \(x(t)\) and \(A\)

To compute the distance between \(x(t)\) and \(A\), we will first find the general expression for the distance between \(x(t)\) and a constant function in \(A\). Given any constant function \(y(t) = c \in A\), $$ \|x(t) - y(t)\| = \sup_{t \in [-1,1]} |t^2 - c| $$ Now, we need to determine the value of \(c\) that minimizes this expression. Let's find the critical points by differentiating the expression inside the absolute value with respect to \(c\) and setting it to zero, $$ \frac{d}{dc} (t^2 - c)= -1 $$ This equation has no solution, which means the distance is minimized at the boundary points \(t=-1,1\). At \(t = -1\), $$ \|x(t) - y(t)\| = \sup_{t \in [-1,1]} |t^2 - 1| $$ and at \(t = 1\), $$ \|x(t) - y(t)\| = \sup_{t \in [-1,1]} |t^2 - 0| $$ Therefore, the distance is minimized when \(c = 0\), $$ \text{dist}(x, A) = \sup_{t \in [-1,1]} |t^2| = 1. $$

Determine if there exists a best approximation of \(x(t)\) in \(A\)

Since we found the distance between \(x(t)\) and \(A\), we can now determine if there exists a best approximation of \(x(t)\) in \(A\). A best approximation exists if and only if $$ \|x(t) - y(t)^*\|= \text{dist}(x, A) $$ where \(y(t)^*\) is a function in \(A\). In our case, we have \(y(t)^* = 0\) which is a constant function in \(A\). We just calculated the distance between \(x(t)\) and \(A\) as 1 based on the given norm, and $$ \|x(t) - y(t)^*\| = \|x(t) - 0\| = \|x(t)\| = \sup_{t \in [-1,1]} |t^2| = 1. $$ So, the best approximation of \(x(t)\) in \(A\) exists, and it is the constant function \(y(t)^* = 0\).

Key concepts

Normed Spaces

Normed spaces are fundamental constructs in functional analysis. A normed space is essentially a vector space that comes equipped with a function called a "norm." The norm is used in measuring the "size" or "length" of vectors within the space.

In the context of the exercise, we are working with the space of continuous functions on the interval \([-1, 1]\). Functions in this space can be thought of as vectors, and the norm we are using is the supremum norm. The supremum norm assesses the maximum absolute value of a function over its domain.

• A norm must satisfy certain properties: non-negativity, the triangle inequality, absolute scalability, and being zero only for the zero vector.
• In the example, the supremum norm is noted as \|x\| = \( \sup \{|x(t)|: t \in[-1,1]\} \), which measures the largest value a function \(x(t)\) takes on that interval.

Best Approximation

Best approximation deals with the idea of finding the "closest" function within a set to a given function in a normed space. The distance between the function and the set is minimized, providing either an "infimum" or exact best approximation.

To solve for the best approximation of \(x(t) = t^2\) from the set of constant functions \(A\), we searched for a constant \(c\) that minimizes the distance \| x(t) - c \|. This assures us that this particular constant function is the nearest to \(x(t)\).

• We calculate by minimizing the supremum norm of the difference between our target function and a trial function from \(A\).
• Finding the best approximation not only helps in reducing approximation error but also provides a practical approach for making predictions using simpler models.

Constant Functions

Constant functions are a simple yet essential class of functions, especially in approximations. In our examination, the set \(A\) comprises only constant functions. These are functions where the output value remains constant regardless of the input value within the specified domain.

Such a function can be expressed as \(y(t) = c\), where \(c\) is constant for all \(t\). Constant functions are particularly useful in functional approximations, providing a base case or simplest form of a function.

• They are completely determined by their constant value, making them easy to handle analytically.
• In our example, checking various values of \(c\), we found the best approximation at \(c = 0\), which practically illustrates the simplicity and utility of constant functions in approximations.

Supremum Norm

The supremum norm is a type of norm that measures the greatest absolute value of a function over its domain. It is particularly useful in functional analysis when dealing with continuous functions.

In our problem, the supremum norm is used to calculate how close \(x(t) = t^2\) is to any constant function over the interval \([-1, 1]\). It determines the worst-case or maximum error between the function \(x(t)\) and a constant function.

• The supremum norm is defined as \(\sup_{t \in [-1,1]} |x(t)|\), representing the peak deviation from zero across the interval.
• In our scenario, it helps us find the minimum distance \dist(x, A)\, which is the greatest discrepancy \(|t^2 - c|\) when \(t\) ranges through \([-1, 1]\).