Is f in the vector space of cubic spline functions?

In summary, the conversation discusses checking if the function $f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$ is in the vector space of cubic spline functions on $[-1,1]$ in respect to the given points. The function is shown to be a piecewise polynomial of degree smaller or equal to $3$, with pieces between each $x_i$ and $x_{i+1}$. However, there is concern about the continuity of $f$ on $[-1,1]$ due to a potential discontinuity at $x=-\frac 13$. A suggestion is made to expand the inner expression and consider different cases.
  • #1
mathmari
Gold Member
MHB
5,049
7
Hey! :eek:

Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$.

We have that \begin{align*}f(x)&=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |\\ & =\begin{cases}|x|^3-\left |x+\frac{1}{3}\right |^3 , & |x|^3-\left |x+\frac{1}{3}\right |^3\geq 0 \\-|x|^3+\left |x+\frac{1}{3}\right |^3 , & |x|^3-\left |x+\frac{1}{3}\right |^3<0\end{cases} \\ & =\begin{cases}|x|^3-\left |x+\frac{1}{3}\right |^3 , & |x|^3\geq \left |x+\frac{1}{3}\right |^3 \\-|x|^3+\left |x+\frac{1}{3}\right |^3 , & |x|^3<\left |x+\frac{1}{3}\right |^3\end{cases} \\ & = \begin{cases}|x|^3-\left |x+\frac{1}{3}\right |^3 , & |x|\geq \left |x+\frac{1}{3}\right | \\-|x|^3+\left |x+\frac{1}{3}\right |^3 , & |x|<\left |x+\frac{1}{3}\right |\end{cases}\end{align*}

The function is piecewise a polynomial of degree smaller or equal to $3$, right?

Now we have to check if $f$ is continuous on $[-1,1]$.

How could we continue to get the definition of $f$ ? (Wondering)
 
Mathematics news on Phys.org
  • #2
Yes. And those pieces should be between those $x_i$.
That is, between each $x_i$ and $x_{i+1}$ we should have a polynomial of degree $\le 3$.
It doesn't look like we will get that, since the derivative at $x=-\frac 13$ will be discontinuous. (Worried)

mathmari said:
Now we have to check if $f$ is continuous on $[-1,1]$.

How could we continue to get the definition of $f$ ?

How about we start with the inner expression $|x|^3-\left |x+\frac{1}{3}\right|$ and expand it for the cases:
\begin{cases} -1 \le x<-\frac 13 \\ -\frac 13\le x < 0 \\ 0 \le x \le 1\end{cases}
Then all the absolute signs should disappear. (Thinking)
 

FAQ: Is f in the vector space of cubic spline functions?

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects, called vectors, and a set of operations, such as addition and scalar multiplication, that can be performed on these vectors. These operations follow certain rules, such as closure and associativity, and allow for the manipulation and combination of vectors.

2. What are cubic spline functions?

Cubic spline functions are a type of mathematical function used in interpolation and approximation. They consist of piecewise cubic polynomials that are connected at certain points, called knots, to create a smooth curve. These functions are commonly used in computer graphics, engineering, and data analysis.

3. How do you determine if a function is in the vector space of cubic spline functions?

To determine if a function is in the vector space of cubic spline functions, it must satisfy certain properties. These include being continuous, having continuous first and second derivatives, and having a specified number of continuous derivatives at each knot. Additionally, the function must be able to be written as a linear combination of other cubic spline functions.

4. What are the benefits of using cubic spline functions?

Cubic spline functions have several benefits, including their ability to accurately represent complex curves and data sets, their smoothness and continuity, and their flexibility in terms of choosing the number and placement of knots. They are also computationally efficient and can be easily manipulated and interpolated.

5. How are cubic spline functions used in real-world applications?

Cubic spline functions are used in a variety of real-world applications, such as computer graphics, engineering, physics, and finance. They are commonly used for data interpolation and approximation, curve fitting, and creating smooth and visually appealing curves and surfaces. They are also used in numerical analysis and optimization algorithms.

Similar threads

Replies
9
Views
2K
Replies
2
Views
761
Replies
7
Views
1K
Replies
4
Views
2K
Replies
4
Views
875
Replies
4
Views
11K
Replies
5
Views
1K
Replies
1
Views
11K
Back
Top