Why do piecewise smooth function spaces require an infinite basis in PDEs?

[Ed Quanta]

The 3 dimensional space that we inhabit must have a basis of 3 vectors which is fair enough.

But in my partial differential equations class in which Fourier series was introduced, it was said that piecewise smooth function space has a basis of an infinite number of vectors. If there is a simple enough answer to this, I am curious to why this is. Does it have to do with the discontinuities that can arise in PWS space?

 

[matt grime]

No, it's because there are an infinite number of linearly independent smooth functions (forget even piecewise)

Take Functions from R to R

set f_i(x) = x^i, i in N

if the space were finite dimensional, of say dimension n, then given any n+1 of those functions, say just the first n+1, for ease, we could find real numbers a_i with

sum 0 to n a_ix^i equal to the zero function on R ie zero for all x in R. But poly of degree n has only n roots, doesn't it? so it can't be identically zero.

 

[M98Ranger]

The concept of vector spaces and bases plays a crucial role in understanding partial differential equations (PDEs). In three-dimensional space, we can easily visualize a basis of three vectors that span the entire space. However, when dealing with PDEs, we often encounter functions that are not as simple as vectors in three-dimensional space. These functions can have discontinuities, non-uniform behavior, and infinite variations. Therefore, to fully understand and solve PDEs, we need to expand our understanding of vector spaces and bases.

In PDEs, we often work with functions that are defined on a specific domain, such as a finite interval or a region in space. These functions are known as piecewise smooth functions, as they may have discontinuities at certain points but are otherwise smooth. Unlike vectors in three-dimensional space, these functions cannot be represented by a finite number of basis vectors. Instead, they require an infinite number of basis functions to span the entire function space.

This is where Fourier series come into play. Fourier series provide a way to represent a piecewise smooth function as an infinite sum of trigonometric functions. These trigonometric functions form a basis for the function space and allow us to expand our understanding of vector spaces to include functions with infinite variations.

The reason why piecewise smooth function space requires an infinite number of basis vectors is due to the nature of these functions. As mentioned earlier, they can have discontinuities and non-uniform behavior, which cannot be captured by a finite number of basis vectors. Therefore, we need an infinite number of basis functions to accurately represent these functions and solve PDEs involving them.

In summary, the basis of vector spaces and Fourier series in PDEs is essential for understanding and solving these complex equations. The use of an infinite number of basis vectors allows us to expand our understanding of vector spaces and include functions with discontinuities and infinite variations. This is crucial in solving PDEs that arise in various fields such as physics, engineering, and mathematics.