How to determine constant to be in Hilbert space

[member 428835]

Hi PF!

I'm trying to solve an ODE through the Ritz method, which is to say approximate the solution through a series $$\Phi = \sum_{i=1}^N a_if_i,\\
f_i = \phi_i-d_i.$$
Here $a_i$ are constants to be determined and $f_i$ are prescribed functions, where $\phi_i$ is a function and $d_i$ is a constant.

A necessary input for this method is to determine $f_i$. For my particular problem the solution can be constructed from even or odd functions $\phi_i$, which I denote $\phi_{i,even}$ and $\phi_{i,odd}$. Since $\phi_i$ are Fourier modes (I've not listed them explicitly but I can) it is known they each form complete orthogonal systems in two mutually orthogonal subspaces, $H_1$ and $H_2$ for odd and even respectively, and together they compose the Hilbert space $H$.

Here's where I get lost: the text states in order for $\phi_i-d_i \in H$, it must be true $d_i = \frac{1}{\Gamma}\int_\Gamma\phi_i$. Why is this? I observe with this condition $\int_\Gamma f_i = 0$, not sure if this is relevant or not though. Any help would be awesome!

 

[jvicens]

Hi there,

The condition $d_i = \frac{1}{\Gamma}\int_\Gamma\phi_i$ is necessary for $\phi_i-d_i$ to be an element of the Hilbert space $H$. This is because the Hilbert space is a complete vector space of functions, and in order for a function to be in this space, it must satisfy certain conditions.

One of these conditions is that the function must have a finite norm, which is defined as $\sqrt{\langle f, f \rangle}$, where $\langle f, g \rangle$ is the inner product of two functions. In this case, the inner product is defined as $\langle f, g \rangle = \int_\Gamma f(x)g(x)dx$.

If we substitute $f_i = \phi_i-d_i$ into the inner product, we get $\langle f_i, f_i \rangle = \int_\Gamma (\phi_i(x)-d_i)^2dx$. In order for this integral to be finite, the constant $d_i$ must be chosen such that $\int_\Gamma d_i^2dx$ is also finite. This is why we have the condition $d_i = \frac{1}{\Gamma}\int_\Gamma\phi_i$, as it ensures that the integral is finite and therefore the function $\phi_i-d_i$ is in the Hilbert space.

As for your observation that $\int_\Gamma f_i = 0$, this is indeed relevant. It means that the function $f_i$ has a mean value of zero over the interval $\Gamma$, which is a useful property for the Ritz method. This is because the Ritz method seeks to minimize the error between the exact solution and the approximate solution, and having a mean value of zero helps to achieve this.

I hope this helps to clarify the condition and its relevance in the Ritz method. Let me know if you have any further questions.