Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with convergence, while Fredholm concerns himself with existence of solutions via the integral equation $I \lambda - K$. There doesn't seem to be any series or convergence in the work of Fredholm, also anyone writing about the work of Fredholms have an abstract kernel $K$ and do not say anything about how this looked in his specific problems which makes it even harder to see the connection.
Is there simple explanation why these problems seem so far apart?
Own idea ;
Fourier suggest solution in term of series. No one knows what this expression represent. Fredholm doesn't investigate the solution itself but rather tries showing that some solution exists. If it doesn't then fourier expression might not even be well defined. It turns out that this has solutions, and then people carry on investigating the convergence.
Am I on the right track?
Fourier's Treatise on heat conduction was first submitted in 1807. In that work he introduced the method of separation of variables, and various expansions in orthogonal functions. Fourier died in 1830.
Gauss looked at Potential Theory at roughly the same time. By 1830, Gauss was heavily involved in the study of Electromagnetism. He gave a minimization principle for finding solutions of the Laplace equation.
In his 1847 work, Dirichlet took Gauss' work a step further in studying a minimization principle associated with the potential equation. This is now known as the "Dirichlet principle."
Riemann adapted the principle of Dirichlet to study Riemann surfaces and holomorphic functions.
Dealing with the Dirichlet principle became a focus of Analysis in the latter part of the 19th century. With several missteps, and mounting applications, there was a push to understand this work.
Schwarz published a paper in 1885 on the vibrating membrane. Using separation of variables to eliminate time reduced the study to the Helmholtz equation $$ \Delta f + \lambda f = g. $$ Using potential theory, this could be reduced to $$ f + \lambda \int K(x,y)f(y)dy = h. $$ This equation is now known as a Fredholm integral equation of the second kind. The parameter $\lambda$ was originally a separation parameter, and would have to do with eigenvalues in the context of Fourier Analysis. Schwarz introduced the integral version of what is now known as the Cauchy-Schwarz inequality in order to study sequential methods for solving minimization problems.
Poincare published a paper in 1890 concerning two distinct topics: (1) The Dirichlet Problem, and (2) The Fourier Cooling off Problem, which was still difficult to address because of the trigonometric expansions associated with unevenly spaced eigenvalues (the eigenvalues were solutions of a transcendental equation.) Properties of solutions of the above integral equations were studied as a function of the parameter $\lambda$.
Fredholm initiated an effective and deep study of integral equations in 1900. Fredholm carried this out by discretizing the equation using Riemann sums, and then took a limit of the determinant of the system as the increment went to $0$. The work of Fredholm is where a linear operator was first defined in an abstract way, and the resolvent operator was introduced $(A-\lambda I)^{-1}$. The behavior of solutions as a function of the parameter $\lambda$ was studied and this, for the first time, gave insight into what is now referred to as Spectral Theory. All of the previous problems mentioned above were naturally connected in this setting. Being limits of finite numbers of equations, the end results retained some of this character. This was one of the first significant rigorous breakthroughs in this complicated thread.
Starting around 1905, Hilbert further abstracted everything into the more modern version of Spectral Theory and Hilbert-Schmidt operators. Hilbert specialized to symmetric operators, and was able to prove the convergence of orthogonal expansions of Fourier Analysis. A few years later, one of the Riesz brothers characterized much of Fredholm's work in terms of compact operators, and this theory remains almost unchanged since its original publication from 100+ years ago.
This is a tedious chain with a lot of independent threads that are related in fundamental ways, but which are also different in important respects. Fourier Analysis motivates a good part of the thread, but by no means is it everything. The Spectral Theory part is almost entirely motivated by Fourier's work, but that's not the whole story.