In regard to $L^p$ spaces, is $L^p \subset L^q$ for $pAttempting to understand if there are relationships between $L^p$ spaces and can't find a conclusive answer in Kreyszig's 'Introduction to Functional Analysis with Applications' nor in a solid browsing of Wikipedia. 
More context:
Dirichlet conditions seem to indicate that if a function is absolutely integrable it can be represented as a fourier series. To me, this means that the function must lie within $L^1$ space, i.e. $\int | f(x) | \ dx < \infty$. 
I began this hunt in an attempt to understand why functions in $L^2([0,1])$ space can be represented as fourier series, or more specifically, why the basis components of $L^2([0,1])$ are functions of $x$ (e.g. $e^{-2 \pi i n x}$). If $L^2([0,1])$ functions are contained within $L^1$ then I think I understand, though this seems like a leap.
Appreciate any help.  
 A: As @Thorgott commented, there is no relation between those spaces in general, but
$$
L^p(\Omega)\subset L^q(\Omega)\qquad\textrm{for}\quad p>q>0,
$$
if $\Omega$ has finite measure, since the Hölder inequality gives
$$
\|u\|_{L^q(\Omega)}^q
=\||u|^q\|_{L^1(\Omega)}
\leq
\|1\|_{L^{p/(p-q)}(\Omega)}
\||u|^q\|_{L^{p/q}(\Omega)}
=|\Omega|^{(p-q)/p}
\|u\|_{L^{p}(\Omega)}^q.
$$
In fact, we do not need the restriction $p,q\geq1$.
To illustrate what happens when $|\Omega|=\infty$, take $\Omega=\mathbb{R}$ with the Lebesgue measure on it. Then the function $u\equiv1$ is in $L^\infty(\mathbb{R})\setminus L^1(\mathbb{R})$. However, the function $u(x)=\min\{0,\log|x|$} is in $L^1(\mathbb{R})\setminus L^\infty(\mathbb{R})$.
A: Alex, when reading the title it looks like you ask for conditions to inclusion between $L^p$-spaces, but in your query, it looks more like you are into the convergence of Fourier series. Since these matters are different I split the answer into two parts.
Inclusion between $L^p$-spaces
When the measure space, i.e. the domain of integration, is

*

*Finite (like $[0,1]$), then as @timur pointed out, we have $$L^p\subset L^q, \quad \text{provided $p>q>0$} $$


*Infinite and non-discrete, then as @timur pointed out there is no inclusion.


*Infinite and discrete, then
$$L^p\subset L^q, \quad \text{provided $q>p>0$} $$
see How do you show monotonicity of the $\ell^p$ norms?
These matters are connected to the Fourier series in a theorem known as the Pontryagin duality. The interval $[0,1]$ can be viewed upon as the circle group, and the dual group of the circle group is the group of integers (a discrete infinite group) - in case of the real numbers, the dual group is the real numbers again.
Convergence of Fourier series
Fourier series are series of the form $f(x)\sim\sum_{n=-\infty}^\infty c_ne^{2\pi inx}$,
with $c_n=\int_0^1f(x)e^{-2\pi inx}dx$.
Such series are defined for $f\in L^p[0,1]$ when $p\geq1$. When talking about convergence of Fourier series one usually means pointwise convergence (almost everywhere since sets of zero measures are unimportant in the Lebesgue sense).
Kolmogorov constructed a function in $L^1$ where the Fourier series, which do exist, diverges almost everywhere (1922).
Note that Kolmogorov's example shows that extra conditions (like the Dini-test) upon $L^1$-functions to have convergent Fourier series are non-superfluous.
For long it was an open problem if the Fourier series of a $L^2$-function converge or not, and it was proved to be true by Carleson in 1962, and was later sharpened to $L^p$ $1<p\leq2$ by Hunt.
Interestingly, it could be mentioned that there are intermediate spaces $A$ and $B$, with $L^1\supset A\supset B\supset L^p$ for all $p>1$, where $A$ contain functions with divergent Fourier series and while $B$ does not which has been studied by Konyagin and others.
