Abstract:
Let $\mathbb T = \mathbb R/2\pi\mathbb Z$, $d\ge2$.
We associate with an integrable function
$f: \mathbb T^d \to \mathbb C$ its trigonometric Fourier series
$$ f \sim \sum_{\mathbf k\in\mathbb Z^d} \hat f(\mathbf k) \exp(i\mathbf k\mathbf x),$$
where $\mathbf k = (k_1,\dots,k_d)\in \mathbb Z^d$, $\mathbf x = (x_1,\dots,x_d)\in \mathbb T^d$,
$\mathbf k \mathbf x = k_1x_1 + \dots + k_dx_d$.
For a bounded set $A\subset\mathbb R^d$
we define partial trigonometric Fourier sums мы with respect to the set $A$ as
$$ S_A(f) (x) = \sum_{\mathbf k\in\mathbb Z^d\cap A} \hat f(\mathbf k) \exp(i\mathbf k\mathbf x).$$
Next we define Pringsheim partial sums of $f$. Let
$\mathbf n = (n_2,\dots,n_d)\in \mathbb N^d.$ Define
$$ S_{\mathbf n} (f) = \sum_{|\mathbf k| \le \mathbf n}
\hat f(\mathbf k) \exp(i\mathbf k\mathbf x),$$
where $|\mathbf k| \le \mathbf n$ means that
$|k_j| \le n_j$ for $j=1,\dots,d$.
Theorem 1.
For any function $f\in L(\mathbb T^d)$ there exists a sequence of
Pringsheim partial sums converging to $f$ in $L^p$ for all $p\in(0,1)$
and almost everythere.
Theorem 2.
For any function $f\in L(\mathbb T^d)$, any set
$E\subset \mathbb T^d$ of positive measure, and any sequence of vectors
$\{\mathbf n^{(j)} = (n_1^{(j)}, \dots, n_d^{(j)})\}$
satisfying the condition
$$ \min(n_1^{(j)}, \dots, n_d^{(j)})\to 0],(j\to\infty) \tag{1},$$
if $S_{\mathbf n^{(j)}} (f)$ converges to a finite on $E$ function $g$, then $g$ coincides with $f$ almost everywhere on $E$.
If $f$ is a real function then its Pringsheim partial sums are also real functions.
In this case the condition of finiteness of $g$ on $E$
can be omitted, and we may allow the function $g$ to be equal to $+\infty$ or to
$-\infty$ at some points. This means that the sequence
$\{S_{\mathbf n^{(j)}} (f)\}$ cannot converge to $+\infty$ or to $+\infty$
on a set of positive measure.
If $f$ is not necessarily a real function, one may ask: is it possible,
assuming the condition (1), to have the equality
$$ \lim_{j\to\infty} |S_{\mathbf n^{(j)}} (f)| = \infty \tag{2}$$
on a set of positive measure? It turns out that this equality can hold almost everywhere.
For various classes of sequences $\{A_j\}$, $j =1,2,\dots$ we consider the question:
is it possible that the sequence $\{|S_{A_j}(f)|\}$ tends to infinity almost
everywhere on $\mathbb T^d$?
|
