Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, denoted by $c_n(f)$ for $n \in \mathbb{Z}^d$, are such that:

1. $c_n(f)$ is real, strictly positive, with $c_{n}(f) = c_m(f)$ for any $n,m \in \mathbb{Z}^d$ with equal Euclidian norms $\lVert n \rVert = \lVert m \rVert$,
2. the sum $\sum_{n \in\mathbb{Z}^d} c_n(f) = \infty$, and
3. the $c_n(f)$ are decreasing in the sense that $c_n(f) \leq c_m(f)$ as soon as $\lVert n \rVert \geq \lVert m \rVert$.

Can we prove that $f$ is necessarily discontinuous and even unbounded at $0$ with these assumptions?

Context: I am especially interested by the situation where $c_n(f)$ behaves asymptotically like (or even is equal to for $n \neq 0$) $\lVert n \rVert^{- \alpha}$ for some $\alpha \in (1/2, 1]$. The case $d = 1$ with $2\pi$ periodic functions is already interesting and may be quite simpler.

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, denoted by $c_n(f)$ for $n \in \mathbb{Z}^d$, are such that:

1. $c_n(f)$ is real, strictly positive, with $c_{n}(f) = c_m(f)$ for any $n,m \in \mathbb{Z}^d$ with equal Euclidian norms $\lVert n \rVert = \lVert m \rVert$,
2. the sum $\sum_{n \in\mathbb{Z}^d} c_n(f) = \infty$, and
3. the $c_n(f)$ are decreasing in the sense that $c_n(f) \leq c_m(f)$ as soon as $\lVert n \rVert \geq \lVert m \rVert$.

Can we prove that $f$ is necessarily discontinuous and even unbounded at $0$ with these assumptions?

Context: I am especially interested by the situation where $c_n(f)$ behaves asymptotically like (or even is equal to for $n \neq 0$) $\lVert n \rVert^{- \alpha}$ for some $\alpha \in (d/2, d]$. The case $d = 1$ with $2\pi$ periodic functions is already interesting and may be quite simpler.

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, denoted by $c_n(f)$ for $n \in \mathbb{Z}^d$, are such that:

1. $c_n(f)$ is real and strictly positive for any $n$,
2. the sum $\sum_{n \in\mathbb{Z}^d} c_n(f) = \infty$, and
3. the $c_n(f)$ are decreasing in the sense that $c_n(f) \leq c_m(f)$ as soon as $\lVert n \rVert \geq \lVert m \rVert$ for the Euclidian norm.

Can we prove that $f$ is necessarily discontinuous and even unbounded at $0$ with these assumptions?

Context: I am especially interested by the situation where $c_n(f)$ behaves asymptotically like $\lVert n \rVert^{- \alpha}$ for some $\alpha \in (1/2, 1]$.

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, denoted by $c_n(f)$ for $n \in \mathbb{Z}^d$, are such that:

1. $c_n(f)$ is real, strictly positive, with $c_{n}(f) = c_m(f)$ for any $n,m \in \mathbb{Z}^d$ with equal Euclidian norms $\lVert n \rVert = \lVert m \rVert$,
2. the sum $\sum_{n \in\mathbb{Z}^d} c_n(f) = \infty$, and
3. the $c_n(f)$ are decreasing in the sense that $c_n(f) \leq c_m(f)$ as soon as $\lVert n \rVert \geq \lVert m \rVert$.

Can we prove that $f$ is necessarily discontinuous and even unbounded at $0$ with these assumptions?

Context: I am especially interested by the situation where $c_n(f)$ behaves asymptotically like (or even is equal to for $n \neq 0$) $\lVert n \rVert^{- \alpha}$ for some $\alpha \in (1/2, 1]$. The case $d = 1$ with $2\pi$ periodic functions is already interesting and may be quite simpler.