**I am retreating back on this statement, after some explorations and calculation**
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this [reference](http://www.mathnet.ru/links/efaa32db37c4234e4e8a2b2d9cd3b330/mzm7569.pdf). But I must mention that my quest for jump discontinuities has not seen a dead end but a new light after this failure. I am very much interested in this class of functions and this kind of jumps, and have found a new way to deal with them in 2 dimensions. My new pursuit: A direction dependent jump can be converted to jump of a 1-d function via Radon transform, reflecting jumps in different directions at some point into separate 1-d jumps via Radon transform. Via Fourier slice theorem, such 1-d jumps can be dealt with different slices of FT of the image using a [1-d technique](https://mathoverflow.net/q/165038/14414) ). Cant wait to lay my hands on image signal processing.


The initial question was in $N$ dimensions which follows after the heading "Initial question". But in view of Willie's comments, I realized it has flaws, but I am still optimistic, and hope they are not fatal.


So I'd like to formulate and pose the question for the simple case of $N=2$, with which I am at comfort, considering my limitations with higher math, and then hope it can be generalized to $N>2$ by mathematicians. I seek their help in this regard.


**Case of $N=2$**


Main motivation is monochromatic image signal or any 2-d signal. Mathematically We represent it as a function $f:\mathbb{R}^2 \to \mathbb{R}$. I know image signal has compact support, but for mathematical convenience, we can always place zeros where ever it is not defined and extend its domain to $\mathbb{R}^2$. (Atleast I do this blindly just for mathematical beauty).


**Definition-1** A function $f:\mathbb{R}^2 \to \mathbb{R}$ is said to be of bounded variation, if for every rectifiable curve in $\mathbb{R}^2$, the 1-d function obtained by
restricting the domain of $f$ to the curve, is a function of bounded variation in 1-d sense.


(Motivation for this definition : For the following definition to make sense, and for the limits defined in it to exist always.)


**Definition-2** : Directional limits. Given a point $\hat{x}\_0 = (x,y) \in \mathbb{R}^2$, for every $\theta \in \mathbb{R}$ we define a directional limit as $$u\_{\theta}(\hat{x}\_0) = \lim\_{r\to 0+} f(x+r\cos(\theta),y+r\cos(\frac{\pi}{2}-\theta))$$ where $r \in \mathbb{R}$.


**Definition-3** : Limit function. We define a limit function $J\_{\hat{x}\_0} : \mathbb{R} \to \mathbb{R}$ for every point $\hat{x}\_0 \in \mathbb{R}^2$ given as $$J\_{\hat{x}\_0}(\theta) = u\_{\theta}(\hat{x}\_0)$$


**Definition-4** Class of functions $\mathcal{V}$ consists of all functions of the form $f:\mathbb{R}^2\to\mathbb{R}$ which satisfy the following two conditions.


1. $f$ should be a function of bounded variation as per **Definition-1**.
2. For every $\hat{x}\_0 \in \mathbb{R}^2$, the associated limit function $J\_{\hat{x}\_0}(\theta)$ (as in def-3) should be a function of bounded variation when its domain is restricted to $[0,2\pi)$.


**Problem**


Let $f \in \mathcal{V}$ be a square integrable function and let its Fourier transform be $\hat{f}$. Given any $\theta \in \mathbb{R}$ and $r\in\mathbb{R}$ we define a directional partial sum $S^{\theta}\_r : \mathbb{R}^2 \to \mathbb{R}$ as $$S^{\theta}\_r(\hat{x}) = \int\limits\_{-rcos(\pi/2-\theta)}^{r\cos(\pi/2-\theta)} \int\limits\_{-rcos\theta}^{r\cos\theta} \hat{f}(k\_x,k\_y) e^{i(xk\_x+yk\_y)} \mathrm{d}{k\_x} \mathrm{d}{k\_y} $$ where $\hat{x} = (x,y) \in \mathbb{R}^2$.


**Is the following statement true?**


Given any $\hat{x}\_0 \in \mathbb{R}^2$ and any $\theta \in [0,\pi)$ $$\lim\_{r\to\infty} S^{\theta}\_r(\hat{x}\_0) = \frac{1}{2}(u\_{\theta}(\hat{x}\_0) + u\_{(\pi-\theta})(\hat{x}\_0) )$$




---


### Higher dimensional case revisited


**Definition 1** Let $\Sigma\subset \mathbb{R}^N$ be a smooth submanifold. A function $f:\Sigma\to \mathbb{R}$ is said to have *bounded variation* if for every rectifiable curve $\gamma\subset \Sigma$, the composition $f\circ\gamma$ has bounded variation in the usual one-dimensional sense.


Now, let $\Sigma \subset\mathbb{R}^N$ be a smooth submanifold and let $x\in \Sigma$. We write $\exp\_x: T\_x\Sigma \to \Sigma$ to denote the [exponential map](http://en.wikipedia.org/wiki/Exponential_map#Riemannian_geometry).


**Definition 2** Let $\Sigma$ be a $d+1$ dimensional submanifold of $\mathbb{R}^N$. Let $f:\Sigma\to \mathbb{R}$. Fix $x\in \Sigma$ and $\omega\in T\_x\Sigma$ a unit vector. We write the *directional limit*
$$ f\_\omega(x) := \lim\_{t \to 0+} f\circ \exp\_x(t\omega) $$
whenever the limit on the right hand side exists. If for every unit vector $\omega\in \mathbb{S}^d \subset T\_x\Sigma$ the directional limit $f\_\omega(x)$ exists, we write $J\_x: \omega \mapsto f\_\omega(x)$. Note that $J\_x:\mathbb{S}^d \to \mathbb{R}$.


**Definition 3** Let $\Omega$ be a smooth submanifold of $\mathbb{R}^N$. We say that a function $f$ belongs to the class $\mathcal{V}(\Omega)$ iff $f$ has bounded variation and that at every point $x\in \Omega$ its blow-up $J\_x$ has bounded variation, both in the sense of Definition 1.


**Definition 4** Given $\omega\in \mathbb{S}^{N-1} \subset \mathbb{R}^N$, we write $\omega\_i$ to be the $i$th coordinate value of $\omega$ relative to the standard rectangular coordinate system. We can define the rectangle $R\_r^\omega$ for $r > 0$ to be
$$ R\_r^\omega = (-r |\omega\_1|, r|\omega\_1|) \times (-r |\omega\_2|,r|\omega\_2|) \times \cdots \times (-r|\omega\_N|, r|\omega\_N|)~,$$
in other words the rectangle with sides parallel to the standard hyperplanes and with diagonal $r\omega$. We also define $\mathrm{sgn}(\omega) = \prod\_{i = 1}^N \mathrm{sgn}(\omega\_i)$.


Now let $f\in \mathcal{V}(\mathbb{R}^N)\cap L^1(\mathbb{R}^N)$. Denote by $\hat{f}$ its Fourier transform. Fix $\omega\in \mathbb{S}^{N-1}$. Write
$$ S\_r^\omega f(x) = \int\_{R\_r^\omega} \hat{f}(\xi) \exp(i x\cdot \xi) \mathrm{d}\xi~. $$


**Question**


Is it true that


$$ \lim\_{r \to\infty} S^\omega\_r f(x) = \frac12 [J\_x(\omega) + J\_x(-\omega)] $$


for every $x\in \mathbb{R}^N$ and $\omega\in \mathbb{S}^{N-1}$?




---


### Initial question


This is a question about pointwise convergence of a Fourier transform of functions of the form $f: \mathbb{R}^N \to \mathbb{R}$, which is potentially a $N$-dimensional generalization to pointwise convergence of $1$ dimensional Fourier transform. This question arose when I am trying to generalize [this statement](https://mathoverflow.net/q/165038/14414) to $N$-dimensions.


**Definition 1** : Functions of bounded variation in $\mathbb{R}^N$.
Given any rectifiable curve in $\mathbb{R}^N$, if the function $f:\mathbb{R}^N \to \mathbb{R}$ evaluated on this curve is a function of bounded variation in the 1-d sense, then we say $f$ is a function of bounded variation.


Definition : Directional Limits.


for a function $f:\mathbb{R}^N \to \mathbb{R}$, given a unit vector $\bf{\hat{a}}$, we define the directional limit of $f$ at a point $\bf{x\_0} \in \mathbb{R}^N$ along $\bf{\hat{a}}$ as $u\_{\bf{\hat{a}}}(\bf{x\_0}) = \lim\_{\alpha \to 0^+}f(\bf{x\_0 + \alpha \hat{a}})$.


Limit function at a point $x\_0$ denoted as $J\_{\bf{x\_0}}:S^{N-1} \to\mathbb{R}$ is defined as $$J\_{\bf{x\_0}}(\bf{\hat{a}}) = u\_{\bf{\hat{a}}}(\bf{x\_0})$$. We denote this jump function in $\theta$-coordinates as $J^{\theta}\_{\bf{x\_0}}:[0,2\pi)^{N-1} \to \mathbb{R}$


**Definition of a class of functions** (This definition is recursive)


Given $\Omega$ an open subset of $\mathbb{R}^N$, we define a set of functions $\mathcal{V}(\Omega)$ with the following properties.


Iff $f \in \mathcal{V}(\Omega)$ then


1. $f:\Omega \to\mathbb{R}$ is square integrable and function and of bounded variation as per **Definition 1**.
2. With an additional constraint that the limit function in $\theta$-coordinates, at any point $P \in \Omega$ , denoted as $J^\theta\_P: [0,2\pi)^{N-1}\to \mathbb{R}$ also belongs to the class of functions $\mathcal{V}([0,2\pi)^{N-1})$.


**Fourier partial sum**


Consider a function $f \in \mathcal{V}(\mathbb{R}^N))$, and let its Fourier transform be $\hat{f}$. Given any unit vector $\bf{\hat{a}} \in \mathbb{R}^N$, and a positive real number $R$, we define Fourier partial sum as $$S^{\bf{\hat{a}}}\_R : \mathbb{R}^N \to \mathbb{R}$$ defined as $$S^{\bf{\hat{a}}}\_R(\bf{x}) = \int\_{-R\cos(\theta\_1)}^{R\cos(\theta\_1)} \int\_{-R\cos(\theta\_2)}^{R\cos(\theta\_2)} ...\int\_{-R\cos(\theta\_{N-1})}^{R\cos(\theta\_{N-1})} \int\_{-R\cos(\phi)}^{R\cos(\phi)} \hat{f}(k\_1,k\_2,k\_3,...k\_N) e^{i(k\_1x\_1+k\_2x\_2+...+k\_Nx\_N)} \mathrm{d}{k\_1}\mathrm{d}{k\_2}...\mathrm{d}{k\_N}$$ where $[\theta\_1,\theta\_2,...\theta\_{N-1}]$ is $\bf{\hat{a}}$ expressed in $\theta$-coordinates, and $\phi = \frac{\Phi\_{N-1}}{2^N} - \sum\limits\_{j = 1}^{N-1} \theta\_j$ where $\Phi\_{N-1}$ is the total solid angle subtended by the full surface of a unit $(N-1)$-sphere given as $$\Phi\_{N-1} = \frac{2\pi^{\frac{N-1}{2}}}{\Gamma(\frac{N-1}{2})}$$


$\bf{k} = [k\_1,k\_2,...k\_N] \in \mathbb{R}^N$ and $\bf{x} = [x\_1,x\_2,...x\_n] \in \mathbb{R}^N$


**Statement**


Question is that whether the following statement is true?


Given any point $\bf{x} \in \mathbb{R}^N$, and any unit vector $\bf{\hat{a}} \in \mathbb{R}^N$, $$\lim\_{R\to \infty} S^{\bf{\hat{a}}}\_R(\bf{x}) = \frac{u\_{\bf{\hat{a}}}(\bf{x}) + u\_{\bf{-\hat{a}}}(\bf{x})}{2}$$



A question about pointwise convergence of Fourier transform in $N$-dimensions
The answer to your question is: **No**. 


Consider the 2D case. Let $\theta = n \pi / 2$ for $n \in \{0,1,2,3\}$. From your definition $S\_r^\theta f = 0$, since on the RHS of its definition you are integrating over a null set. But clearly the corresponding $u\_\theta(x) + u\_{-\theta}(x)$ doesn't always vanish: consider in particular any non-trivial Schwartz function. 


