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. 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 ). 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.
- $f$ should be a function of bounded variation as per Definition-1.
- 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.
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 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
$f:\Omega \to\mathbb{R}$ is square integrable and function and of bounded variation as per Definition 1.
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 $\bf{\hat{a}} \in \mathbb{R}^N$, $$\lim_{R\to \infty} S^{\bf{\hat{a}}}_R(\bf{x}) = \frac{u_{\bf{\hat{a}}}(\bf{x}) + u_{\bf{-\hat{a}}}(\bf{x})}{2}$$
