For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\limits_{-\omega}^{\omega} \hat{f}(\theta) e^{j \theta t} d\theta $$

Let the Wigner Ville Distribution of $e_{\omega}(t)$ be $W_{e_{\omega}}(t,\theta)$. I want to know that the following statement is valid for some $p,q \in \mathbb{N}$

$$\lim\limits_{\omega\to\infty}\frac{\|W_{e_{\omega}}\|_{L^p}}{\|W_{e_{\omega}}\|_{L^q}} = 1$$

What are the conditions on $f$ under which this is possible. Also, comment on is this convergence really important?

PS: The Wigner Ville Distribution (time frequency distribution) of a real function $f$ is defined as the Wigner Distribution of the associated complex function $z(t) = f(t) + if_h(t)$, where $f_h$ is the Hilbert transform of $f$.


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