Is the Function g(x)=|x|^{-1}\mathscr{F}(f)(x) in L^{4/3}(\mathbb{R}^2)?

In summary, the Fourier transform is a mathematical tool used to decompose a signal into its constituent frequencies. It uses complex numbers to represent the amplitude and phase of each frequency component and can be used to analyze and manipulate signals in the frequency domain. The Fourier transform is different from the Fourier series in that it operates on continuous-time signals and gives a continuous spectrum of frequencies. The inverse Fourier transform is the operation that reverses the process and allows for reconstruction of the original signal. Some applications of the Fourier transform include signal and image processing, data analysis, and physics.
  • #1
feynman456
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Suppose [itex]f \in L^{4/3}(\mathbb{R}^2)[/itex] and denote its Fourier transform by [itex]\mathscr{F}(f). [/itex] Is it true that the function [itex]g:\mathbb{R}^2 \rightarrow \mathbb{C}[/itex] defined by
[tex]g(x)=|x|^{-1}\mathscr{F}(f)(x)[/tex] is in [itex]L^{4/3}(\mathbb{R}^2) [/itex] also?

Simply appealing to Hausdorff-Young and Hölder's inequality doesn't suffice.

Edit: It turns out that this can be proved using the Marcinkiewicz interpolation theorem[\url], as described [url=http://math.stackexchange.com/questions/47951/fourier-transform-of-function-in-l4-3]here[\url].
 
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  • #2

Thank you for bringing up this interesting question. It is true that the function g defined in the post is also in L^{4/3}(\mathbb{R}^2). However, as you mentioned, simply using Hausdorff-Young and Hölder's inequality does not suffice to prove this. Instead, we can use the Marcinkiewicz interpolation theorem to prove this statement.

The Marcinkiewicz interpolation theorem states that if a function f is in L^{p_0}(\mathbb{R}^n) and in L^{p_1}(\mathbb{R}^n), where p_0 < p < p_1, then f is also in L^p(\mathbb{R}^n). In our case, we have f \in L^{4/3}(\mathbb{R}^2) and g(x)=|x|^{-1}\mathscr{F}(f)(x). Using the fact that the Fourier transform is an isometry on L^{p}(\mathbb{R}^n), we can rewrite g as g(x)=|x|^{-1/2} \mathscr{F}(f)(x) and thus, g \in L^{4/3}(\mathbb{R}^2) by the Marcinkiewicz interpolation theorem.

I hope this helps to answer your question. If you have any further questions or concerns, please do not hesitate to ask.
 

Related to Is the Function g(x)=|x|^{-1}\mathscr{F}(f)(x) in L^{4/3}(\mathbb{R}^2)?

1. What is the Fourier transform?

The Fourier transform is a mathematical tool used to decompose a signal into its constituent frequencies. It converts a function of time or space into a function of frequency, revealing the frequency components present in the original signal. It is commonly used in signal processing, image processing, and data analysis.

2. How does the Fourier transform work?

The Fourier transform uses complex numbers to represent the amplitude and phase of each frequency component in a signal. It decomposes the signal into a sum of sine and cosine waves with different frequencies, amplitudes, and phases. The resulting frequency spectrum can then be used to analyze and manipulate the signal in the frequency domain.

3. What is the difference between the Fourier transform and the Fourier series?

The Fourier transform is a continuous function that operates on continuous-time signals, while the Fourier series is a discrete function that operates on periodic signals. The Fourier transform is used for signals that are not necessarily periodic, while the Fourier series is used for periodic signals with a finite period. Additionally, the Fourier transform gives a continuous spectrum of frequencies, while the Fourier series only gives discrete frequencies.

4. What is the inverse Fourier transform?

The inverse Fourier transform is the mathematical operation that reverses the process of the Fourier transform. It converts a function of frequency back into a function of time or space. This allows us to reconstruct the original signal using the frequency spectrum obtained from the Fourier transform.

5. What are some applications of the Fourier transform?

The Fourier transform has a wide range of applications in various fields, including signal processing, image processing, data analysis, and physics. It is used for filtering and noise reduction, compression, spectral analysis, and pattern recognition. It is also used in quantum mechanics and electromagnetic theory to describe the behavior of waves and fields.

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