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Wikipedia tells that in 2 dimensions the correct formula is $sinc_C(x,y)=sinc(x)*sinc(y)$.

The problem I see with such expression is that if you draw a plot of this function, it has orthogonal "ripples" that do not look natural. Is there exists a 2-dimensional function with bandlimiting properties that looks like having round ripples?

Or should I just stick with product of sinc functions in 2 dimensions and assume it is correct - e.g. to perform 2D image rescaling.

aleksv
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  • If you want a two-dimensional solution to the wave equation, they're given by https://en.wikipedia.org/wiki/Bessel_function . The Fourier transform of $J_0$ is related to the rectangle function. – Chappers May 13 '18 at 20:36
  • Not sure I need a solution to wave equation. I need a 2D band-limiting function that looks maybe more natural than product of 2 sinc functions. I admit I may be wrong and correct approach is indeed to have orthogonal ripples, it just does not "look" right. – aleksv May 13 '18 at 20:41
  • The Fourier transform of the function that is $1$ on the unit disk and $0$ elsewhere is $J_1(2\pi k)/k$ (where the factors may vary depending on convention). – Chappers May 13 '18 at 21:13
  • On Wikipedia page on Bessel functions, I do not see a finalized 2-dimensional $J_1$ expression, so can't readily draw it to see how it looks in 2D. – aleksv May 13 '18 at 21:24
  • It's a radial function, so it's rotationally symmetric. – Chappers May 13 '18 at 21:27
  • OK, but $J_1$ won't fit image resizing for sure as it has zero at 0. Should I use spherical $J_0$ maybe? It's basically sinc function. Is it OK to use it in radial form? – aleksv May 13 '18 at 21:30
  • I'm afraid that's outside my area of expertise. – Chappers May 13 '18 at 21:32

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