Is it possible to blur an image in such way that a person with sight problems could see it sharp? If someone has short or long sight, is it possible to tune image on a computer monitor in such way, that a person could see it sharp as if they were wearing glasses? If not, will 3d monitor make it possible?
 A: This is a cold thread, but I think some of the answers are off base.  Blurring of an incoherent image can be mathematically represented as a convolution, and if the blurring kernel is known and is invertible (no spectral zeros)  then a deconvolution filter can be developed.  Mathematically it does not matter the order in which the two linear filters are applied, so you could apply the deconvolution filter first.  The problem is that a real image cannot have negative intensities, whereas the pre-deconvolved image is not guaranteed to be positive.  One would have to bias the image upwards to preserve positivity.  Then the observer would see a sharp image it would appear whitewashed.
A: The light source with this properties exists in technology named "Adaptive Optics". 
So the answer is yes, if your monitor (not existing on market yet) has controllable phase source for each pixel on screen (like phased array antenna), and even more. Each pixel actually needs multiples of phase values, depending on angle of look. So for megapixel display you may need gigapixel of phasing elements which follow and compensate phase exactly to errors on the ray trace to location of retina cell of each individual eyeball, looking at screen. 
A: No, if you are someone who needs glasses the effect without them would be like a blur effect which can not be reversed the same strongly blurred picture can not be rendered to its original detailed focus.
Taking a picture without focusing the lens would have the same effect. You can neither create an image that will be sharp with an unfocused lens nor reconstruct the detailed image taken with an unfocused lens.
A: Let's take a simple original picture to look at - just two nearby dots on a white background.  If you have bad vision, the dots look blurred.  
The way good vision works is to ensure that all the light hitting any particular small area of your retina comes from the same direction in front of you.  Conversely, all the light coming from one direction hits one specific spot on your retina.
When you have bad vision, the light from a locus of nearby directions all hits on the same part of your retina, and the light from a particular direction is smeared out over an area on your retina.  Hence, blurred vision is an averaging effect.  When you look at the dots, you'll see them smear out into each other.
You might try to compensate for this by making a "counter-blurred" image where the source dots are smaller, but if the original dots are close enough that light from the center of one dot is spilling over to overlap light from the center of the second dot, making the dots smaller won't fix that problem.  Hence, the dots will always appear blurred.  You can't create the impression that the original has for someone with good vision.
A photograph is really just a bunch of nearby dots, and so the same problem applies.
I don't know about the 3D monitor, though.  I suppose if it can control the direction of light coming off it, it could be modified to focus the light some and create a sharp image for someone with blurred vision.
A: Not on regular monitor screen. The technology necessary to achieve such effect would be holographic display, holographic in the sense of wavefront synthesis. Although this device would be a 3D display, not all 3D display are holographic. You would need technologies such as spatial light modulator. Which only exists as low specs laboratory devices.
A: Apparently there is software solutions for regular displays that somehow distorts the image in a way that makes it sharper for people with eyes out of focus:
New Scientist: Good looking: Phone screens that do the focusing for you

...
Few handheld devices have 3D screens so far, but something similar can
  be done with software alone. "We know how the eye works so we
  mathematically invert that and compute the image," says Daniel Aliaga,
  a computer vision researcher at Purdue University in West Lafayette,
  Indiana, who helped develop the idea. The software distorts images on
  the screen in such a way that it mimics the effect of a lens bending
  the light rays before they reach the eye. "It looks weird to anyone
  else but to you it looks sharp," says Aliaga.
The approach works with both text and images, he says, so with
  pre-focused screens people would no longer need reading glasses.
...

There are some examples at the website of the start-up, including a video: https://youtu.be/Vqd5VjXmKMw
A: As user1631 explains, deconvolution will yield the required image, but this will contain negative intensities. However, if you just raise the background gray value then it should be possible to deal with this issue. So, if we imagine a picture of the night sky and me watching it with my glasses removed. If the stars are deconvolved with the correct point spread function they turn into fringes that alternate in brightness dipping below and rising above the background gray value. If I watch this with my glasses removed, then the blurring will cause the fringes to cancel out relative to the background leaving only the center pixel sticking above the background.
A: Consider the system: monitor -> eyeglasses -> eye. Assuming these are the correct glasses, the monitor is clearly in focus. Now move the eyeglasses towards the monitor, transforming them as necessary to keep the optical path the same. For that person, I'd argue the monitor remains sharply focused.
A: Well, I think that it is possible to create a "blurred" image from a real one in order to be correct for someone with short/long sight.
It is "only" an optical problem, and, as long as we can apply the paraxial approximation, it is a linear system, very suitable to a computer program. We know the laws that focus the rays in one point, so we can disturb these rays so that they arrive at a different point, and it would be possible to "tune" this point to be correct to every people.
I think that the point is that, from a correct and well-defined image, we can do a lot of things with it. If we have a blurred image, and we want to make it correct, it will be more difficult, because now we haven't got all the information as in the first case.
