What is meant by Jensens function on the real line exactly F(x+y)+F(x-y)=2F(x); jensens equation on a R rather than just a continuousinterval(a closed and bounded real valued (continuum interval such as [0,1] and likewise for the range [0,1] );
I presume this is when the equation is allowed to take on negative values; otherwise if werewould have to be restricted on [0,1] to [0,1], there would issues with negative values and so forth and so I presume that on a real valued inverval one just uses jensens standard equation. I presume that this 'real line version' has no distinct analogue when considering real valued intervals; is that correct. Ie that jensens standard equation should be used instead (as jensen's standard equation allows for real valued intervals, it does not just deal with discrete cases); and that trying to adapt the equation in this real line form to the real valued closed interval case, would be a waste of time (ie, the only applicable equation what one would be Jensen equation tradition can be seen to entail this real line version.). So am I correct in presuming that form is not really distinct but for models which are whose domain is not confined to an interval such as [0,1]
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e.See F(x-y)+F(x+y)=2F(x) see https://books.google.com.au/books?id=JzS7CgAAQBAJ&pg=PA200&lpg=PA200&dq=jensen%27s+functional+equation+on+groups&source=bl&ots=gQa2kWMRYF&sig=3QlrUcn6pnDwNSGOInXJUDFg7fQ&hl=en&sa=X&ved=0ahUKEwj0t_6TubrTAhUDFpQKHfMTAD84ChDoAQgwMAM#v=snippet&q=real%20line&f=false;
Or does this just mean the general case where the domain covers all real values (not just an interval. Negative values and all real values >1, and negative. 2.I note that there is.
However I note that there are forms such as f(xy) + f(xy−1) = 2f(x) and f(xy) + f(y−1x) = 2f(x); what is meant y-1 here; does this have functional interpretation. I believe that it is a group interpretation. Is there a functional version of this ,see http://link.springer.com/article/10.1007/s00010-011-0089-7 "Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy−1) = 2f(x) and f(xy) + f(y−1x) = 2f(x"),
and if so, do these apply to interval and how is the inverse statement on y, ie y-1 to be interpreted (as an inverse function).or as 1-y. not 1/x I presume?.Something like the unit element of the structure (the identity/idempotent, max,unit element or some such which would be just, 1 in a probabilistic case; ie for F(x)=x on [0,1] to [0,1]?) or rather the element of the domain which corresponds to the max element of the co-domain of F-1(1) - y; which on [0,1] to [0,1] would correspond generally 1\in domain as F(1)=1
see also