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I am writing in IEEE double column environment. I have some equations with cases. For example, this equation seems to have too much space after the brace and also before and after the commas.

I was using the array environment which seems to cause those large spacing. I still prefer to have some independent control over the alignment of the three parts of the equation, since in some cases for each column, the length may be quite different, and I may choose to center, left, or right align for each column. What would be a good way to make those spacings smaller but still have control over the alignment?

The code I used for the equation is

beginequation
theta_pk=left{ beginarrayccc
omega t+alpha+beta & , & n=1,2,3,4,5,6,ldots\
-omega t-alpha-beta & , & n=7,8,9,10,11,12,ldots
endarrayright.
endequation

as mentioned @marmot in comment, cases from amsmath is right tool for your job. it care for right spacing between building block of your equation:

documentclassIEEEtran
usepackageamsmath

usepackagelipsum % for dummy text

begindocument
lipsum[1]
beginequation
theta_pk =
 begincases
omega t+alpha+beta & , n=1,2,3,4,5,6,ldots\
-omega t-alpha-beta & , n=7,8,9,10,11,12,ldots
 endcases
endequation
lipsum[2-4]
enddocument

addendum:
as mentioned @manooooh in his valued comment, AMS (American Mathematical Society) prefer style, where comas follows math expression, i.e:

 begincases
omega t+alpha+beta, & n=1,2,3,4,5,6,cdots,\
-omega t-alpha-beta, & n=7,8,9,10,11,12,cdots.
 endcases

which use gives

but some people prefer the following style:

 begincases
phantom-
 omega t+alpha+beta, & n=1,2,3,4,5,6,cdots\
-omega t-alpha-beta, & n=7,8,9,10,11,12,cdots
 endcases

giving:

however, documentation "amsldoc", as mentioned barbara beeton in her comment, is given the following example, how mathematicians usually use the cases math environment:

 begincases
 omega t+alpha+beta & n=1,2,3,4,5,6,cdots,\
-omega t-alpha-beta & n=7,8,9,10,11,12,cdots.
 endcases

which gives:

A variant, with the fleqn environment from nccmath. I improvedged the alignment in the first column of the cases environment, using a phantom — sign in the first row.

You also can save some space – to a certain extent, with the mathrlap command from mathtools (which loads amsmath). I didn't need it here.

documentclassIEEEtran
usepackageamsmath, nccmath

usepackagelipsum % for dummy text

begindocument

lipsum[1]
beginfleqn
beginequation
theta_pk =
 begincases
phantom-omega t+alpha+beta, & n=1,2,3,4,5,6,ldots\
-omega t-alpha-beta, & n=7,8,9,10,11,12,ldots
 endcases
endequation
endfleqn
lipsum[2-4]

enddocument