A NOTE ON A FUNCTIONAL EQUATION RELATED TO DIGITAL FILTERING

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Published: Apr 30, 2021
Keywords:
functional equation translation (shift) operators functional equation related to digital filtering wave equation

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Khanithar Naenudorn

Abstract

This research was a study to determine the functional equation corresponding to the following functional equation:


gif.latex?f(x+s,&space;y+t)+f(x-s,&space;y)+f(x,&space;y-t)=f(x-s,&space;y-t)+f(x+s,&space;y)+f(x,&space;y+t)                  (M)


for all gif.latex?x,&space;y,&space;t,&space;s&space;\in&space;G, and gif.latex?f:G\times&space;G\rightarrow&space;\mathbb{C}, where a 2-divisible abelian group and gif.latex?\mathbb{C} were a set of complex numbers. The research results revealed that there were two corresponding functional equations could be obtained as the followings:


gif.latex?f(x+2s,&space;y+2t)+f(x-2s,&space;y-t)+f(x-s,&space;y-2t)+f(x,&space;y+t)+f(x+s,&space;y)=f(x-2s,&space;y-2t)+f(x+2s,&space;y+t)+f(x+s,&space;y+2t)+f(x,&space;y-t)+f(x-s,&space;y)               (M1)


and


gif.latex?f(x+2s,&space;y)+f(x,&space;y+2t)+f(x-s,&space;y)+f(x,&space;y-t)+f(x-2s,&space;y+t)+f(x+s,&space;y-2t)=f(x-2s,&space;y)+f(x,&space;y-2t)+f(x+s,&space;y)+f(x,&space;y+t)+f(x+2s,&space;y-t)+f(x-s,&space;y+2t).                (M2)          

Article Details

Issue
Vol. 9 No. 1 (2021): January - April
Section
Research Article

References

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