quality:corruption, information:extraction
Relationship to the Kronecker delta

The Kronecker delta δ is the quantity defined by

δ i j = { 1 i = j 0 i ≠ j {\displaystyle \delta _{ij}={\begin{cases}1&i=j\\0&i\not =j\end{cases}}}

for all integers i, j. This function then satisfies the following analog of the sifting property: if a (for i in the set of all integers) is any doubly infinite sequence, then

∑ i = − ∞ ∞ a i δ i k = a k. {\displaystyle \sum _{i=-\infty }^{\infty }a_{i}\delta _{ik}=a_{k}.}

Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property

∫ − ∞ ∞ f (x) δ (x − x 0) d x = f (x 0). {\displaystyle \int _{-\infty }^{\infty }f(x)\delta (x-x_{0})\,dx=f(x_{0}).}

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.