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Faltung (Form)

Let $A$ and $B$ be bilinear forms

$\displaystyle A$ $\textstyle =$ $\displaystyle A(x,y)=\sum\sum a_{ij}x_iy_i$  
$\displaystyle B$ $\textstyle =$ $\displaystyle B(x,y)=\sum\sum b_{ij}x_iy_i$  

and suppose that $A$ and $B$ are bounded in $[p,p']$ with bounds $M$ and $N$. Then

F=F(A,B)=\sum\sum f_{ij}x_iy_j,

where the series

f_{ij}=\sum_k a_{ik}b_{kj}

is absolutely convergent, is called the faltung of $A$ and $B$. $F$ is bounded in $[p,p']$, and its bound does not exceed $MN$.


Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 210-211, 1988.

© 1996-9 Eric W. Weisstein