Faltung (Form)

Let $A$ and $B$ be bilinear forms

$$A = A(x,y)=\sum\sum a_{ij}x_iy_i$$

$$B = B(x,y)=\sum\sum b_{ij}x_iy_i$$

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 $MN$.

References

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