this paper is an investigation of $l$-dual frames with respect to a function-valued inner product, the so called $l$-bracket product on $l^{2}(g)$, where g is a locally compact abelian group with a uniform lattice $l$. we show that several well known theorems for dual frames and dual riesz bases in a hilbert space remain valid for $l$-dual frames and $l$-dual riesz bases in $l^{2}(g)$.

We prove the existence of HK density function for a graded pair ( R , I ) where is an N -graded domain finite type over perfect field and ⊂ ideal colength. This generalizes our earlier result one proves such where, in addition standard graded. Other properties functions also hold pairs: example, it multiplicative Segre products, its maximum support F -threshold m -primary provided Proj smooth, ...