There has recently been much interest in approximation by linear combinations of functions taken from a redundant set 79. That is, the elements of 79 are not linearly independent. Perhaps the first example of this type was considered by Schmidt in 1907 [6] who considered the approximation of functions f ( x , y ) of two variables by bilinear forms ~-~tfl=lUi(X)Vi(y ) in L2([0,112). This problem...

In this work we prove Hardy space estimates for bilinear Littlewood-Paley-Stein square function and Calderón-Zygmund operators. Sufficient Carleson measure type conditions are given for square functions to be bounded from H p1 ×H p2 into Lp for indices smaller than 1, and sufficient BMO type conditions are given for a bilinear Calderón-Zygmund operator to be bounded from H p1 ×H p2 into H p for...

We prove a quadrilinear integral estimate in space-time for solutions of the homogeneous wave equation on R. This estimate is a generalization of a previously known bilinear L estimate, and it arises naturally in the study of the local regularity properties of a hyperbolic model equation connected with wave maps from Minkowski space R into a sphere. The scale invariant data space for this equat...

In this paper the notion of an M -th order invariant bilinear differential pairing is introduced and a formal definition is given. If the manifold has an AHS structure, then various first order pairings are constructed. This yields a classification of all first order invariant bilinear differential pairings on homogeneous spaces with an AHS structure except for certain totally degenerate cases....

The $L$-fuzzy approximation operator associated with an $L$-fuzzy approximation space $(X,R)$ turns out to be a saturated $L$-fuzzy closure (interior) operator on a set $X$ precisely when the relation $R$ is reflexive and transitive. We investigate the relations between $L$-fuzzy approximation spaces and $L$-(fuzzy) topological spaces.

For the Kac-Wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine Lie algebra D (1) n , we compute the coefficients of the corresponding Hirota bilinear equations, and verify the coincidence of these bilinear equations with the ones that are satisfied by Givental’s total descendant potential of the Dn singularity, as conjectured ...

The diagonalization of the Heun–Racah operator is studied with help modified algebraic Bethe ansatz. This most general bilinear expression in two generators Racah algebra. A presentation this algebra given terms dynamical operators and allows construction vectors for operator. associated equations are derived both homogeneous inhomogeneous cases.

For the Kac-Wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine Lie algebra D (1) n , we compute the coefficients of the correspondent Hirota bilinear equations, and verify the coincidence of these bilinear equations with the ones that are satisfied by Givental’s total descendant potential of the Dn singularity, as conjectured ...