Sentence examples for dual transforms from inspiring English sources

Starting with the study of the classical Laplace convolution and a cosine convolution, along with associated dual transforms, natural algebra homomorphisms are introduced which capture the convoluted semigroup and cosine function properties.

We also consider its dual transforms associated with the Schrödinger operator L defined by R ˜ = L − 1 / 2 ∇ and the higher order commutator R ˜ b m f ( x ) = ∫ R d ( b ( x ) − b ( y ) ) m K ˜ ( x, y ) f ( y ) d y, where K ˜ ( x, y ) is the kernel of R ˜ and m = 1, 2, …  .

We obtain new inversion formulas for the Radon transform and the corresponding dual transform acting on affine Grassmann manifolds of planes in Rn.

We show that this transform intertwines the actions of certain M n -invariant differential operators on G(p, n) and G(q, n), and we prove an inversion forM n -invarianttransform when n is odifferentialzing the Radoperatorsion formula, and obtaininG pn pandicular an inversion formula for the dual transform.

To obtain an expression for the dual transform the Lagrangian function of (P1) is needed, which has the following form: (16).

Furthermore, Liu and Wang investigated the boundedness of the dual Riesz transforms and its commutators on the Morrey spaces related to the nonnegative potential V belonging to the reverse Hölder class in [2].

In this section we recall some estimates for the kernels of Riesz transform (mathcal{R}) and the dual Riesz transform (tilde{mathcal{R}}), which have been proved in [3].

Also, the dual Riesz transform associated with the Schrödinger operator L is defined by T ∗ = L − 1 2 ∇ and the commutator operator [ b, T ∗ ] ( f ) ( x ) = T ∗ ( b f ) ( x ) − b ( x ) T ∗ f ( x ), x ∈ R n. (2).

Let L = − Δ + V be a Schrödinger operator, where Δ is the Laplacian on R n and the nonnegative potential V belongs to the reverse Hölder class B q for q ≥ n / 2. The Riesz transform associated with the operator L is denoted by T = ∇ ( − Δ + V ) − 1 2 and the dual Riesz transform is denoted by T ∗ = ( − Δ + V ) − 1 2 ∇.

Motivated by [3], our aim is to establish the boundedness for the dual Riesz transform associated with Schrödinger operators and its commutators on weighted Morrey spaces related to the certain nonnegative potentials, where the condition on the potential is weaker than that in [3].

Dual SBP-RA21 and single SBP or UH-RA.21 display phage were produced from the double transformed DH5αF′ bacterial cells.