In his paper "Group C*-algebras without the completely bounded approximation property", Haagerup proves several important results about the weak amenability of locally compact groups. Among these, is the result that a lattice in a second-countable, unimodular, locally compact group is weakly amenable if and only if the surrounding group itself is weakly amenable. A key ingredient in his proof is a method of using (linear) completely bounded Fourier multipliers on the lattice to construct (linear) completely bounded Fourier multipliers on the surrounding group. We use a similar approach to construct multilinear completely bounded Fourier multipliers on the group from multilinear completely bounded Fourier multipliers on the lattice. Our construction is both bounded in the norm of completely bounded Fourier multipliers and preserves uniform convergence on compact sets for bounded nets. We also prove an equivalent characterization of weak amenability where the Fourier algebra is replaced by the space of continuous and compactly supported $n$-linear Fourier multiplier symbols.

It was first shown by D. Potapov and F. Sukochev in 2009 that Lipschitz functions are also operator-Lipschitz on Schatten class operators S^p, 1<p<∞, which is related to a conjecture by M. Krein. Their proof combined Schur multiplication, a generalisation of component-wise matrix multiplication, with the so-called first order divided difference of a function, an approximation of its derivative. Showing that the Schur multipier associated with a divided difference function is bounded relies on a so-called transference technique, the boundedness of certain Schur multipliers can be inferred from the boundedness of associated Fourier multipliers. Soon after, this boundedness result was extended by D. Potapov, A. Skripka, and F. Sukochev to multilinear Schur multipliers of divided differences of arbitrary order, i.e. approximations of higher derivatives.

In this thesis, we offer an alternative boundedness proof for bilinear Schur multipliers of second order divided differences, in which we use recent results of multilinear harmonic analysis towards a multilinear transference proof, as well as recently found sufficient conditions for the boundedness of linear Schur multipliers which cannot be studied by transference. These methods were not known at the time Potapov, Skripka, and Sukochev proved their result.

Moreover, we show that this new proof improves the growth of the bound on the norm of the considered Schur multiplier for p→∞ significantly. Finally, we give an outlook on further steps towards an alternative boundedness proof of multilinear Schur multipliers of divided differences of arbitrary order.

In this thesis, we derive a multivariate analogue of Ruijsenaars’s 2F1-generalisation R. We use Hopf algebra representation theory of the modular double of sl.2/, a Hopf algebra structure strongly related to quantum groups, to relate the function R to overlap coefficients of eigenfunctions. Using properties of the algebra and the representation, we derive an Askey-Wilson type difference equation. We moreover recover Ruijsenaars’s unitary transformation kernel E.

Expanding on the Hopf algebra structure, we extend our derivations to the multivariate version of R. Employing representation theory, we obtain multivariate difference equations. Furthermore, we demonstrate that the multivariate function enables the definition of a unitary transformation on multivariate functions in L2..0; ∞/N/.

In 2017 Martijn Caspers, Fedor Sukochev and Dmitriy Zanin published a paper which generalises the proof of Davies' 1988 paper, and thus resolves the Nazarov-Peller conjecture. The proofs of these papers have been presented in this thesis. They have been expanded with a proof that generalises the conjecture to arbitrary Schatten classes. The optimality of the estimates in the conjecture is also studied, following the example of the 2016 paper by Coine et al. In addition, the quantum mechanical context is provided to interpret the presented results.

The spherical transform maps the orthogonal basis of symmetric Jacobi-type polynomials to an orthogonal basis of (symmetric) Wilson polynomials. The spherical transform is closely related to the Cherednik-Opdam transform, as it is essentially its symmetric version. The symmetric Jacobi-type polynomials can be composed from the non-symmetric Jacobi-type polynomials. These relations, between the symmetric and non-symmetric theory, give an incentive to consider the Cherednik-Opdam transform of non-symmetric Jacobi-type polynomials. This work gives an overview of the symmetric theory about the spherical transform of Jacobi-type polynomials and lays down the groundwork for the Cherednik-Opdam transform of the non-symmetric Jacobi-type polynomials.