Quadrature domains in the plane are those on which a given test class of functions (say, the class of holomorphic functions) satisfies a generalized mean value property. The purpose of the talk will be to see how the Schwarz reflection principle leads to an understanding of these domains. The talk will be elementary.

The epsilon-pseudospectrum ?(a) of an element a of an arbitrary Banach algebra A is studied. Its relationships with the spectrum and numerical range of a are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of a. Suppose for some_x000f_ epsilon > 0 and a,b \in A, ?(ax) = ?(bx) for all x \in A. It is shown that a = b if: (I) a is invertible. (ii) a is Hermitian idempotent. (iii) a is the product of a Hermitian idempotent and an invertible element. (iv) A is semisimple and a is the product of an idempotent and an invertible element. (v) A = B (X) for a Banach space X. (vi) A is a C*-algebra. (vii) A is a commutative semisimple Banach algebra.

A famous theorem of Norbert Wiener states that for a function f in L^1(R), span of translates f(x?a) with complex coefficients is dense in L^1(R) if and only if the Fourier transform of f is nonvanishing on R. That is the ideal generated by f in L^1(R) is dense in L^1(R) if and only if the Fourier transform of f is nonvanishing on R. This theorem is well known as the Wiener Tauberian theorem. This theorem has been extended to abelian groups. The hypothesis (in the abelian case) is on a Haar integrable function which has nonvanishing Fourier transform on all unitary characters. However, back in 1955, Ehrenpreis and Mautner observed that Wiener Tauberian theorem fails even for the commutative Banach algebra of integrable radial functions on SL(2,R). In this talk we shall discuss about a genuine analogue of the theorem for real rank one, connected noncompact semisimple Lie groups with finite centre.

Let f be a non-zero Bernstein function, i.e., f(s) = a+bs+\int_0^{\infty}(1?e?xs)d?(x), a,b \geq 0,? \geq 0.There exists a uniquely determined product convolution semigroup (\rho_c)_c>0 on (0,\infty) such that (1) \int_0^{\infty} x^n d\rho_c(x) = (f(1)…...f(n))_c, c>0 , n= 0,1,...,... The Stieltjes moment sequence in (1) is always determinate when c \leq 2as an easy consequence of the Carleman criterion. However, for c >2, it can be determinate or indeterminate depending on f. In fact, in the case f(s) =s,where the moment sequence is (n!)c, it was proved that the moment sequence is indeterminate. In this case \rho_c=e_c(t)dm(t), where m is Lebesgue measure on the half-line and (2) ec(t) = 12\pi \int_{-\infty}^{\infty} t^{ix?1} \Gamma(1?ix)^c dx, t >0. The proof of the indeterminacy was quite delicate based on asymptotic formulas for stable distributions due to Skorokhod. In recent work with Jose Lopez, Spain, we have found the asymptotic behaviour of e_c at infinity. Then, it is easy to derive the indeterminacy of e_c from a criterion of Krein. In the lecture I will give the necessary background for the above.

The classical moment problem for the half-line and the real line were formulated and solved by Stieltjes (1894) and Hamburger (1920) respectively, and the theory has been treated in influential monographs by Shohat-Tamarkin (1943) and Akhiezer (1965). The refined theory of the indeterminate moment problem, i.e., the case where two and hence infinitely many different measures have the same moments, is a delicate blend of complex function theory and spectral theory. It has gained renewed interest with the study of orthogonal polynomials associated with q-basic hypergeometric functions important for the representation theory of quantum groups, because many of these new orthogonal polynomials come from indeterminate moment problems. Also several remarkable formulas of Ramanujan can be understood in the light of indeterminate moment problems. It is only during the last two decades that one has been able to make rather complete calculations of all the relevant ”characters” appearing in a concrete indeterminate moment problem.The log-normal distribution is an example of an indeterminate measure. We will give a review of the theory together with new results about the relation between the growth of P and summability properties of the sequence (Pn(z)). The order of the function P is called the order of the moment problem.The multidimensional moment problem will also be discussed.

To a probability distribution \mu with moments of any order (1) s_n = \int x^n d?(x), n = 0,1,..., we consider the orthonormal polynomials P_n, i.e., \int P_n(x)P_m(x) d\mu(x) = \delta_{nm}. They satisfy P_2(z) := \sum_n |P_n(z)|^2<\infty for all complex z precisely in the indeterminate case, where there are different probability measures with the same moments (1). This leads to a study of entire functions like K(z,w) = \sum_n P_n(z)P_n(w), z,w \in C and L(z) = \sum_n z_n/\sqrt{s_{2n}}. During the last 20 years there has been a general study of these entire function as well as many concrete examples, often related to q-series. We will give a review of some of these results together with new results about the relation between the growth of P and summability properties of P_n. The order of the function P is called the order of the moment problem. It is shown that under suitable conditions on the recurrence coefficients in the three term recurrence relation zPn(z)=bnPn+1(z) +anPn(z)+bn?1Pn?1(z), the order of the moment problem is equal to the exponent of convergence of the sequence (bn). Similar results are obtained for logarithmic order and for more general types of slow growth. The new results are based on joint work with Ryszard Szwarc, Wroclaw.

A. Duran and the speaker have studied some non-linear transformations from Hausdorff moment sequences (an) to Stieltjes moment sequences (sn), namely sn= (a0a1···an)?1, sn= (a0+a1+···+an)?1. Several examples will be given leading to surprising new moment sequences. The “sum” transformation has a fixed point (mn) defined by the recursive equation (m0+m1+···+mn) mn= 1, n \geq 0. The representing measure \omega of (mn) has an increasing and convex density on ]0,1[ and is best characterized via its Bernstein transform f. It turns out that f has a meromorphic extension to the whole complex plane satisfying the functional equation \psi(f(z+ 1)) = f(z), and that f can be completely described in terms of iterations of the rational function \psi(z) = z?1/z. The function f is also characterized by a theorem analogous to the Bohr-Mollerup theorem for the Gamma function.

Let f be a smooth function on R. The divided difference matrices whose (i,j) entries are [f(\lambda_i)-f(\lambda_j)]/[\lambda_i-\lambda_j], \lambda_1,...,\lambda_n \in R are called Loewner matrices. In a seminal paper published in 1934 Loewner used properties of these matrices to characterise operator monotone functions. In the same paper he established connections between this matrix problem, complex analytic functions, and harmonic analysis. These elegant connections sent Loewner matrices into the background. Some recent work has brought them back into focus. In particular, characterisation of operator convex functions in terms of Loewner matrices has been obtained. In this talk we describe some of this work.

Starting with a positive definite kernel $K$ defined on a bounded open connected subset $\Omega$ of $\mathbb C^d,$ we give several canonical constructions for producing new positive definite kernels on $\Omega,$ possibly taking values in $Hom(E)$ for some normed linear space $E$ of dimension $d.$ Specifically, this includes the curvature defined as the $d\times d$ matrix of real analytic functions $$\big ( \!\! \big ( \tfrac{\partial}{\partial_i \bar{\partial}_j} \log K \big ) \!\!\big ).$$ These kernels define an inner product on a submodule (over the polynomial ring) functions holomorphic on $\Omega.$ The completion is a Hilbert space on which the polynomials act by point-wise multiplication making it into a "Hilbert module". We will discuss hereditary properties, sub and quotient of these Hilbert modules.