By using Riemann–Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants detN [ cla−mb [ f ] ] generated by holomorhpic symbols, where la = a (resp. mb = b) except for a finite subset of indices a = h1, . . . , hn (resp. b = t1, . . . , tr). In addition to the usual Szegö asymptotics, our answer involves a determinant of size n + r.

Contrary to the second order case, biharmonic heat kernels are sign-changing. A deep knowledge of their behaviour may however allow to prove positivity results for solutions of the Cauchy problem. We establish further properties of these kernels, we prove some Lorch-Szegö-type monotonicity results and we give some hints on how to obtain similar results for higher polyharmonic parabolic problems.

The classical Szegö theorems study the asymptotic behaviour of the determinants of the finite sections PnT (a)Pn of Toeplitz operators, i.e., of operators which have constant entries along each diagonal. We generalize these results to operators which have almost periodic functions on their diagonals.

An analogue of Krein’s extension theorem is proved for operatorvalued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szegö parameters. One singles out a distinguished completion, called central, which is related to quasi-multiplicative positive definite functions. An application is given to factorization o...

We use the existence of localized eigenfunctions of the Laplacian on the Sierpiński gasket (SG) to formulate and prove analogues of the strong Szegö limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences.

In this research contribution, we have constructed a subclass of analytic bi-univalent functions using Horadam polynomials. Bounds for certain coefficients and Fekete- Szegö inequalities been estimated.