The weighted semigroup algebra Mb (S, w) is studied via its identification with Mb (S) together with a weighted algebra product *w so that (Mb (S, w), *) is isometrically isomorphic to (Mb (S), *w). This identification enables us to study the relation between regularity and amenability of Mb (S, w) and Mb (S), and improve some old results from discrete to general case.

Let G be a locally compact Hausdorff topological group and H be a compact subgroup of G. Then, the homogeneous space G/H possesses a specific Radon measure, which is called a relatively invariant measure. We show that the concepts of convolution and involution can be extended to the integrable functions defined on this homogeneous space. We study the properties of convolution and prove that the space of integrable functions is an involutive Banach algebra with an approximate identity. We also find a necessary and sufficient condition on a closed subspace of this Banach algebra to make it a left ideal.

We show that the second-order Hochschild cohomology groups of measure algebra M(G) and group algebra L-1(G) with coefficients in C-phi are Banach spaces where G is a locally compact group and phi is the augmentation character. 

Let G be a locally compact group or a discrete group and let w be a weight function on G. We define a new convolution on the Beurling algebra L1(G, w) and show that it is also a Banach algebra with this new convolution. We show that there is a one to one correspondence between the unitary representations of G and the non-degenerate ?-representations of L1(G, w). We show that L1(G, w) is amenable if and only if G is amenable. This improves earlier results which require w to be diagonally bounded.

In this paper, seepage analysis in isotropic environment is presented by discrete singular convolution (DSC) method. This method has been used for solving numerical problems since 1999. The theoretical basis of the method is distribution and wavelet theory. DSC exhibits accuracy for integration of global methods and flexibility for handling complex geometries and boundary conditions of local methods. At first, a simple case governed by Laplace equation and regularized Shannon’s kernel is considered to show the accuracy and efficiency of DSC method through L? and L2 errors. Seepage under concrete dam with or without sheet-pile is then solved through two different case studies. The results of this method against finite difference results are considered. The results show that this method is more accurate than finite difference method with reception of more operation value and optimization of R parameter.

We introduce a new convolution operator L-a(lambda) (b, c; beta). Several subordination and superordination results involving this operator are proved. 

In this paper, we introduce a convolution-type product for  strongly integrable Hilbert $C^*$-module valued maps on locally compact groups. We investigate various properties of this product related to uniform continuity, boundless, etc. For instance, we prove a convolution theorem. Also, we study the boundless of the  related convolution operator in various settings.