In this paper we consider the system of functions G_(r,n)^α (x) (r∈N,n=0,1,…) which is orthogonal with respect to Sobolev-type inner product on (-1,1) and generated by Gegenbauer polynomials. The main goal work study some properties related {φ_(k,r) (x)}_(k≥0) {G_(r,n)^α (x)} functions. We conditions a function f(x) given in generalized for it be expandable into mixed Fourier series form f(x)~∑...

Abstract We develop product space theory of singular integrals with mild kernel regularity. study these regularity questions specifically in situations that are very tied to the T 1 type arguments and corresponding structural theory. In addition, our results multilinear.

let $(m^{2n},g)$ be a real hypersurface with recurrent shapeoperator and tangent to the structure vector field $xi$ of the sasakian space form$widetilde{m}(c)$. we show that if the shape operator $a$ of $m$ isrecurrent then it is parallel. moreover, we show that $m$is locally a product of two constant $phi-$sectional curvaturespaces.

Stochastic games and product-games. An n-player stochastic game is given by (1) a set of players N = 1 n , (2) a nonempty and finite set of states S, (3) for each state s ∈ S, a nonempty and finite set of actions As for each player i, (4) for each state s ∈ S and each joint action as ∈×i∈N As , a payoff r i s as ∈ to each player i, (5) for each state s ∈ S and each joint action as ∈×i∈N As , a ...