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The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formulais a real sequence and {d n } ∞ n=1 is a positive chain sequence. We establish that there exists an unique nontrivial probability measure µ on the unit circle for which {R n (z) − 2(1 − m n )R n−1 (z)} gives the sequence of orthogonal polynomials. Here, {m n } ∞ n=0 is the minimal parameter sequence of the positive chain sequence {d n } ∞ n=1 . The element d 1 of the chain sequence, which does not effect the polynomials R n , has an influence in the derived probability measure µ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {M n } ∞ n=0 is the maximal parameter sequence of the chain sequence, then the measure µ is such that M 0 is the size of its mass at z = 1. An example is also provided to completely illustrates the results obtained.

Abstract. Based on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725-732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature K N ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.

Abstract. We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in N × R, where N is an n-dimensional complete Riemannian manifold with radial sectional curvature K N ≤ κ. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces have a positive fundamental tone.