Mathematics

Discrete analogs of the index Whittaker transform are introduced and investigated. It involves series and integrals with respect to a second parameter of the Whittaker function W μ,in (x), x>0, μ∈R, n∈N, i is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established.

In this note, we express explicitly the Dunkl kernel and generalized Bessel functions of type A n−1 by the Humbert's function Φ (n) 2 , with one variable specified. The obtained formulas lead to a new proof of Xu's integral expression for the intertwining operator associated to symmetric groups, which was recently reported in [21].

We present in this paper some embeddings of various dyadic martingale Hardy-amalgam spaces H S p,q , H s p,q , H ??p,q , Q p,q and P p,q of the real line. In the same settings, we characterize the dual of H s p,q for large p and q . We also introduce a Garsia-type space G p,q and characterize its dual space.

We unite two themes in dyadic analysis and number theory by studying an analogue of the failure of the Hasse principle in harmonic analysis. Explicitly, we construct an explicit family of measures on the real line that are p -adic doubling for any finite set of primes, yet not doubling, and we apply these results to show analogous statements about the reverse Hölder and Muckenhoupt A p classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties.

We introduce a novel approach via dynamical system to primal-dual projection method and investigate the existence, uniqueness and extendability of solutions in a Hilbert space X with right-hand side which is continuous and bounded on a bounded and closed subset D ^ ?�X and locally Lipschitz on set D ^ ?�{ z ¯ } , where z ¯ is the only stationary point of the differential equation. We apply the results to dynamical system associated with the problem of finding the zeros of the sum of maximally monotone operators.

This paper targets to study the effect of the Riemann-Liouville fractional integral operator on unbounded variation points of a continuous function. In particular, we show that the fractional integral preserves the bounded variation points of a function. We also prove that the fractional integral operator is a bounded linear operator on the class of bounded variation and continuous functions.

We illustrate with several new applications the power and elegance of the Bendixson Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function related with the curvature of the orbits of the vector field as a Dulac function. We get some general results for Lienard type equations and for rigid planar systems. We also present a remarkable phenomenon: for each integer m greater than one, we provide a simple one parametric differential system for which we prove that it has limit cycles only for the values of the parameter in a subset of an interval that decreases exponentially when m grows. One of the strengths of the results presented in this work is that although they are obtained with simple calculations, that can be easily checked by hand, they improve and extend previous studies. Another one is that, for certain systems, it is possible to reduce the question of the number of limit cycles to the study of the shape of a planar curve and the sign of an associated function in one or two variables.

As a contribution to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has shown how analogues of the Jacobian elliptic functions may be derived from incomplete hypergeometric integrals in signatures three and four. We determine precisely the signatures in which the Jacobian analogues or their squares are indeed elliptic.

We study L p - L q estimate for the spectral projection operator Π λ associated to the Hermite operator H=|x | 2 −Δ in R d where Π λ is the projection to the subspace spanned by the Hermite functions which are the eigenfunctions of H with the eigenvalue λ . Such estimates were previously available only for q= p ′ , equivalently p=2 or q=2 (by T T ∗ argument) except for the estimates which are consequences of interpolation between those estimates. As shown in the works of Karadzhov, Thangavelu, and Koch and Tataru, the local and global estimates are of different nature. Especially, Π λ exhibits complicated behavior near the set λ − − √ S d−1 . Compared with the spectral projection operator associated with the Laplacian, the L p - L q boundedness of Π λ is not so well understood up to now for general p,q . In this paper we consider the L p - L q estimate for Π λ in a general framework including the local and global estimates with 1≤p≤2≤q≤∞ and undertake the work of characterizing the sharp bounds on Π λ . We establish various new sharp estimates in an extended range of p,q . First of all, we provide a complete characterization of the local estimate for Π λ which was first considered by Thangavelu. Secondly, for d≥5 , we prove the endpoint L 2 - L 2(d+3)/(d+1) estimate for Π λ which has been left open since the work of Koch and Tataru. Thirdly, we extend the range of p,q for which the operator Π λ is uniformly bounded from L p to L q . As an application, we obtain new L p - L q resolvent estimates for the Hermite operator H and Carleman estimates for the heat operator. This allows us to prove the strong unique continuation property of the heat operator for the potentials contained in L ∞ t L d/2,∞ x .

This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew orthogonal polynomials and then give different examples of Catalan-Hankel Pfaffians with continuous and q -moment sequences.