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Let with , and , and let
where
We establish the asymptotic expansion
where stands for the Bernoulli polynomials. Further, we prove that the functions and are completely monotonic in on for every if and only if and , respectively. This not only unifies the two known results but also yields some new results.
It is proved that if the increasing sequence kn n=0..∞
n=0 of nonnegative integers has density greater than 1/2 and D is an arbitrary simply
connected subregion of CRthen the system of Hermite associated functions
Gkn(z) n=0..∞ is complete in the space H(D) of complex functions holomorphic in D.
We define and investigate the conjugate operator for Fourier-Bessel expansions. Weighted norm and weak type (1,1) inequalities are proved for this operator by using a local version of the Calderón-Zygmund theory, with weights in most cases more general than weights. Also results on Poisson and conjugate Poisson integrals are furnished for the expansions considered. Finally, an alternative conjugate operator is discussed.
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