We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.
We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh’s type. We reformulate these equations giving a better description in terms of continuous operators on spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series which converges in the -norm and almost everywhere, where denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution....
One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by and as an “exceptional family”...
Let w be a generalized Jacobi weight on the interval [-1,1] and, for each function f, let Sf denote the n-th partial sum of the Fourier series of f in the orthogonal polynomials associated to w. We prove a result about uniform boundedness of the operators S in some weighted L spaces. The study of the norms of the kernels K related to the operators S allows us to obtain a relation between the Fourier series with respect to different generalized Jacobi weights.
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