Index Transforms

The Fourier–Stieltjes transform of Minkowskiʼs ?(x) function and an affirmative answer to Salemʼs problem

“…[18]). But finally we came to the conclusion that an application of index transforms [16] will be a rather effective tool to achieve the goal and give an affirmative answer to the question. Namely, we will explore composition and asymptotic properties of the Kontorovich-Lebedev transform with respect to the Haar measure dx…”

“…Hence we treat the integral over (−∞, M], appealing again to inequality (16) in order to justify a passage to the limit under the integral sign when λ → π…”

On the least values of Lp-norms for the Kontorovich–Lebedev transform and its convolution

“…The purpose of this paper is to obtain an analog of the Hausdorff-Young inequality [2] and the norm estimates for the following convolution operator (cf. [6,7,11]) (1.2) where K i (x) is the modified Bessel function of the second kind [1] with respect to the pure imaginary index = i . We will consider these operators in appropriate Lebesgue spaces.…”

“…We will consider these operators in appropriate Lebesgue spaces. In particular, the convolution operator (1.1) is well defined in the Banach ring L (R + ) ≡ L 1 (R + ; K (x) dx), ∈ R (see [11,7]), i.e. the space of all summable functions f : R + → C with respect to the measure K (x) dx for which…”

A convolution for the Hankel–Kontorovich–Lebedev transformation

“…The convolution corresponding with this transform has been investigated in great detail by S. B. Yakubovich [22] from a classical point of view and by H.-J. Glaeske and A. Hess [5] in spaces of generalized functions.…”