In this paper we present a method of studying convolution operator under the Sonin conditions imposed on kernel. The particular case kernel is fractional integral Riemman–Liouville operator, other various types kernels are Bessel-type function, functions with power-logarithmic singularities at origin e.t.c. We pay special attention to study close power type functions. main our aim Sonin–Abel eq...

In this paper, we present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella functions, by using the extended Bessel due to Boudjelkha [?]. Some recurrence relations, transformation formulas, Mellin transform integral representations are obtained for these generalizations. Further, an extension Riemann-Liouville fractional deri...

In a rectangular domain, boundary-value problem is considered for mixed-type equation with regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using method separation variables, Laplace transform, unique solvability has been established. Moreover, we have found explicit solution initial problems derivative non-zero startin...

Controlled g-atomic subspace for a bounded linear operator is being presented and characterization has been given. We give an example of controlled K-g-fusion frame. construct new frame the Hilbert space H ? X using frames spaces X. Several useful resolutions identity on theory g-fusion have discussed. Frame pair Bessel sequences introduced.

We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Paley– Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrary smooth functions. Furtherm...

In this paper, we introduce the concept of \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials, where \(\mathbb{B}_{\alpha}\) is raising operator \(\mathbb{B}_{\alpha}:=x^2 \cdot {d}/{dx}+\big(2(\alpha-1)x+1\big)\mathbb{I}\), with nonzero complex number \(\alpha\) and \(\mathbb{I}\) representing identity operator. We show that Bessel polynomials \(B^{(\alpha)}_n(x),\ n\geq0\), \(\alpha\neq...

A dual frames multiplier is an operator consisting of analysis, multiplication and synthesis processes, where the analysis are made by two in a Hilbert space, respectively. In this paper we investigate spectra some multipliers giving, particular, conditions to be at most countable. The contribution extends results available literature about Bessel with symbol decaying zero Riesz bases.

A countable collection X of functions in L2(IR ) is said to be a Bessel system if the associated analysis operator T ∗ X : L2(IR ) → `2(X) : f 7→ (〈f, x〉)x∈X is well-defined and bounded. A Bessel system is a fundamental frame if T ∗ X is injective and its range is closed. This paper considers the above two properties for a generalized shift-invariant system X. By definition, such a system has t...

The variation of photonic orbital angular momentum at Compton scattering is 7 characterized. We determine scattering matrix of a twisted light based on the fundamental 8 conservation of orbital angular momenta. Numerical values for two different twisted light modes: 9 Laguerre Gaussian and Bessel Gaussian, are generated and illustrated. Our analysis indicate that 10 states of photonic orbital a...