Λεπτομέρειες βιβλιογραφικής εγγραφής
| Τίτλος: |
Bessel operator with Jacobi polynomial integration. |
| Συγγραφείς: |
Bouzeffour, Fethi1 (AUTHOR) fbouzaffour@ksu.edu.sa |
| Πηγή: |
Integral Transforms & Special Functions. Sep2026, Vol. 37 Issue 9, p763-778. 16p. |
| Θεματικοί όροι: |
*Bessel functions, *Jacobi polynomials, *Fractional calculus, *Error analysis in mathematics, *Taylor's series, *High-order derivatives (Mathematics), *Approximation theory, *Mathematical formulas |
| Περίληψη: |
This work extends approximation formulas for n-th order derivatives, originally defined via integrals involving Jacobi polynomials, to higher-order Bessel operators $ \mathscr {L}_\alpha $ L α , defined by \[ \mathscr{L}_\alpha = \frac{\textrm{ d}^2}{\textrm{ d}x^2} + \frac{2\alpha + 1}{x} \frac{\textrm{ d}}{\textrm{ d}x}, \quad \alpha \geq -\frac{1}{2}. \] L α = d 2 d x 2 + 2 α + 1 x d d x , α ≥ − 1 2. The proposed approach enables the estimation of n-th powers of Bessel operators from noisy observations of a smooth function, using a truncated Delsarte-type Taylor expansion that introduces a slight temporal shift. For even functions $ f \in C^{2n+2} $ f ∈ C 2 n + 2 , we establish that the bias error is of order $ O(h^2) $ O (h 2). As an application, we derive a new pointwise formula for the fractional Bessel operator $ (-\mathscr {L}_\alpha)^{\nu } $ (− L α) ν , valid for $ \nu > 0 $ ν > 0. [ABSTRACT FROM AUTHOR] |
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