Umbral Interpolation: A Survey
Abstract
:1. Introduction
- X be a linear space of real functions defined on the interval , continuous and with continuous derivatives of all necessary orders;
- be the space of polynomials of degree less than or equal to n;
- L be a linear functional on X of the following type
- , being the set of natural numbers.
2. Preliminaries
3. Umbral Interpolation
4. Examples
4.1. Umbral Interpolation Related to
4.1.1. Generalized Taylor Polynomial
4.1.2. Bernoulli of the First Kind Umbral Interpolation
4.1.3. Euler Umbral Interpolation
4.1.4. First Kind Bernoulli-Type Umbral Interpolation
4.2. Umbral Interpolation Related to
4.2.1. Generalized Bernoulli of the Second Kind Umbral Interpolation
4.2.2. Generalized Boole Umbral Interpolation
4.3. Umbral Interpolation Related to
-Umbral Interpolation
5. Numerical Examples
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Umbral Interpolation Formulas
Appell-Taylor | |||
Q | interpolatory conditions | polynomials | |
Appell-Bernoulli | |||
Q | interpolatory conditions | polynomials | |
Appell-Euler | |||
Q | interpolatory conditions | polynomials | |
Appell-Bernoulli of the first kind | |||
Q | interpolatory conditions | polynomials | |
Newton interpolation | |||
Q | interpolatory conditions | polynomials | |
ine | |||
Generalized Bernoulli of the second kind interpolation | |||
Q | interpolatory conditions | polynomials | |
Generalized Boole interpolation | |||
Q | interpolatory conditions | polynomials | |
interpolation | |||
Q | interpolatory conditions | polynomials | |
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Costabile, F.A.; Gualtieri, M.I.; Napoli, A. Umbral Interpolation: A Survey. Mathematics 2025, 13, 271. https://doi.org/10.3390/math13020271
Costabile FA, Gualtieri MI, Napoli A. Umbral Interpolation: A Survey. Mathematics. 2025; 13(2):271. https://doi.org/10.3390/math13020271
Chicago/Turabian StyleCostabile, Francesco Aldo, Maria Italia Gualtieri, and Anna Napoli. 2025. "Umbral Interpolation: A Survey" Mathematics 13, no. 2: 271. https://doi.org/10.3390/math13020271
APA StyleCostabile, F. A., Gualtieri, M. I., & Napoli, A. (2025). Umbral Interpolation: A Survey. Mathematics, 13(2), 271. https://doi.org/10.3390/math13020271