On some constants in simultaneous approximation
Pointwise estimates for the error which is feasible in simultaneous approximation of a function and its derivatives by an algebraic polynomial were originally pursued from theoretical motivations, which did not immediately require the estimation of the constants in such results. However, recent nume...
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| Format: | Article |
| Language: | English |
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Wiley
1995-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171295000342 |
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| _version_ | 1832556494664499200 |
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| author | K. Balázs T. Kilgore |
| author_facet | K. Balázs T. Kilgore |
| author_sort | K. Balázs |
| collection | DOAJ |
| description | Pointwise estimates for the error which is feasible in simultaneous approximation of a
function and its derivatives by an algebraic polynomial were originally pursued from theoretical
motivations, which did not immediately require the estimation of the constants in such results.
However, recent numerical experimentation with traditional techniques of approximation such as
Lagrange interpolation, slightly modified by additional interpolation of derivatives at ±1, shows that
rapid convergence of an approximating polynomial to a function and of some derivatives to the
derivatives of the function is often easy to achieve. The new techniques are theoretically based upon
older results about feasibility, contained in work of Trigub, Gopengauz. Telyakovskii, and others, giving
new relevance to the investigation of constants in these older results. We begin this investigation here.
Helpful in obtaining estimates for some of the constants is a new identity for the derivative of a
trigonometric polynomial, based on a well known identity of M. Riesz. One of our results is a new proof
of a theorem of Gopengauz which reduces the problem of estimating the constant there to the question
of estimating the constant in a simpler theorem of Trigub used in the proof. |
| format | Article |
| id | doaj-art-b536232dbcd346f59519cb9a547ab099 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1995-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-b536232dbcd346f59519cb9a547ab0992025-02-03T05:45:16ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251995-01-0118227928610.1155/S0161171295000342On some constants in simultaneous approximationK. Balázs0T. Kilgore1Budapest University of Economics, Pf. 489, Budapest 5 H-1828, HungaryDivision of Mathematics, Auburn University, Auburn 36849, Alabama, USAPointwise estimates for the error which is feasible in simultaneous approximation of a function and its derivatives by an algebraic polynomial were originally pursued from theoretical motivations, which did not immediately require the estimation of the constants in such results. However, recent numerical experimentation with traditional techniques of approximation such as Lagrange interpolation, slightly modified by additional interpolation of derivatives at ±1, shows that rapid convergence of an approximating polynomial to a function and of some derivatives to the derivatives of the function is often easy to achieve. The new techniques are theoretically based upon older results about feasibility, contained in work of Trigub, Gopengauz. Telyakovskii, and others, giving new relevance to the investigation of constants in these older results. We begin this investigation here. Helpful in obtaining estimates for some of the constants is a new identity for the derivative of a trigonometric polynomial, based on a well known identity of M. Riesz. One of our results is a new proof of a theorem of Gopengauz which reduces the problem of estimating the constant there to the question of estimating the constant in a simpler theorem of Trigub used in the proof.http://dx.doi.org/10.1155/S0161171295000342simultaneous approximationtheorem of Gopengauz. |
| spellingShingle | K. Balázs T. Kilgore On some constants in simultaneous approximation International Journal of Mathematics and Mathematical Sciences simultaneous approximation theorem of Gopengauz. |
| title | On some constants in simultaneous approximation |
| title_full | On some constants in simultaneous approximation |
| title_fullStr | On some constants in simultaneous approximation |
| title_full_unstemmed | On some constants in simultaneous approximation |
| title_short | On some constants in simultaneous approximation |
| title_sort | on some constants in simultaneous approximation |
| topic | simultaneous approximation theorem of Gopengauz. |
| url | http://dx.doi.org/10.1155/S0161171295000342 |
| work_keys_str_mv | AT kbalazs onsomeconstantsinsimultaneousapproximation AT tkilgore onsomeconstantsinsimultaneousapproximation |