## On determination of best-possible constants in integral inequalities involving derivatives

HTML articles powered by AMS MathViewer

- by Beny Neta PDF
- Math. Comp.
**35**(1980), 1191-1193 Request permission

## Abstract:

This paper is concerned with the numerical approximation of the best possible constants ${\gamma _{n,k}}$ in the inequality \[ {\left \| {{F^{(k)}}} \right \|^2} \leqslant \gamma _{n,k}^{ - 1}\;\left \{ {{{\left \| F \right \|}^2} + {{\left \| {{F^{(n)}}} \right \|}^2}} \right \},\] where \[ {\left \| F \right \|^2} = \int _0^\infty |F(x){|^2}\;dx.\] A list of all constants ${\gamma _{n,k}}$ for $n \leqslant 10$ is given.## References

- N. P. Kupcov,
*Kolmogorov estimates for derivatives in $L_{2}[0,\infty )$*, Trudy Mat. Inst. Steklov.**138**(1975), 94–117, 199 (Russian). Approximation of functions and operators. MR**0393388** - S. B. Stečkin,
*Inequalities between norms of derivatives of arbitrary functions*, Acta Sci. Math. (Szeged)**26**(1965), 225–230 (Russian). MR**185064** - S. B. Stečkin,
*Best approximation of linear operators*, Mat. Zametki**1**(1967), 137–148 (Russian). MR**211169** - V. V. Arestov,
*Precise inequalities between the norms of functions and their derivatives*, Acta Sci. Math. (Szeged)**33**(1972), 243–267 (Russian). MR**320729** - V. V. Arestov,
*Some extremal problems for differentiable functions of one variable*, Trudy Mat. Inst. Steklov.**138**(1975), 3–28, 199 (Russian). Approximation of functions and operators. MR**0415165** - Ju. N. Subbotin and L. V. Taĭkov,
*Best approximation of a differentiation operator in the space $L_{2}$*, Mat. Zametki**3**(1968), 157–164 (Russian). MR**228892** - J. S. Bradley and W. N. Everitt,
*On the inequality $\parallel f^{\prime \prime }\parallel ^{2}\leq K\parallel f\parallel \parallel f^{(4)}\parallel$*, Quart. J. Math. Oxford Ser. (2)**25**(1974), 241–252. MR**349930**, DOI 10.1093/qmath/25.1.241 - S. D. Conte,
*Elementary numerical analysis: An algorithmic approach*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1965. MR**0202267** - William N. Everitt and Brian D. Sleeman (eds.),
*Ordinary and partial differential equations*, Lecture Notes in Mathematics, Vol. 564, Springer-Verlag, Berlin-New York, 1976. MR**0477213**

## Additional Information

- © Copyright 1980 American Mathematical Society
- Journal: Math. Comp.
**35**(1980), 1191-1193 - MSC: Primary 26D15; Secondary 46E30, 65J99
- DOI: https://doi.org/10.1090/S0025-5718-1980-0583496-X
- MathSciNet review: 583496