### APPROXIMATION BY LOCAL PARABOLIC SPLINES CONSTRUCTED ON THE BASIS OF INTERPOLATION IN THE MEAN

#### Abstract

The paper deals with approximative and form-retaining properties of the local parabolic splines of the form \(S(x)=\sum\limits_j y_j B_2 (x-jh),\) \( (h>0),\) where \(B_2\) is a normalized parabolic spline with the uniform nodes and functionals \(y_j=y_j(f)\) are given for an arbitrary function \(f\) defined on \(\mathbb{R}\) by means of the equalities $$y_j=\frac{1}{h_1}\int\limits_{\frac{-h_1}{2}}^{\frac{h_1}{2}}

f(jh+t)dt \quad (j\in\mathbb{Z}). $$ On the class \(W^2_\infty\) of functions under \(0<h_1\leq 2h\), the approximation error value is calculated exactly for the case of approximation by such splines in the uniform metrics.

#### Keywords

#### Full Text:

PDF#### References

Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L. Spline-functions methods. Moscow: Nauka, 1980. 355 p. [in Russian]

Piegl L., Tiller W. The NURBS Book. New York: Springer, 1997. 646 p.

Zavyalov Yu.S. On formulas of local approximation exact on the cubic splines // Comp. systems, 1998. Vol. 128. P. 75–88. [in Russian]

Korneychuk N.P. Splines in the approximation theory. Moscow: Nauka, 1984. 352 p. [in Russian]

Subbotin Yu.N. Heritance of monotonisity and convexity properties under local approximation // J. Comp. Math. and Math. Physics, 1993. Vol. 37, no. 7. P. 996–1003. [in Russian]

Subbotin Yu.N. Extremal problems of functional interpolation and interpolation of splines in the mean // Trudy Steklov Math. Institute of RAS, 1975. Vol. 109. P. 35–60. [in Russian]

Subbotin Yu.N. Extremal functional interpolation in the mean with the minimal value of the n-th derivative on large intervals of meaning // Math. zametki, 1996. Vol. 59, no. 1. P. 114–132. [in Russian]

Subbotin Yu.N. Extremal \(L_p\)-interpolation in the mean on intersecting intervals of meaning // Izv. RAS Ser. Math., 1997. Vol. 61, no. 1. P. 177–198. [in Russian] DOI: 10.4213/im110

Shevaldin V.T. Some problems of extremal interpolation in the mean for linear differential operators // Trudy Steklov Math. Institute of RAS, 1983. Vol. 164. P. 203–240. [in Russian]

Shevaldin V.T. Extremal interpolation in the mean on intersecting intervals of meaning and L-splines // Izv. RAS Ser. Math., 1998. Vol. 62, no. 4. P. 201–224. [in Russian] DOI: 10.4213/im193

#### Article Metrics

### Refbacks

- There are currently no refbacks.