## Cardinal Hermite spline interpolation: convergence as the degree tends to infinity

HTML articles powered by AMS MathViewer

- by M. J. Marsden and S. D. Riemenschneider PDF
- Trans. Amer. Math. Soc.
**235**(1978), 221-244 Request permission

## Abstract:

Let ${\mathcal {S}_{2m,r}}$, denote the class of cardinal Hermite splines of degree $2m - 1$ having knots of multiplicity*r*at the integers. For $f(x) \in {C^{r - 1}}(R)$, the cardinal Hermite spline interpolant to $f(x)$ is the unique element of ${\mathcal {S}_{2m,r}}$ which interpolates $f(x)$ and its first $r - 1$ derivatives at the integers. For $y = ({y^0}, \ldots ,{y^{r - 1}})$ an

*r*-tuple of doubly-infinite sequences, the cardinal Hermite spline interpolant to

*y*is the unique $S(x) \in {\mathcal {S}_{2m,r}}$ satisfying ${S^{(s)}}(\nu ) = {y^s},s = 0,1, \ldots ,r - 1$, and $\nu$ an integer. The following results are proved: If $f(x)$ is a function of exponential type less than $r\pi$, then the derivatives of the cardinal Hermite spline interpolants to $f(x)$ converge uniformly to the respective derivatives of $f(x)$ as $m \to \infty$. For functions from more general, but related, classes, weaker results hold. If

*y*is an

*r*-tuple of ${l^p}$ sequences, then the cardinal Hermite spline interpolants to

*y*converge to ${W_r}(y)$, a certain generalization of the Whittaker cardinal series which lies in the Sobolev space ${W^{p,r - 1}}(R)$. This convergence is in the Sobolev norm. The class of all such ${W_r}(y)$ is characterized. For small values of

*r*, the explicit forms of ${W_r}(y)$ are described.

## References

- Richard R. Goldberg,
*Restrictions of Fourier transforms and extension of Fourier sequences*, J. Approximation Theory**3**(1970), 149–155. MR**261269**, DOI 10.1016/0021-9045(70)90023-7 - S. L. Lee,
*Exponential Hermite-Euler splines*, J. Approximation Theory**18**(1976), no. 3, 205–212. MR**435665**, DOI 10.1016/0021-9045(76)90015-0 - S. L. Lee,
*Fourier transforms of $B$-splines and fundamental splines for cardinal Hermite interpolations*, Proc. Amer. Math. Soc.**57**(1976), no. 2, 291–296. MR**420074**, DOI 10.1090/S0002-9939-1976-0420074-8 - Peter R. Lipow and I. J. Schoenberg,
*Cardinal interpolation and spline functions. III. Cardinal Hermite interpolation*, Linear Algebra Appl.**6**(1973), 273–304. MR**477565**, DOI 10.1016/0024-3795(73)90029-3 - M. Marsden and R. Mureika,
*Cardinal spline interpolation in $L_{2}$*, Illinois J. Math.**19**(1975), 145–147. MR**377357** - M. J. Marsden, F. B. Richards, and S. D. Riemenschneider,
*Cardinal spline interpolation operators on $l^{p}$ data*, Indiana Univ. Math. J.**24**(1974/75), 677–689. MR**382925**, DOI 10.1512/iumj.1975.24.24052 - M. J. Marsden, F. B. Richards, and S. D. Riemenschneider,
*Erratum: “Cardinal spline interpolation operators on $l^p$ data” (Indiana Univ. Math. J. 24 (1974/75), 677–689)*, Indiana Univ. Math. J.**25**(1976), no. 9, 919. MR**410176**, DOI 10.1512/iumj.1976.25.25072 - M. J. Marsden and S. D. Riemenschneider,
*Convergence results for cardinal Hermite splines*, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 457–462. MR**0435667**
D. J. Newman, - Franklin Richards,
*Uniform spline interpolation operators in $L_{2}$*, Illinois J. Math.**18**(1974), 516–521. MR**358155** - Franklin Richards,
*The Lebesgue constants for cardinal spline interpolation*, J. Approximation Theory**14**(1975), no. 2, 83–92. MR**385391**, DOI 10.1016/0021-9045(75)90080-5 - F. B. Richards and I. J. Schoenberg,
*Notes on spline functions. IV. A cardinal spline analogue of the theorem of the brothers Markov*, Israel J. Math.**16**(1973), 94–102. MR**425439**, DOI 10.1007/BF02761974 - S. D. Riemenschneider,
*Convergence of interpolating cardinal splines: power growth*, Israel J. Math.**23**(1976), no. 3-4, 339–346. MR**420076**, DOI 10.1007/BF02761812 - I. J. Schoenberg,
*Notes on spline functions. I. The limits of the interpolating periodic spline functions as their degree tends to infinity*, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math.**34**(1972), 412–422. MR**0316944** - I. J. Schoenberg,
*Cardinal spline interpolation*, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR**0420078** - I. J. Schoenberg,
*Notes on spline functions. III. On the convergence of the interpolating cardinal splines as their degree tends to infinity*, Israel J. Math.**16**(1973), 87–93. MR**425438**, DOI 10.1007/BF02761973 - I. J. Schoenberg,
*Cardinal interpolation and spline functions. VI. Semi-cardinal interpolation and quadrature formulae*, J. Analyse Math.**27**(1974), 159–204. MR**493057**, DOI 10.1007/BF02788646 - I. J. Schoenberg,
*On the remainders and the convergence of cardinal spline interpolation for almost periodic functions*, Studies in spline functions and approximation theory, Academic Press, New York, 1976, pp. 277–303. MR**0481739** - I. J. Schoenberg,
*On spline interpolation at all integer points of the real axis*, Mathematica (Cluj)**10(33)**(1968), 151–170. MR**237997** - François Trèves,
*Topological vector spaces, distributions and kernels*, Academic Press, New York-London, 1967. MR**0225131**
J. M. Whittaker, - A. H. Zemanian,
*Distribution theory and transform analysis. An introduction to generalized functions, with applications*, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR**0177293** - A. Zygmund,
*Trigonometric series: Vols. I, II*, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR**0236587**

*Some remarks on the convergence of cardinal splines*, MRC Report #1474, Math. Research Center, Univ. of Wisconsin, Madison, 1974.

*Interpolatory function theory*, Cambridge Univ. Press, London, 1935.

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**235**(1978), 221-244 - MSC: Primary 41A05
- DOI: https://doi.org/10.1090/S0002-9947-1978-0463752-3
- MathSciNet review: 0463752