A note on optimal Hermite interpolation in Sobolev spaces

This paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] , $n\in \mathbb{N}$ n ∈ N in space $L_{\infty }[a,b]$ L ∞ [ a , b ] and weighted spaces $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p< \infty $ 1 ≤ p < ∞ with ω a continuous-integrable weight function in $(a,b)$ ( a , b ) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in $L_{\infty }$ L ∞ (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ) are optimal for $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] in $L_{\infty }[a,b]$ L ∞ [ a , b ] (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

Encoding Post-Spectral Sound

This article examines computer-based music (ca. 1982–87) created by Finnish composer Kaija Saariaho at the Institut de Recherche et Coordination Acoustique/Musique (IRCAM) in Paris. A detailed account of archival materials for an early étude in voice synthesis, Vers le blanc (1982), demonstrates the music-theoretical import of software to Saariaho’s development of a robust compositional method that resonated with the emergent aesthetics of a post-spectral milieu. Subsequent analyses of two additional works from this period—Jardin secret II (1984–86) for harpsichord and tape, and IO (1987) for large ensemble and electronics—serve to illustrate Saariaho’s extension of this method into instrumental settings. Specific techniques highlighted include the use of interpolation systems to create continuous processes of transformation, the organization of individual musical parameters into multidimensional formal networks, and the exploration of harmonic structures based on the analysis of timbral phenomena. Relating these techniques to the affordances of contemporaneous IRCAM technologies, including CHANT, FORMES, and Saariaho’s own customized program, “transkaija,” this article adopts a transductive approach to archival research that is responsive to the diverse media artifacts associated with computer-based composition.