The Relative Sizes of the Terms in Chebyshev and other Ultraspherical Expansions

Lanczos (1952) claims that expansion of a function in a seriesof Chebyshev polynomials is usually superior to expansion ina series of ultraspherical polynomials for any possible non-zerovalue of the parameter . We look for conditions on the functionunder which this superiority can be proved and measured; weachieve some success on the assumption that is positive, butthe results for negative are less satisfactory.     

Spline Representation of Incompressible Flow

We describe a scheme for reconstructing a distribution of fluidvelocities from given values of the net flow across the facesof a rectangular mesh in two or three dimensions. Assuming thedata to be consistent, the reconstructed velocities will havethe properties that the continuity condition is everywhere satisfied,that the track of any fluid particle is a smooth curve and thatthere is no flow across any part of the boundary that has beenspecified as representing a solid wall.     

Vibration of a Submerged Window

The problem considered is that of the transmission of a planeacoustic wave through an aperture in an infinite plane baffle,when the aperture is covered by a thin flexible plate. The idealizedconfiguration, in which the aperture is an infinitely long slit,is examined analytically and numerically.