Fractals and generalized inner products

The generation of fractals by a generalization of the matrix product is described. These matrix products allow visualization of additional structure of the fractals as well as producing fractals of new types.     

Middle points and inner products

Let X be a real normed space with unit sphere S. It is provedthat X is an inner product space if and only if there is a realnumber , 0 < < 1, and , such that every chord of S that supports S touches S at itsmiddle point.     

Generalized Frobenius inner products

Mathematische Annalen -     

Middle points, medians and inner products

Let be a real normed space with unit sphere . Gurari and Sozonov proved that is an inner product space if and only if, for any , . We prove that it suffices to consider points such that .

Making use of the above result we also prove that if , is smooth, and 0 is a Fermat-Torricelli median of any three points such that , then is an inner product space.

     

Regularized inner products of modular functions

In this note, we give an explicit basis for the harmonic weak forms of weight two. We also show that their holomorphic coefficients can be given in terms of regularized inner products of weight zero weakly holomorphic forms.     

Computation of inner products of multivariate B-splines

In this paper we derive recurrence relations for inner products of multivariate B-splines of arbitrary spatial dimension and arbitrary knot sets.     

Shift-invariant bi-inner product functionals are inner products

Bi-inner product functionals generated by a pair of Bessel sequences of L² functions are introduced. It is shown that these functionals are constant multiples of the inner products of L² and l², if and only if they are shift-invariant both in space (or time) and in phase. This result is then applied to characterize dual frames and bi-orthogonal Riesz bases of L².     

Shift-invariant bi-inner product functionals are inner products

Bi-inner product functionals generated by a pair of Bessel sequences of L² functions are introduced. It is shown that these functionals are constant multiples of the inner products of L² and l², if and only if they are shift-invariant both in space (or time) and in phase. This result is then applied to characterize dual frames and bi-orthogonal Riesz bases of L². The research of this author was supported by NSF Grant # DMS-95-05460 and ARO Contract #DAAH-04-95-10193, This author is currently visiting the Department of statistics, Stanford University, Stanford, CA 94305. The research of this author was supported by the Texas Coordinating Board of Higher Education under Grant # 999903-109.     

Sobolev inner products and orthogonal polynomials of Sobolev type

This survey paper deals with polynomials which are orthogonal with respect to scalar products of the form _R F ^T[A]G withF ^T=[f(x), f(Ⅎ(x),...f ^(y)(x)], [A] A _ji =A _ji =A _ij =d _ji (I _ji ) where d _ji is a measure of supportI _ij and [A] is positive semi-definite. Basic properties are indicated or proved in particular cases.     

A note on semibounded pairs of inner products

Using the convexity of the 2-dirnensionai field of values we reprove the well-known result that an inner product that is semibounded on the neutral set of an indefinite inner product of arbitrary dimension is so on the whole space. With this field of values approach we can also remedy some errors in the literature on indefinite inner product spaces.     

An orthogonality type based on invariant inner products

Aequationes mathematicae - A new orthogonality type in normed linear spaces, which is based on invariant inner products, is introduced. It is shown that this orthogonality has properties of...