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In this paper, we study an interesting property of L 1-approximation. For many subspaces M, there exist α ∗(M)>0 with the following property: if f vanishes off a set of measure at most α ∗(M), then the zero function is a best L 1-approximant to f from M. We explain this phenomenon, provide estimates for α ∗(M) in many cases, and present some open questions.

Let H be any complex inner product space with inner product . We say that f : C -->C is Hermitian positive definite on H if the matrix $$(f())_{r,s=1}^n \eqno(*)$$ is Hermitian positive definite for all choice of z^1,...,z^n in H, all n. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinct z^1,...,z^n in H. In this article we prove that if dim H >= 3, then f is Hermitian positive definite on H if and only if $$f(z) = \sum_{k,m =0}^\infty b_{k,m} z^k \oz^m \eqno(**)$$ where \oz is the conjugate of z, b_{k,m}>= 0, all k,m in Z_+, and the series converges for all z in C. We also prove that f of the form (**) is strictly Hermitian positive definite on any H if and only if the set $$J={(k, m) : b_{k,m}> 0}$$ is such that (0,0) is in J, and every arithmetic sequence in Z intersects the values {k-m : (k,m)\in J} an infinite number of times., Comment: 25 pages

If $$f(t) = \sum\nolimits_{k = 0}^\infty {a_k t^k } $$ converges for all $$.......CONVERSION........$$ with all coefficients $$a_k \geqslant {\text{0}}$$ , then the function $$f(\left\langle {x,\left. y \right\rangle } \right.)$$ is positive definite on H×H for any inner product space H. Set K={k: a k >0}. We show that $$f(\left\langle {{\text{x,}}\left. {\text{y}} \right\rangle } \right.)$$ is strictly positive definite if and only if K contains the index 0 plus an infinite number of even integers and an infinite number of odd integers.

We characterize the set of linear operators of the form $$Tx^n = x^n + \sum\limits_{k = 0}^{n - 1} {b_{n,k} x^k , n = 0,1, \ldots ,}$$ which satisfy Z(T p ) ≧ Z(p) for every polynomial p, where Z counts the number of zeros on all of R. We also consider the analogous question on [0,∞)].

The main result of this paper is the following. If 9 is any given polynomial of two variables, then There are many different methods of approximating multivariate functions. We have, for example, the classic methods of polynomials, Fourier series, or tensor products, and more modern methods using wavelets, radial basis functions, multivariate splines, or ridge functions. Many of these are natural generalizations of methods developed for approximating univariate functions. However, functions of many variables are fundamentally different from functions of one variable, and approximation techniques for such problems are much less developed and under- stood. We discuss in these pages one approach to this problem which is truly multivariate in character. To motivate the approach considered, we recall Hilbert's 13th problem. Although not formulated in the following terms, Hilbert's 13th problem was interpreted by some as conjecturing that not all functions of three variables could be represented as superpositions (compositions) and sums of functions of two variables. Surprisingly it transpired that all functions could be so represented, and even more was true. Kolmogorov and his student Arnold proved in a series of papers in the late 1950s that fixed continuous one-variable functions hii exist such that every continuous function f of d variables on (0, 1) d could be represented in the form

This paper demonstrates that Descartes Systems can be conveniently generated from matrix subdivision algorithms determined by totally positive matrices.

C1(K) is the space of real continuous functions onK endowed with the usualL1,-norm where\(K = \overline {\operatorname{int} K}\) is compact inRm · U is a finite-dimensional subspace ofC1,(K). The metric projection ofC1,(K) ontoU contains a continuous selection with respect toL1, -convergence if and only ifU is a unicity (Chebyshev) space forC1,(K). Furthermore, ifK is connected andU is not a unicity space forC1,(K), then there is no continuous selection with respect toL∞-convergence. An example is given of aU and a disconnectedK with no continuous selection with respect toL1-convergence, but many continuous selections with respect toL∞-convergence.

The asymptotic limit of minimum norm quadrature formulae for some Hilbert spaces of functions regular and analytic in a domainB is studied, asB expands.

Representation theorems for Tchebycheff polynomials with homogeneous boundary conditions are proved, and a number of extremal problems are solved.

LetWp(r)={f:f∈Cr−1[0, 1],f(r−1) abs.cont., ∥f(r)∥p