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In a previous paper, we developed lower and upper bounds for the generalized hypergeometric functions p F q , p = q , p = q + 1, and certain confluent forms under appropriate restrictions on the variable and parameters. In the present paper, we extend these notions and obtain similar inequalities for certain generalized hypergeometric functions of two variables.

In this paper we consider the problem of characterizing those situations under which the best uniform linear approximation to an arbitrary continuous function is unique. The problem has been solved by Haar where the set of approximants is a finite dimensional subspace, but in this paper we generalize this by allowing the set of approximants to be any subset of a finite dimensional space. Some previous work has been done on this problem by Rice [2, p. 87 ff.] for a number of partial results.

This paper is concerned with nonlinear optimization problems in normed linear spaces. Necessary and sufficient conditions for optimal points are given and the range of applicability of these conditions is studied. The results are applied to nonlinear approximation problems.

This paper is concerned with Chebyshev approximation by spline functions with free knots. If a zero of a Chebyshev spline function occurs at a knot, the multiplicity of the zero is suitably extended. Theorems on uniqueness on the whole approximation interval and on subintervals are stated in terms of alternation properties.

In this paper we establish the existence of a best Lp approximation, 1 ⩽ p ⩽ ∞, to a given function f∈Lp( D , where D ⊂ Rm is a bounded domain, from the family Vn(S) of all nth order exponential sums in m independent variables for which the corresponding exponential parameters lie in the closed set S ⊆ C. In so doing we extend the previously known existence theorem which corresponds to the special case where m = 1 and D is a finite interval.

The purpose of this paper is twofold. First, we prove the uniqueness of a polynomial of best uniform approximation, in a certain class P of “monotone” polynomials, to a given continuous function. This is the content of Theorem 3.1 which complements the results of Lorentz and Zeller [l]. Secondly, we prove (Theorem 6.1) that a polynomial of best L, approximation in the class 8, to a given continuous function is also unique. This is the analog of Jackson’s theorem for general polynomials. As a preliminary to Theorem 6.1, we give a necessary condition for a polynomial in B to be a polynomial of best L, approximation to an integrable function.

Birkhoff and de Boor first posed the question of the existence of a convergent bicubic spline interpolation scheme for non-rectangular domains. In this paper that query is answered affirmatively for L-shaped domains. Specifically, it is shown that ∥sf − f∥= O(hr) where sf is the bicubic spline interpolant associated with a smooth function f, h is the maximum mesh spacing, r − 4 for uniform partitions, and r = 3 for nonuniform partitions.

Let R be a region in n -space and Q a linear quadrature formula for R of the form (f)= ∑ r=1 k r f(x r ) . It is known that if Q(ƒ) = ∝ R ƒ whenever ƒ is a polynomial of degree 3 or lower, then k ⩾ n + 1. It is known that the minimum possible value of k depends on the region R , being 2 n for the n -cube and n + 2 for the n -simplex ( n > 1). In 1956 Hammer and Stroud conjectured that k ⩾ n + 2 for every R , when n > 1. In this paper we construct an R , and a Q with the required property, with k = n + 1.

This paper deals with the pointwise saturation problem for the Abel means of Fourier series. Most of the known results on pointwise approximation in connection with summation processes of Fourier series are Jackson-type theorems. (See the books of N. Achieser, P. P. Korovkin, and I. P. Natanson on approximation theory. We wish also to refer to the results due to G. Alexits and his school, cf. [I] and [5]). Here we shall study a Bernstein-type problem for the Abel means, i.e., a converse of the Jackson-type problems. But we shall study it only in case of saturated approximation. Although this type of question has been studied for several years, only a few results are available. BajSanski-BojaniC [3] proved a pointwise “0”-theorem for the Bernstein polynomials, and recently V. A. Andrienko [2] studied this problem for the FejCr means of Fourier series. A generalization of his result was given by the author [4]. On the other hand, these problems are closely connected with theorems concerning generalized derivatives of functions which have their origin in Schwas we denote

In this paper we shall show that each ƒϵ L p [0,1] (1 ⩽ p ⩽ ∞) has a best L p approximation from the set of exponential sums, V n ( S ), provided S is closed. Here V n ( S ) denotes the set of all solutions of all n -th order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomial all lie in S . We thus extend the previously known existence theorems which apply only in the special cases where S is compact or where S = R .

It is the purpose of this paper to present a method for the computation of best uniform approximation, through the replacement of the uniform norm by another norm designated by ∥ · ∥ n and also by two pseudonorms designated by ps ∥ · ∥ n and ( A ) ∥ · ∥ n . Existence, uniqueness, characterization and computation of best approximations for ∥ · ∥ n , ps ∥ · ∥ n , and ( A ) ∥ · ∥ n are examined.

for which there exists an x E C([O, 11) such that the polynomial of best approximation of degree j to x (in the sense of CebySev) is pj (j = 0, 1, . . ., n 1). What is the answer in the particular case II = 2? In the present paper we shall consider the following more general problem: Let {Gk} be a sequence of linear subspaces of a normed linear space E. Characterize those sequences (gk) in E, with g, E Gk (k = 1,2,. . .), for which there exists an x E E such that

It is shown in this paper (Theorem 1) that if α and β are real numbers such that α gb is irrational and algebraic, then there exist two (necessarily distinct) distributions S and T on R, both with compact supports, such that δ′ = ΔαS + ΔβT. Here ΔαS means S − Sα and Sα denotes the translate of S by α. It is also shown that δ′ has no such representation if α gb has rational or certain transcendental values, and that S and T can be chosen to have order two and no lower order. If ϑ belongs to any of the spaces D, E, S or their duals then ϑ ∗ S and ϑ ∗ T belong to the same space as ϑ, and ϑ′ = Δ α (ϑ ∗ S) + Δ β (ϑ ∗ T) . The formula δ′ = ΔαS + ΔβT is generalized to Rn by means of the tensor product of distributions, and it follows from this formula that there is no discontinuous translation-invariant linear form on any of the spaces D (Rn), E (Rn), S (Rn) or their duals. The same thing is also proved for E (Tn) and its dual where Tn denotes the n-dimensional torus group.

In this paper, we show the existence of nonnegative interpolation formulas for functions which are solutions of second-order uniformly elliptic equations over bounded domains.