Vol. 84, No. 1, 1979

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Uniform and Lp approximation for generalized integral polynomials

Le Baron O. Ferguson

Vol. 84 (1979), No. 1, 53–62

Let X be a compact Hausdorff space and denote the space of all real valued continuous functions on X by C(X,R). With pointwise operations this space becomes a linear space which we norm by defining f= supxX|f(x)| whenever f C(X,R). A subset of C(X,R) is said to be point separating if for any two distinct points x and y in X there is an f in with f(x)f(y). Let Z[F] denote the ring of all polynomials in elements of which have integral coefficients. Thus an element q of Z[] is an element of C(X,R) with the special form

    ∑r1   ∑rk         j1    jk
q =     ⋅⋅⋅    aj1 ⋅⋅⋅jk fk ⋅⋅⋅fk
j1=0   jk=0

where the a’s are integers, the f’s belong to and the r’s are nonnegative integers. Such q are our integral polynomials. If X is a subset of n-dimensional Euclidean space and is taken to be the set of n coordinate projections, then the elements of Z[] are polynomials in the usual sense.

Mathematical Subject Classification 2000
Primary: 41A10
Secondary: 41A29
Received: 29 December 1978
Published: 1 September 1979
Le Baron O. Ferguson