AN IMPROVED CONSTANT FOR THE MUNTZ--JACKSON THEOREM

H. N. Odogwu

Abstract: We improve a Newman result [2,3] from 1974 concerning approximation of a continuous function by generalized polynomials. He proved that every $f\in C[0,1]$ there exists a generalized polynomial $P(x)=\sum_{k=0}^N c_kx^{\lambda k}$ such that $$ |f(x)-P(x)| \leq Aw_f(\varepsilon),\qquad x\in [0,1]\tag 1 $$ holds. Here $0=\lambda_0<\lambda_1<\cdots\lambda_N$ are given numbers $w_f$ is the modulus of continuity of $f$, $\varepsilon=\max\{|B(z)/z|: Re\, z=1\}$, $B(z)$ is the Blaschke product corresponding to the above set of $\lambda_k$'s and $A$ is a constant. Newman [2] proved that (1) holds with $A=368$. We show that (1) is valid with $A=66$. We prove this by slightly modifying Newman's proof and choosing the size of an interval, to which a suitable contradiction is extended, optimally.

Classification (MSC2000): 41A30, 41A25

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