Degree of polynomial approximation of entire functions

[1/0%]
Show
LaTeX
Problem 4.208 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 6 "Supplementary Problems"

We consider the approximation of the entire function g(z)g(z) by polynomials. We define the integer

k=k(r,α) k = k(r,\alpha )

(termed the degree of approximation of the function g(z)g(z) on the circle ∣z∣=r\left|z\right|=r to accuracy α\alpha) as the degree of the polynomial P(z)P(z) of the lowest possible degree for which

∫02π∣g(reiφ)−P(reiφ)∣2 dφ<α−2∫02π∣g(reiφ)∣2 dφ; \int _{0}^{2\pi } \left|g(re^{i\varphi })-P(re^{i\varphi })\right|^{2} \, d\varphi < \alpha ^{-2} \int _{0}^{2\pi } \left|g(re^{i\varphi })\right|^{2} \, d\varphi ;

(r>0r>0, α>1\alpha >1). Then

lim⁡α→∞klog⁡klog⁡α=lim⁡r→∞log⁡klog⁡r=ρ, \lim _{\alpha \to \infty } \frac{k\log k}{\log \alpha } = \lim _{r\to \infty } \frac{\log k}{\log r} = \rho ,

where we keep rr fixed when computing the first limit and α\alpha fixed when computing the second limit; ρ\rho is the order of g(z)g(z).

?
Remark.
?

Cf. Problem 123 of Part III. See problem 51 for the definition of order. G. Polya, problem: Jber. deutsch. Math. Verein., Section 2, 34, pp. 97-98 (1925). Solution by G. Szego, op. cit. 35, pp. 86-89 (1926).

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.