Direct theorems for the approximation of functions, regular in convex polygons, by exponential polynomials in the integral metric (Q1119768)

Let M be a closed convex polygon with the vertices \(\gamma_ 1,...,\gamma_ N\), \(N\geq 3\), let \(C=\partial M\), let \(E^{(p)}(M)\), \(p\geq 1\), be the Smirnov space in M with the norm \[ \| f\| =(\int_{C}| f(z)|^ p dz)^{1/p}, \] and let \(L(\lambda)=\sum_{1\leq k\leq N}d_ k \exp \gamma_ k\lambda\). Given a function \(f\in E^{(p)}(M)\), the author constructs, in the well-known way, the sequence of quasi-polynomials \(S_ n(f)=\sum^{n}_{k=1}c_ k \exp \lambda_ kz\), \(L(\lambda_ k)=0\), and proves several upper estimates for \(\| f-S_ n(f)\|\) in terms of an integral continuity module.