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Fourier expansions of entire functions. (English) Zbl 0938.30020

Let \(E\subset\mathbb C\) be a compact Jordan region with positive transfinite diameter, let \(\omega\) be a positive continuous function on \(E\), and let \(H(E)\) be the Hilbert space of functions holomorphic in int\(E\) with the scalar product \((g_1,g_2):=\int_{E}g_1\overline g_2\omega dx dy\). Fix a complete orthogonal system \((p_{n})_{n=0}^\infty\subset H(E)\) consisting of polynomials with \(\deg p_{n}\leq n\). Let \(f\) be an entire function. Define the Fourier coefficients \(C_{n}:=\int_{E}f\overline p_{n}\omega dx dy\), \(n=0,1,2,\dots\). Let \(M(r):=\sup\{|f(z)|: |z|=r\}\) and let \(\log^{[t]}\) be the \(t\)-th iterate of \(\log\). The authors characterize the \((p,q)\)–order \(\rho:=\limsup_{r\to+\infty}\frac{\log^{[p]}M(r)}{\log^{[q]}r}\) and the generalized \((p,q)\)–type \(T^\ast:=\limsup_{r\to+\infty} \frac{\log^{[p-1]}M(r)}{(\log^{[q-1]}r)^\rho}\) in terms of \((C_{n})_{n=0}^\infty\).

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
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