Approximation by Dirichlet series with nonnegative coefficients (Q5959032)

\(\forall \{\alpha_k\}\subset 0,+\infty)\) such that \(0=\alpha_0 <\alpha_k \uparrow+ \infty\) and \(\forall \{a_k\} \subset[0, +\infty)\), NEWLINE\[NEWLINEf_n(t)= \sum^n_{k=0} a_k\exp (-\alpha_kt)\tag{1}NEWLINE\]NEWLINE is a Dirichlet polynomial with positive coefficients. The author proves: 1) A function \(f(t)\in C[0,+\infty)\) can be approximated arbitrarily close by Dirichlet polynomials (1) in the norm NEWLINE\[NEWLINE\sup \biggl\{\bigl|f(t)-f_n(t) \bigr|: 0\leq t<+\infty \biggr\} \tag{2}NEWLINE\]NEWLINE if and only if \(f(t)\) is completely monotonic in \([0,+\infty)\). 2) If a function NEWLINE\[NEWLINEf(t)\in L_p(0,+ \infty)\cap C[0,+\infty)NEWLINE\]NEWLINE \((1\leq p<+\infty)\) can be approximated arbitrarily close by Dirichlet polynomials (1) in the norm NEWLINE\[NEWLINE\left[ \int_0^{+\infty} \bigl|f(t)-f_n(t) \bigr|^p dt\right]^{1/p}, \tag{3}NEWLINE\]NEWLINE then \(f(t)\) is completely monotonic in \([0,+\infty)\). 3) If \(\{\alpha_k \}\) is fixed, \(\{a_k\}\subset \mathbb{R}\) and NEWLINE\[NEWLINE\sum^{+\infty}_{k=1} (1/\alpha_k) <+\infty \quad\text{or}\quad \beta>11 \left(1+ \sum^{+\infty}_{k=1} \alpha_k \right)\left( \sum^{+\infty}_{k=1} \alpha_k<+ \infty\right),NEWLINE\]NEWLINE then the completely monotonic function \(e^{-\beta t}(0<\beta \overline\in \{\alpha_k\})\) cannot be approximated arbitrarily close in the norm (2) by Dirichlet polynomials of the form (1) with real coefficients. The definition of a completely monotonic function can be found in \textit{D. V. Widder}, The Laplace transform (1941; Zbl 0063.08245)]. Some of the above results can be applied to rheology and to the molecular theory.