Abstract
We establish a theorem combining the estimates of Ingham and Muntz--Szasz. Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Muntz--Szasz part. A very special case of this theorem allows us to prove the simultaneous observability of some string--heat and beam--heat systems.
Highlights
Non-harmonic Fourier series proved to be very useful in control theory of partial differential equations [8], [9], [21]
For parabolic systems an powerful method is based on the Muntz–Szasz generalization [19], [24], [6] of the Weierstrass approximation theorem, see, e.g., [22]
In this paper we establish a theorem combining the estimates of Ingham and Muntz–Szasz
Summary
Non-harmonic Fourier series proved to be very useful in control theory of partial differential equations [8], [9], [21]. In the case of reversible linear evolutionary systems these methods are often based on various generalizations of a classical theorem of Ingham [11], itself a generalization of Parseval’s equality, see, e.g., [10], [14], [16] and their references. For parabolic systems an powerful method is based on the Muntz–Szasz generalization [19], [24], [6] of the Weierstrass approximation theorem, see, e.g., [22]. We allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Muntz–Szasz part. Observability, non-harmonic Fourier series, Ingham’s theorem, Muntz– Szasz theorem, wave equation, heat equation, beam equation
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