Eigenfunction of all shift operators

  • Thread starter montyness
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montyness
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Prove that if a continuous function [itex] e\left( x \right) [/itex] on [itex]\mathbb{R}[/itex] is eigenfunction of all shift operators, i.e. [itex] e\left( x+t \right) = \lambda_t e\left( x \right) [/itex] for all x and t and some constants [itex] \lambda_t [/itex], then it is an exponential function, i.e. [itex] e\left( x \right)= Ce^{ax}[/itex] for some constants C and a.

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Ray Vickson
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Prove that if a continuous function [itex] e\left( x \right) [/itex] on [itex]\mathbb{R}[/itex] is eigenfunction of all shift operators, i.e. [itex] e\left( x+t \right) = \lambda_t e\left( x \right) [/itex] for all x and t and some constants [itex] \lambda_t [/itex], then it is an exponential function, i.e. [itex] e\left( x \right)= Ce^{ax}[/itex] for some constants C and a.

Thanks in Advance.

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