The Classical Limit of Quantum Mechanics and the Fejer Sum of the Fourier Series Expansion of a Classical Quantity

In quantum mechanics, the expectation value of a quantity on a quantum state, provided that the state itself gives in the classical limit a motion of a particle in a definite path, in classical limit goes over to Fourier series form of the classical quantity. In contrast to this widely accepted point of view, a rigorous calculation shows that the expectation value on such a state in classical limit exactly gives the Fej\'{e}r's arithmetic mean of the partial sums of the Fourier series.