By Harald Bohr
Influenced by way of questions about which capabilities may be represented by way of Dirichlet sequence, Harald Bohr based the idea of virtually periodic features within the Twenties. this gorgeous exposition starts with a dialogue of periodic services sooner than addressing the just about periodic case. An appendix discusses virtually periodic features of a posh variable. it is a attractive exposition of the idea of just about Periodic features written through the writer of that concept; translated by way of H. Cohn.
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Extra info for Almost Periodic Functions (Ams Chelsea Publishing)
The (periodic) convolution of two L−periodic functions is defined as (f1 ∗ f2 ) (x) = 1 L L f1 (ξ )f2 (x − ξ )dξ. 0 Expanding both functions in their Fourier series, one observes that (f1 ∗ f2 ) (x) = = 1 L 1 L L 0 +∞ k=−∞ +∞ k,n=−∞ +∞ fˆ1 (k)e2πikξ/L fˆ2 (n)e2πin(x−ξ )/L dξ n=−∞ L fˆ1 (k)fˆ2 (n)e2πinx/L 0 e2πi(k−n)ξ/L dξ. 6) 18 ENGINEERING APPLICATIONS OF NONCOMMUTATIVE HARMONIC ANALYSIS The integral in the above expression has value Lδk,n , and so the following simplification results: +∞ fˆ1 (k)fˆ2 (k)e2πikx/L .
Or, simply taking the derivative of the Fourier series, df = dx one finds that df = dx +∞ k=−∞ +∞ d fˆ(k) e2πikx/L dx fˆ(k) · 2π ik/L e2πikx/L , k=−∞ and so the Fourier coefficients of the derivative of f (x) are Fk df dx = fˆ(k) · 2π ik/L. If f (x) can be differentiated multiple times, the corresponding Fourier coefficients are multiplied by one copy of 2πik/L for each derivative. Note that since each differentiation results in an additional power of k in the Fourier coefficients, the tendency is for these Fourier coefficients to not converge as quickly as those for the original function, and they may not even converge at all.
Substitution of this solution into the wave equation yields the Helmholtz equation ∇ 2 + k2 U = 0 where k = ω/c = 2π/λ is called the wave number, and λ is the wave length of the light. A particular solution to the Helmholtz equation for the case of a spherical point source at x = 0 is given by eik|x| U (x) = . |x| We now may consider what happens to light from this point source as it passes through an opening, or aperture, in an otherwise opaque screen, as illustrated in Fig. 1. According to the HuygensFresnel principle, the observable light on the other side of the screen is the same as if a continuum of spherical sources had been placed at the opening.