Integrals of High-Pass Functions

A function $h(x)$ whose Fourier transform vanishes over the interval ( - Ω ,Ω ) is called a high-pass function. This paper develops topics in the mathematical theory of high-pass functions. The definition of high-pass functions is extended to functions $h(x)$ which do not have Fourier transforms by requiring that (i) ∫ₓx + 1 |h(t)|dt is uniformly bounded for - ∞ < x < ∞, (ii) ∫- ∞^∞ h(x)g(x)dx = 0, for all $g(x)$ in L₁ of the form g(x) = ∫- Ω^Ω G(ω )eiω x dω. For such $g(x)$, the condition (i) insures that the integral (ii) is absolutely convergent. Within this extended definition, it is shown that a high-pass function $h(x)$ has a bounded unbiased integral h( - 1) (x), which is also a high-pass function, for which h( - 1) (x) - h( - 1) (y) = ∫yˣ h(t) dt. It is given by convolution of $h(x)$ with any L₁ function $K(x)$ of the form (iii) K(x) = 1/2sgn x - g(x), where $g(x)$ is the restriction to the real line of an entire function of exponential type Ω, or, what is shown to be equivalent, by convolution of $h(x)$ with any function in L₁ of the form (iv) K(x) = (1 / 2π )∫- ∞^∞ K (ω )eiω x dω where K(ω ) = 1 / iω for |ω | ≥ Ω. From the convolution representation of h( - 1) (x), the following results are obtained. If $h(x)$ is a high-pass function: (v) The Fourier integral of $h(x)$ is summable $(C,1)$ to zero in the spectral gap ( - Ω ,Ω ). (vi) If lim x → ∞ ∫ₓx + 1 |h(t)|dt = 0, then lim x → ∞ ∫₀ˣ h(t)dt exists. (vu) |h( - 1) (x)| ≤ (1/2 + 1 / 2π ) ₜ ∫₀^π / Ω |h(t + u)|du.