Fourier Transform and Sampling Theory

Evaluate \[ I=2\int _{-\infty }^{\infty }\frac {x\sin x}{x^2+4}\,dx. \] You must indicate the contour of integration and treat integration over each contour.

Evaluate the integrals \[ \begin {array}{rcll} \text {(a)}&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx &&\text {(\(a\) is a real parameter)},\\[2.2ex] \text {(b)}&\displaystyle \int _0^\infty \frac {x^\alpha }{x+1}\,dx &&\text {(\(\alpha \) is a real parameter, \(-1<\alpha <0\))}. \end {array} \]

Consider the global Cauchy problem \[ u_y=Du_{xx}+bu_x+cu,\quad x\in \mathbb {R},\ t>0;\qquad u(0,x)=g(x), \] where \(b\), \(c\) \(D\) are constants such that \(D>0\).

1. Convert to the standard heat equation by setting \(v(t,x)=u(t,x)e^{hx+kt}\) for appropriately chosen \(h\) and \(k\).
2. Solve the heat equation for \(v\) using a transform method.
3. Hence show that if \(c<0\) and \(g\) is bounded, then \(y(t,x)\to 0\) as \(t\to \infty \).

Calculate the following integrals (\(a\) and \(b\) are positive parameters) \[ \begin {array}{rcl} \text {(a)}\quad I(a)&=&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx,\\[2.2ex] \text {(b)}\quad I(a,b)&=&\displaystyle \int _{-\infty }^{\infty } \frac {\cos ax-\cos bx}{x^2}\,dx. \end {array} \]

\(F(\omega )\) is the Fourier transform of \(f(x)\) (which is continuous and not identical zero), \(x\) and \(\omega \) are real variables. Explain why it is impossible that both functions \(f(x)\) and \(F(\omega )\) have finite support.

Integrate

1. \[ \int _0^\infty \frac {\sin \alpha x}{x}\,dx, \]
2. \[ \int _0^\infty \frac {x^\alpha }{1+x}\,dx, \]
3. \[ \int _0^\infty \sin x^2\,dx \]

[Explain the logic (in particular, the choice of contour), but do not worry about getting the exact numbers; \(\alpha \) is a real parameter.]

Evaluate the integrals

1. \[ \int _{-\infty }^{\infty }\frac {\sin x}{x}\,dx, \]
2. \[ \int _0^\infty \frac {x^\alpha }{x+1}\,dx. \]

Show that Fourier transform \[ f(x)\to F(\mu )=\int _{-\infty }^{\infty } f(x)e^{i\mu x}\,dx \] preserves the inner product and the \(L_2\) norm (Plancherel’s equations): \[ \int _{-\infty }^{\infty }\bar {f}(x)g(x)\,dx = \int _{-\infty }^{\infty }\bar {F}(\mu )G(\mu )\,\frac {d\mu }{2\pi }, \] \[ \text {and}\qquad \int _{-\infty }^{\infty }\lvert f(x)\rvert ^2\,dx = \int _{-\infty }^{\infty }\lvert F(\mu )\rvert ^2\,\frac {d\mu }{2\pi }. \] (\(\overline {\phantom {x}}\) denotes complex conjugation).

Integrate \[ \int _{-\infty }^{\infty }\frac {\sin \alpha x}{\sinh \pi x}\,dx. \]

1. Calculate the Fourier transform of a Gaussian \(f(x)=e^{-x^2}\) (\(x\in \mathbb {R}\)).
2. Is it possible that a function \(f(x)\) (\(x\in \mathbb {R}\)) and its Fourier transform \(F(\mu )\) (\(\mu \in \mathbb {R}\)) both have finite support?

Evaluate

1. \[ \int _{-\infty }^{\infty } e^{ix^2}\,dx \]
2. \[ \int _0^\infty \frac {\sin x}{x}\,dx. \]

Consider the global Cauchy problem \[ u_t=Du_{xx}+bu_x+cu,\qquad x\in \mathbb {R},\ t>0,\qquad u(x,0)=g(x), \] where \(D>0\), \(b\) and \(c\) are constant coefficients.

1. Convert to a diffusion equation by setting \(v(x,t)=u(x,t)\exp (hx+kt)\) for appropriately chosen \(h,k\).
2. Solve the diffusion equation for \(v\) using a transform method.
3. Hence, show that if \(c<0\) and \(g\) is bounded, then \(u(x,t)\to 0\) as \(t\to \infty \).

Evaluate the integrals:

1. \[ I=\int _0^\infty \frac {\sin x}{x}\,dx, \]
2. \[ I=\int _0^\infty \frac {x^\alpha }{2+x}\,dx \qquad (-1<\alpha <0). \]

1. Calculate the Fourier transform of a Gaussian \(f(x)=e^{-x^2}\).
2. Formulate and prove Heisenberg’s uncertainty principle.

Evaluate the integrals:

1. \[ I=\int _0^\infty \frac {\sin x}{x}\,dx, \]
2. \[ I=\int _0^\infty \frac {x^\alpha }{2+x}\,dx \qquad (-1<\alpha <1). \]

The operator \(Lu=-u''\) on \(L^2(-\infty ,\infty )\) has a Green’s function defined by \(-G''-\lambda G=\delta (x-\xi )\) given by \[ G(x,\xi ;\lambda )=-\frac {e^{i\sqrt {\lambda }|x-\xi |}}{2i\sqrt {\lambda }}. \] Assume the representation \[ -\frac {1}{2\pi i}\int _{C_\infty }G(x,\xi ;\lambda )\,d\lambda =\delta (x-\xi ) \] and show how it leads to the Fourier Transform Theorem that \[ f(x)=\frac {1}{2\pi }\int _{-\infty }^{\infty }e^{-i\mu x}F(\mu )\,d\mu , \qquad F(\mu )=\int _{-\infty }^{\infty }e^{i\mu x}f(x)\,dx. \]

Use Fourier transforms to solve the integral equation \[ \int _{-\infty }^{\infty } k(x-y)u(y)\,dy-u(x)=f(x) \] with \(k(x)=H(x)\), the Heaviside function.

Use the \(z\)-transform to solve the system of equations \[ \frac {du_n}{dt}=\frac {1}{h^2}\left (u_{n+1}-2u_n+u_{n-1}\right ), \qquad -\infty <n<\infty \] with \(u_n(t=0)=\sin \frac {2\pi n}{k}\), with \(k\) an integer.

The windowed Fourier transform of a function \(f(x)\in L^2(-\infty ,\infty )\) is defined by \[ Gf(\omega ,\mu )=\int _{-\infty }^{\infty }g(x-\mu )f(x)e^{-\omega x}\,d\mu \,dx. \] Suppose \(g(x)\) is a real valued function. State and verify the formula for the reconstruction of \(f(x)\) from \(Gf\), including any additional conditions on \(g(x)\). You may assume the validity of the Fourier transform.

Evaluate the integral \[ \int _{-\infty }^{\infty }\frac {\sin \alpha x}{x}\,dx \] (\(\alpha \) is a constant parameter). Explain your steps.

Evaluate the integral \[ \int _0^\infty \frac {\sin x}{x}\,dx. \] Explain your reasoning.

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2}, \qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=f(t). \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\widehat {F}(s)\sinh sx}{\sinh sl} \] where \(\widehat {F}(s)\) is the Laplace transform of \(f(t)\).
2. Suppose that \(\phi _0(x,t)\) is the solution when \(f(t)=1\). Show that the solution for general \(f\) is given by \[ \phi (x,t)=\int _0^t \frac {\partial \phi _0}{\partial t'}(x,t')f(t-t')\,dt'. \]

1. Introducing the discrete Fourier transform \(U(k,t)=\sum _{n=-\infty }^{\infty }e^{ikn}u_n(t)\), \(0\le k<2\pi \), show that the general solution to the discretized heat equation \[ \frac {du_n}{dt}=\frac {1}{h^2}\{u_{n+1}-2u_n+u_{n-1}\},\qquad -\infty <n<\infty , \] can be written in the form \[ u_n(t)=\int _0^{2\pi } U(k,0)e^{-\epsilon (k)t}e^{-ikn}\,\frac {dk}{2\pi } \] and calculate the function \(\epsilon (k)\).
2. Using the fact that \(U(k,t)\) and \(u_n(t)\) form a transform pair for fixed \(t\), prove the identity \[ \sum _{n=-\infty }^{\infty }e^{i(k-k')n}=2\pi \delta (k-k'). \] Hence, find \(u_n(t)\) given the initial data \(u_n(0)=\sin (2\pi n/p)\) for some fixed integer \(p\).

Use Fourier transforms to show that the solution of the diffusion equation \[ \frac {\partial \phi }{\partial t}=\frac {\partial ^2\phi }{\partial x^2},\qquad -\infty <x<\infty \] with initial data \(\phi (x,0)=h(x)\) takes the form \[ \phi (x,t)=\int _{-\infty }^{\infty } g(x-x',t)h(x')\,dx' \] where \[ g(x,t)=\frac {e^{-x^2/4Dt}}{2\sqrt {\pi Dt}}. \] Determine \(\lim _{t\to 0}g(x,t)\).

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2},\qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=1. \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\sinh sx}{s\sinh sl}. \]
2. Obtain the solution \(\phi (x,t)\) by inverting the Laplace transform using a Bromich integral and show that it can be expressed as the infinite series \[ \phi (x,t)=\frac {x}{l}+\sum _{n=1}^{\infty }\frac {2(-1)^n}{n\pi } \sin \left (\frac {n\pi x}{l}\right )\cos \left (\frac {n\pi t}{l}\right ). \]

Use Fourier transforms to show that the solution of the diffusion equation \[ \frac {\partial \phi }{\partial t}=\frac {\partial ^2\phi }{\partial x^2},\qquad -\infty <x<\infty \] with initial data \(\phi (x,0)=h(x)\) takes the form \[ \phi (x,t)=\int _{-\infty }^{\infty } g(x-x',t)h(x')\,dx' \] where \[ g(x,t)=\frac {e^{-x^2/4Dt}}{2\sqrt {\pi Dt}}. \] Determine \(\lim _{t\to 0}g(x,t)\).

Remark: For this problem you may use without proof that \[ \frac {1}{2\pi }\int _{-\infty }^{\infty }e^{i\mu x}\,d\mu =\delta (x). \]

1. Suppose \(f(x)\) has the Fourier transform \(F(\mu )\). Evaluate \[ \int _{-\infty }^{\infty } f(x)\overline {f(x-k)}\,dx \] in terms of \(F(\mu )\).
2. Use this identity to prove that the set \(\{\operatorname {sinc}(t-k)\}_{k=-\infty }^{\infty }\) forms an orthonormal set.

Suppose the function \(\phi (x)\) is Lipschitz continuous on the unit interval \(0<x<2\pi \) and that \(\lim _{x\to 0}\phi (x)=\phi (0^+)\) exists. Suppose \(\int _0^\infty \frac {\sin x}{x}\,dx=\alpha \) is known. Evaluate \(\lim _{k\to \infty }\int _0^{2\pi }\phi (x)\frac {\sin kx}{x}\,dx\) (with \(k\) restricted to be integer-valued).

Evaluate \[ \int _{-\infty }^{\infty }\frac {\sin x}{x}\,dx. \]

1. Find the Fourier transform of \(\dfrac {\sin x}{x}\).
2. Suppose that the Fourier transform of the function \(f(t)\) is 0 for frequencies whose absolute value is greater than \(\pi /h\). Find the representation of \(f(t)\) in terms of sinc functions.

Evaluate the integral \[ I=\int _0^\infty \frac {\sin x}{x}\,dx. \]

Prove that the Hermite polynomials form a complete set on \(-\infty <x<\infty \) with the inner product \(\langle u,v\rangle =\int _{-\infty }^{\infty }u(x)v(x)e^{-x^2/2}\,dx\).

Hint: The following steps will prove useful:

1. Show that if the Hermite-Bessel coefficients of \(f(x)\) are all zero, then \(\int _{-\infty }^{\infty }e^{-x^2/2}x^nf(x)\,dx=0\) for all \(n\).
2. Show that the Fourier transform \(F(z)\) of the function \(f(x)e^{-x^2/2}\) has no singularities in the complex plane and is therefore an entire function.
3. Calculate the Taylor series of \(F(z)\) and show that it is zero. What conclusion can you draw about \(f(x)\) and what does this say about Hermite polynomials?

The (generalized) Hilbert transform \(H\) of a periodic function \(f\) can be defined as follows: For any simply connected two-dimensional domain \(\Omega \), let \(\phi \) satisfy \(\nabla ^2\phi =0\) with \(\nabla \phi \cdot \mathbf {n}=f\) on \(\partial \Omega \). Then \(Hf=\nabla \phi \cdot \mathbf {t}\) on \(\partial \Omega \). (\(\mathbf {n}\) is the exterior unit normal, \(\mathbf {t}\) is the unit tangent vector.)

1. Prove that \(H(Hf)=-f\).
2. Find a relationship between the Fourier transform and the Hilbert transform, using a unit circle as the domain \(\Omega \).
3. What is the Hilbert transform of \(f(x)=a\sin (x)+b\cos (2x)\)?