Fourier Analysis and Convolution

Define the Fourier transform on \(\mathbb {R}\) (do not worry about the choice of normalization). Show that the Fourier transform of an \(L^1(\mathbb {R},\lambda )\) function is in \(C_0(\mathbb {R})\). You may use that the Fourier transform of a continuous compactly supported function is in \(C_0(\mathbb {R})\).

In this problem, let \(\lambda \) denote Lebesgue measure on \(\mathbb {R}\) and \(\widehat {\cdot }\) denote the Fourier transform (on \(\mathbb {R}\)). Show that if \(f,g\) are Lebesgue integrable and \(\hat f=\hat g\) then \(f=g\) \(\lambda \)-a.e.

Using the methods of complex analysis compute \[ \int _0^\infty \frac {1}{x^4+16}\,dx. \]

Compute \[ \int _0^{2\pi }\frac {d\theta }{4+\cos (\theta )} \] using the methods of complex analysis.

Using the methods of complex analysis, compute \[ \int _0^\infty \frac {x^2}{x^4+1}\,dx \] and simplify your answer.

Use the methods of complex analysis to compute \[ \int _{-\infty }^{\infty }\frac {2}{(x^2+4)(x^2+1)}\,dx. \]

Prove that the convolution of two \(L^1(\lambda )\) functions is in \(L^1(\lambda )\).

Using methods of complex analysis, evaluate \[ \int _0^\infty \frac {x^2}{x^4+x^2+1}\,dx. \]

Let \(f:U\to \mathbb C\) be a holomorphic function, where \(U\) is an open set. Let \(a\in U\). A standard result in complex analysis is that if \(f'(a)\ne 0\), then there is an open neighborhood \(V\subset U\) of \(a\) on which \(f\) is injective and the image \(f(V)\) is a neighborhood of \(f(a)\). Prove this result!

(a) Let \(A\subset [0,1]\) and \(A\in \mathcal B\). Show that for all \(\epsilon >0\) there exists a finite number of intervals \(I_1,\ldots ,I_n\) so that \[ \lambda \left (A\Delta \left (\cup _{j=1}^n I_j\right )\right )<\epsilon . \] Note: \(\Delta \) denotes symmetric difference. So \[ A\Delta B=\{x\in A\cup B:x\notin A\cap B\}. \]

(b) Show that we cannot assume that \(A\subset \cup _{j=1}^n I_j\).

Complex Analysis

Let \(\mathbb D\) denote the open unit disk in \(\mathbb C\). A region is an open connected subset of \(\mathbb C\).

Consider \(\mathbb R\) with the Lebesgue measure, and recall the definitions of convolution and the Fourier transform, for \(f,g\in L^1(\mathbb R)\): \[ f*g(x)=\int f(x-y)g(y)\,dy,\qquad \hat f(\xi )=\int f(x)e^{-2\pi ix\xi }\,dx. \] Show that for \(f,g\in L^1(\mathbb R)\), \(f*g\in L^1(\mathbb R)\), and that \[ \widehat {f*g}=\hat f\,\hat g. \]

Recall that the Fourier transform of a function \(f\in L^1(\mathbb R)\) (with respect to Lebesgue measure) is given by \[ \hat f(x)=\int _{-\infty }^{\infty } e^{-2\pi ixy}f(y)\,dy \] Show that \(\hat f\) is a bounded, continuous function on \(\mathbb R\).

Let \(f\in L^1([-\pi ,\pi ])\), and let \(c_n\) be the nth Fourier coefficient of \(f\): \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi } f(x)e^{-inx}\,dx \quad n\in \mathbb Z \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

Let \(f\in L^1([-\pi ,\pi ])\) and for \(n\in \mathbb Z\) let \(c_n\) be the nth Fourier coefficient of \(f\), that is \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

Let \(\alpha \) be an irrational number. Let \(f\) be a real valued measurable function on \(\mathbb R\) such that \(f(x)=f(x+1)=f(x+\alpha )\). Use Fourier series to show that \(f\) is constant almost everywhere.

Consider a measurable subset \(A\) of the circle \(\{z\in \mathbb C\mid |z|=1\}\). Fix a real number \(\theta \) so that \(\theta /\pi \) is irrational. Define \[ A_\theta =\{e^{i\theta }z\mid z\in A\}, \] the rotation of \(A\) through \(\theta \) radians. Prove that if \[ A=A_\theta \] then either \(A\) has measure zero or else \(A\) is the complement of a set of measure zero. (Hint: consider the Fourier series of \(\chi _A-\chi _{A_\theta }\); here \(\chi _S\) denotes the characteristic function of \(S\).)

Consider \(L^1(T)\) to be the space of periodic functions on \(\mathbb R\) of period \(2\pi \) which are integrable on \([0,2\pi ]\). Let \(M\) be a proper closed translation invariant subspace of \(L^1(T)\).

(a) Prove that there exists a non-zero \(h\in L^\infty (T)\) such that \(f*h(x)=0\) for all \(f\in M\), \[ \text {where } f*h(x)=\frac {1}{2\pi }\int _0^{2\pi } f(x-y)h(y)\,dy; \]

(b) use part (a) to prove there must be an \(n\in \mathbb Z\) such that \(\hat f(n)=0\) for every \(f\in M\), where \(\hat f(n)\) is the \(n\) th Fourier coefficient of \(f\).

Show that the parallelogram law fails for \(L^1([0,1])\) so it is not a Hilbert space.

Hint: Use \(f=\chi _{[0,1/2]}\) and \(g=\chi _{[1/2,1]}\).

Prove that if \(f,g\in L^2(\mathbb R)\) and the convolution product \(f*g\) is defined by \[ f*g(x)=\int f(y)g(x-y)\,dy, \] then \(f*g(x)\) is a continuous function which vanishes at infinity.

Let \(f\) be a continuous positive valued function on \(\mathbb R\), and let \(g\) be the characteristic function of a finite nonempty open interval in \(\mathbb R\). Show that the function \(h=fg\) is in \(L^1(\mathbb R)\), but its Fourier transform \(\hat h\) is not.

(a) State the definition of a Fourier transform of \(f\in L^1(\mathbb R)\);

(b) State the Fourier Inversion Theorem;

(c) Show that the Fourier transform is one-to-one on \(L^1(\mathbb R)\).

Using methods of complex analysis, evaluate \[ \int _0^\infty \frac {x^2\,dx}{x^4+x^2+1} \]

Suppose that \(f\in L^1(\mathbb R^n)\) and \(\hat f\in L^1(\mathbb R^n)\) (\(\hat f\) denoting the Fourier transform of \(f\)). Then show that \[ f\in L^1(\mathbb R^n)\cap L^2(\mathbb R^n). \]

Using the definitions of convolution and of Fourier transform show that, for \(f,g\in L^1(\mathbb R)\), \[ \widehat {(f*g)}(t)=\hat f(t)\hat g(t) \] for \(t\in \mathbb R\).

Recall the \(k\)th Fourier coefficient of \(f\in L^1([-\pi ,\pi ])\) is \[ \hat f(k)=\frac {1}{2\pi }\int _{-\pi }^{\pi }f(t)e^{-ikt}\,dt. \]

(1) If \(f(t)=t\) for \(-\pi \le t<\pi \), show that \(\hat f(k)=(-1)^{k+1}\frac {1}{ik}\) for \(k\ne 0\) and \(\hat f(0)=0\);

(2) Show that \(\sum _{k=1}^{\infty }k^{-2}=\frac {\pi ^2}{6}\).

Use methods of complex analysis to find \[ \int _{-\infty }^{\infty }\frac {dx}{x^4+1}. \]