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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2023 • Question 3

Question

Math 6210 • Real Analysis Fall 2023 • Question 3

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 \(C_0(\mathbb {R})\).

Raw LaTeX

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 \(C_0(\mathbb{R})\).

Topic Tags

Compactness in Metric and Normed Spaces Fourier Analysis and Convolution Lp Spaces and Sequence Spaces