386 Index

complex number

argument of, 11

complex conjugate of, 10

imaginary part, 10

modulus of, 10

real part of, 10

unimodular, 10

conjugate Dirichlet kernel, 84

continuity, 4

absolute, 8

modulus of, 175

uniform, 4, 262, 264

convolution

Cauchy, 241, 295

of functions in

L1(R),

254

of functions in

L1(R2),

287

of functions in

L2(R),

264, 268–270

of functions with period 1, 69–75

in several variables, 282

of periodic arithmetic functions, 39

in several variables, 280

Cooley, James William, 52, 378

Corput, Johannes Gualtherus van der

inequality, 233

theorem, 233

Crane, Edward Thomas, 290

Darboux, Jean-Gaston, 84

Darst, Richard Brian, 194, 378

difference, forward, 298

difference–quotient, 309

Dini, Ulisse, 134, 378

test, 196, 209

Dirac, Paul Adrien Maurice

delta function with

period 1, 60, 134

Dirichlet, Peter Gustav Lejeune, 54, 88,

134, 378

kernel, 77, 92

conjugate, 84

modified, 84

modified conjugate, 85

test, 111–112

Discrete Fourier Transform, 36–52

Multiple, 279–280

discriminant, 2

distribution, 60

dominated convergence

for integrals, 7

for series, 6

Dumitrescu, Bogdan, 286, 378

Duren, Peter Larkin, 134, 378

Egorov, Dmitri Fyodorovich

theorem, 7

entire function, 22

Euler, Leonhard, 30, 88, 247

constant, 226

Euler–Maclaurin summation, 226

formula for

∑

n−2, 83

identity, 23

exponential function, 22–24

Fast Fourier Transform (FFT), 48–52

Fatou, Pierre Joseph Louis, 148, 378

Fej´ er, Lip´ ot, 134, 163, 182, 378

kernel, 92, 131, 134

conjugate, 107, 134

lemma, 101

theorem, 102, 121, 125

Fej´ er–Riesz theorem, 163

false in higher dimensions, 285

Ferguson, Ronald Aubrey, 182, 377

Ferreira, Paulo Jorge S. G., 278, 377

Fischer, Ernst Sigismund, 144, 148, 378

Fourier Transforms, 249–278

Multiple, 286–290

Fourier, Jean Baptiste Joseph, 54, 246,

378

coeﬃcients, 54

Fourier Series, 54–247

in several variables, 280–286

study of heat, 211–213

Fubini, Guido

theorem, 8

function

almost periodic, 286

arithmetic, 36

periodic, 36

of bounded variation, 198–205

exponential, 22–24

harmonic, 211

Lipschitz, 184

logarithm, 25

nowhere differentiable, 215–216

sawtooth, 58, 62, 78, 79, 151

discrete, 42

square wave, 59, 62, 64

support of, 272

theta, 276

functions

hyperbolic, 369–374

trigonometric

in calculus, 357–369

in geometry, 351–357