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
Previous Page Next Page