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A First Course in Fourier Analysis.

By: Publisher: Cambridge : Cambridge University Press, 2008Copyright date: ©2008Edition: 2nd edDescription: 1 online resource (863 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780511375958
Subject(s): Genre/Form: Additional physical formats: Print version:: A First Course in Fourier AnalysisDDC classification:
  • 515.2433
LOC classification:
  • QA403.5 -- .K36 2007eb
Online resources:
Contents:
Cover -- Half-title -- Title -- Copyright -- Contents -- Selected Applications -- Preface -- To the Student -- Synopsis -- To the Instructor -- Acknowledgments -- The Mathematical Core -- 1 Fourier's representation for functions on R, Tp, Z and Pn -- 1.1 Synthesis and analysis equations -- Introduction -- Functions on R -- Functions on Tp -- Functions on Z -- Functions on PN -- Summary -- 1.2 Examples of Fourier's representation -- Introduction -- The Hipparchus-Ptolemy model of planetary motion -- Gauss and the orbits of the asteroids -- Fourier and the .ow of heat -- Fourier's representation and LTI systems -- Schoenberg's derivation of the Tartaglia-Cardan formulas -- Fourier transforms and spectroscopy -- 1.3 The Parse validentities and related results -- The Parseval identities -- The Plancherel identities -- Orthogonality relations for the periodic complex exponentials -- Bessel's inequality -- The Weierstrass approximation theorem -- A proof of Plancherel's identity for functions on Tp -- 1.4 The Fourier-Poisson cube -- Introduction -- Discretization by h -sampling -- Periodization by p -summation -- The Poisson relations -- The Fourier-Poisson cube -- 1.5 The validity of Fourier's representation -- Introduction -- Functions on PN -- Absolutely summable functions on Z -- Continuous piecewise smooth functions on Tp -- The sawtooth singularity function on T1 -- The Gibbs phenomenon for w0 -- Piecewise smooth functions on Tp -- Smooth functions on R with small regular tails -- Singularity functions on R -- Piecewise smooth functions on R with small regular tails -- Extending the domain of validity -- Further reading -- Exercises -- 2 Convolution of functions on R, Tp, Z, and Pn -- 2.1 Formal definitions of… -- Correlation and conjugation are closely related -- 2.2 Computation of f*g -- Direct evaluation -- The sum of scaled translates.
The sliding strip method -- Generating functions -- 2.3 Mathematical properties of the convolution product -- Introduction -- The Fourier transform of… -- Algebraic structure -- Translation invariance -- Differentiation of… -- 2.4 Examples of convolution and correlation -- Convolution as smearing -- Echo location -- Convolution and probability -- Convolution and arithmetic -- Further reading -- Exercises -- 3 Thecalculus for finding Fourier transforms of functions on R -- 3.1 Using the definition to find Fourier transforms -- Introduction -- The box function -- The Heaviside step function -- The truncated decaying exponential -- The unit gaussian -- Summary -- 3.2 Rules for finding Fourier transforms -- Introduction -- Linearity -- Translation and modulation -- Dilation -- Inversion -- Convolution and multiplication -- Summary -- 3.3 Selected applications of the Fourier transform calculus -- Evaluation of integrals and sums -- Evaluation of convolution products -- The Hermite functions -- Smoothness and rates of decay -- Further reading -- Exercises -- 4 The calculus for finding Fourier transforms of functions on Tp, Z, and Pn -- 4.1 Fourier series -- Introduction -- Direct integration -- Elementary rules -- Poisson's relation -- Bernoulli functions and Eagle's method -- Laurent series -- Dilation and grouping rules -- 4.2 Selected applications of Fourier series -- Evaluation of sums and integrals -- The polygon function -- Rates of decay -- Equidistribution of arithmetic sequences -- 4.3 Discrete Fourier transforms -- Direct summation -- Basic rules -- Dilation -- Poisson's relations -- 4.4 Selected applications of the DFT calculus -- The Euler-Maclaurin sum formula -- The discrete Fresnel function -- Further reading -- Exercises -- 5 Operator identities associated with Fourier analysis -- 5.1 The concept of an operator identity -- Introduction.
Operators applied to functions on PN -- Blanket hypotheses -- 5.2 Operators generated by powers of F -- Powers of F -- The even and odd projection operators -- The normalized exponential transform operators -- The normalized cosine transform and sine transform operators -- The normalized Hartley transform operators -- Connections -- Tag notation -- 5.3 Operators related to complex conjugation -- The bar and dagger operators -- The real, imaginary, hermitian, and antihermitian projection operators -- Symmetric functions -- Symmetric operators -- 5.4 Fourier transforms of operators -- The basic definition -- Algebraic properties -- 5.5 Rules for Hartley transforms -- 5.6 Hilbert transforms -- Defining relations -- Operator identities -- The Kramers-Kronig relations -- Further reading -- Exercises -- 6 The fast Fourier transform -- 6.1 Pre-FFT computation of the DFT -- Introduction -- Horner's algorithm for computing the DFT -- Other pre-FFT methods for computing the DFT -- How big is 4N2? -- The announcement of a fast algorithm for the DFT -- 6.2 Derivation of the FFT via DFT rules -- Decimation-in-time -- Decimation-in-frequency -- Recursive algorithms -- 6.3 The bit reversal permutation -- Introduction -- A naive algorithm -- The reverse carry algorithm -- The Bracewell-Buneman algorithm -- 6.4 Sparse matrix factorization of F when N=2m -- Introduction -- The zipper identity -- Exponent notation -- Sparse matrix factorization of F -- The action of B2m -- An FFT algorithm -- An alternative FFT algorithm -- Precomputation of… -- Application of Q4M -- 6.5 Sparse matrix factorization of H when N=2m -- The zipper identity and factorization -- Application of T4M using precomputed… -- 6.6 Sparse matrix factorization of F whenN=P1P2···Pm -- Introduction -- The zipper identity for FMP -- Factorization of… -- An FFT -- The permutation… -- The permutation….
Closely related factorizations of F,H -- 6.7 Kronecker product factorization of F -- Introduction -- The Kronecker product -- Rearrangement of Kronecker products -- Parallel and vector operations -- Stockham's autosort FFT -- Further reading -- Exercises -- 7 Generalized functions on R -- 7.1 The concept of a generalized function -- Introduction -- Functions and functionals -- Schwartz functions -- Functionals for generalized functions -- 7.2 Common generalized functions -- Introduction -- The comb III -- The functions… -- Summary -- 7.3 Manipulation of generalized functions -- Introduction -- The linear space G -- Translate, dilate, derivative, and Fourier transform -- Reflection and conjugation -- Multiplication and convolution -- 7.4 Derivatives and simple differential equations -- Differentiation rules -- Derivatives of piecewise smooth functions with jumps -- Solving differential equations -- 7.5 The Fourier transform calculus for generalized functions -- Fourier transform rules -- Basic Fourier transforms -- Support- and bandlimited generalized functions -- 7.6 Limits of generalized functions -- Introduction -- The limit concept -- Transformation of limits -- 7.7 Periodic generalized functions -- Fourier series -- The analysis equation -- Convolution of p-periodic generalized functions -- Discrete Fourier transforms -- Connections -- 7.8 Alternative definitions for generalized functions -- Functionals on S -- Other test functions -- Further reading -- Exercises -- Selected Applications -- 8 Sampling -- 8.1 Sampling and interpolation -- Introduction -- Shannon's hypothesis -- 8.2 Reconstruction of f from its samples -- A weakly convergent series -- The cardinal series -- Recovery of an alias -- Fragmentation of… -- 8.3 Reconstruction of f from samples of a1*f,a2*f,... -- Filters -- Samples from one filter -- The Papoulis generalization.
8.4 Approximation of almost bandlimited functions -- Further reading -- Exercises -- 9 Partial differential equations -- 9.1 Introduction -- 9.2 The wave equation -- A physical context: Plane vibration of a taut string -- The wave equation on R -- The wave equation on Tp -- Each point on the string vibrates with the frequency -- 9.3 The diffusion equation -- A physical context: Heat .ow along a long rod -- The diffusion equation on R -- The diffusion equation on Tp -- 9.4 The diffraction equation -- A physical context: Diffraction of a laser beam -- The diffraction equation on R -- The diffraction equation on Tp -- 9.5 Fast computation of frames for movies -- Further reading -- Exercises -- 10 Wavelets -- 10.1 The Haar wavelets -- Interpretation of F[m,k] -- Arbitrarily good approximation -- Successive approximation -- Coded approximation -- 10.2 Support-limited wavelets -- The dilation equation -- Smoothness constraints -- Order of approximation -- Orthogonality constraints -- Daubechies wavelets -- 10.3 Analysis and synthesis with Daubechies wavelets -- Coefficients for frames and details -- The operators… -- Samples for frames and details -- The operators… -- 10.4 Filter banks -- Introduction -- Factorization of… -- Fourier analysis of a filter bank -- Perfect reconstruction .lter banks -- Compression and reconstruction -- Further reading -- Exercises -- 11 Musical tones -- 11.1 Basic concepts -- Introduction -- Perception of pitch and loudness -- Ohm's law -- Scales -- Musical notation -- 11.2 Spectrograms -- Introduction -- The computation -- Slowly varying frequencies -- 11.3 Additive synthesis of tones -- Introduction -- Amplitude envelopes -- Synthesis of a bell tone -- Synthesis of a brass tone -- 11.4 FM synthesis of tones -- Introduction -- The spectral decomposition -- Dynamic spectral enrichment -- 11.5 Synthesis of tones from noise.
Introduction.
Summary: This book introduces applied mathematics through Fourier analysis, with applications to studying sampling theory, PDEs, probability, diffraction, musical tones, and wavelets.
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Cover -- Half-title -- Title -- Copyright -- Contents -- Selected Applications -- Preface -- To the Student -- Synopsis -- To the Instructor -- Acknowledgments -- The Mathematical Core -- 1 Fourier's representation for functions on R, Tp, Z and Pn -- 1.1 Synthesis and analysis equations -- Introduction -- Functions on R -- Functions on Tp -- Functions on Z -- Functions on PN -- Summary -- 1.2 Examples of Fourier's representation -- Introduction -- The Hipparchus-Ptolemy model of planetary motion -- Gauss and the orbits of the asteroids -- Fourier and the .ow of heat -- Fourier's representation and LTI systems -- Schoenberg's derivation of the Tartaglia-Cardan formulas -- Fourier transforms and spectroscopy -- 1.3 The Parse validentities and related results -- The Parseval identities -- The Plancherel identities -- Orthogonality relations for the periodic complex exponentials -- Bessel's inequality -- The Weierstrass approximation theorem -- A proof of Plancherel's identity for functions on Tp -- 1.4 The Fourier-Poisson cube -- Introduction -- Discretization by h -sampling -- Periodization by p -summation -- The Poisson relations -- The Fourier-Poisson cube -- 1.5 The validity of Fourier's representation -- Introduction -- Functions on PN -- Absolutely summable functions on Z -- Continuous piecewise smooth functions on Tp -- The sawtooth singularity function on T1 -- The Gibbs phenomenon for w0 -- Piecewise smooth functions on Tp -- Smooth functions on R with small regular tails -- Singularity functions on R -- Piecewise smooth functions on R with small regular tails -- Extending the domain of validity -- Further reading -- Exercises -- 2 Convolution of functions on R, Tp, Z, and Pn -- 2.1 Formal definitions of… -- Correlation and conjugation are closely related -- 2.2 Computation of f*g -- Direct evaluation -- The sum of scaled translates.

The sliding strip method -- Generating functions -- 2.3 Mathematical properties of the convolution product -- Introduction -- The Fourier transform of… -- Algebraic structure -- Translation invariance -- Differentiation of… -- 2.4 Examples of convolution and correlation -- Convolution as smearing -- Echo location -- Convolution and probability -- Convolution and arithmetic -- Further reading -- Exercises -- 3 Thecalculus for finding Fourier transforms of functions on R -- 3.1 Using the definition to find Fourier transforms -- Introduction -- The box function -- The Heaviside step function -- The truncated decaying exponential -- The unit gaussian -- Summary -- 3.2 Rules for finding Fourier transforms -- Introduction -- Linearity -- Translation and modulation -- Dilation -- Inversion -- Convolution and multiplication -- Summary -- 3.3 Selected applications of the Fourier transform calculus -- Evaluation of integrals and sums -- Evaluation of convolution products -- The Hermite functions -- Smoothness and rates of decay -- Further reading -- Exercises -- 4 The calculus for finding Fourier transforms of functions on Tp, Z, and Pn -- 4.1 Fourier series -- Introduction -- Direct integration -- Elementary rules -- Poisson's relation -- Bernoulli functions and Eagle's method -- Laurent series -- Dilation and grouping rules -- 4.2 Selected applications of Fourier series -- Evaluation of sums and integrals -- The polygon function -- Rates of decay -- Equidistribution of arithmetic sequences -- 4.3 Discrete Fourier transforms -- Direct summation -- Basic rules -- Dilation -- Poisson's relations -- 4.4 Selected applications of the DFT calculus -- The Euler-Maclaurin sum formula -- The discrete Fresnel function -- Further reading -- Exercises -- 5 Operator identities associated with Fourier analysis -- 5.1 The concept of an operator identity -- Introduction.

Operators applied to functions on PN -- Blanket hypotheses -- 5.2 Operators generated by powers of F -- Powers of F -- The even and odd projection operators -- The normalized exponential transform operators -- The normalized cosine transform and sine transform operators -- The normalized Hartley transform operators -- Connections -- Tag notation -- 5.3 Operators related to complex conjugation -- The bar and dagger operators -- The real, imaginary, hermitian, and antihermitian projection operators -- Symmetric functions -- Symmetric operators -- 5.4 Fourier transforms of operators -- The basic definition -- Algebraic properties -- 5.5 Rules for Hartley transforms -- 5.6 Hilbert transforms -- Defining relations -- Operator identities -- The Kramers-Kronig relations -- Further reading -- Exercises -- 6 The fast Fourier transform -- 6.1 Pre-FFT computation of the DFT -- Introduction -- Horner's algorithm for computing the DFT -- Other pre-FFT methods for computing the DFT -- How big is 4N2? -- The announcement of a fast algorithm for the DFT -- 6.2 Derivation of the FFT via DFT rules -- Decimation-in-time -- Decimation-in-frequency -- Recursive algorithms -- 6.3 The bit reversal permutation -- Introduction -- A naive algorithm -- The reverse carry algorithm -- The Bracewell-Buneman algorithm -- 6.4 Sparse matrix factorization of F when N=2m -- Introduction -- The zipper identity -- Exponent notation -- Sparse matrix factorization of F -- The action of B2m -- An FFT algorithm -- An alternative FFT algorithm -- Precomputation of… -- Application of Q4M -- 6.5 Sparse matrix factorization of H when N=2m -- The zipper identity and factorization -- Application of T4M using precomputed… -- 6.6 Sparse matrix factorization of F whenN=P1P2···Pm -- Introduction -- The zipper identity for FMP -- Factorization of… -- An FFT -- The permutation… -- The permutation….

Closely related factorizations of F,H -- 6.7 Kronecker product factorization of F -- Introduction -- The Kronecker product -- Rearrangement of Kronecker products -- Parallel and vector operations -- Stockham's autosort FFT -- Further reading -- Exercises -- 7 Generalized functions on R -- 7.1 The concept of a generalized function -- Introduction -- Functions and functionals -- Schwartz functions -- Functionals for generalized functions -- 7.2 Common generalized functions -- Introduction -- The comb III -- The functions… -- Summary -- 7.3 Manipulation of generalized functions -- Introduction -- The linear space G -- Translate, dilate, derivative, and Fourier transform -- Reflection and conjugation -- Multiplication and convolution -- 7.4 Derivatives and simple differential equations -- Differentiation rules -- Derivatives of piecewise smooth functions with jumps -- Solving differential equations -- 7.5 The Fourier transform calculus for generalized functions -- Fourier transform rules -- Basic Fourier transforms -- Support- and bandlimited generalized functions -- 7.6 Limits of generalized functions -- Introduction -- The limit concept -- Transformation of limits -- 7.7 Periodic generalized functions -- Fourier series -- The analysis equation -- Convolution of p-periodic generalized functions -- Discrete Fourier transforms -- Connections -- 7.8 Alternative definitions for generalized functions -- Functionals on S -- Other test functions -- Further reading -- Exercises -- Selected Applications -- 8 Sampling -- 8.1 Sampling and interpolation -- Introduction -- Shannon's hypothesis -- 8.2 Reconstruction of f from its samples -- A weakly convergent series -- The cardinal series -- Recovery of an alias -- Fragmentation of… -- 8.3 Reconstruction of f from samples of a1*f,a2*f,... -- Filters -- Samples from one filter -- The Papoulis generalization.

8.4 Approximation of almost bandlimited functions -- Further reading -- Exercises -- 9 Partial differential equations -- 9.1 Introduction -- 9.2 The wave equation -- A physical context: Plane vibration of a taut string -- The wave equation on R -- The wave equation on Tp -- Each point on the string vibrates with the frequency -- 9.3 The diffusion equation -- A physical context: Heat .ow along a long rod -- The diffusion equation on R -- The diffusion equation on Tp -- 9.4 The diffraction equation -- A physical context: Diffraction of a laser beam -- The diffraction equation on R -- The diffraction equation on Tp -- 9.5 Fast computation of frames for movies -- Further reading -- Exercises -- 10 Wavelets -- 10.1 The Haar wavelets -- Interpretation of F[m,k] -- Arbitrarily good approximation -- Successive approximation -- Coded approximation -- 10.2 Support-limited wavelets -- The dilation equation -- Smoothness constraints -- Order of approximation -- Orthogonality constraints -- Daubechies wavelets -- 10.3 Analysis and synthesis with Daubechies wavelets -- Coefficients for frames and details -- The operators… -- Samples for frames and details -- The operators… -- 10.4 Filter banks -- Introduction -- Factorization of… -- Fourier analysis of a filter bank -- Perfect reconstruction .lter banks -- Compression and reconstruction -- Further reading -- Exercises -- 11 Musical tones -- 11.1 Basic concepts -- Introduction -- Perception of pitch and loudness -- Ohm's law -- Scales -- Musical notation -- 11.2 Spectrograms -- Introduction -- The computation -- Slowly varying frequencies -- 11.3 Additive synthesis of tones -- Introduction -- Amplitude envelopes -- Synthesis of a bell tone -- Synthesis of a brass tone -- 11.4 FM synthesis of tones -- Introduction -- The spectral decomposition -- Dynamic spectral enrichment -- 11.5 Synthesis of tones from noise.

Introduction.

This book introduces applied mathematics through Fourier analysis, with applications to studying sampling theory, PDEs, probability, diffraction, musical tones, and wavelets.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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