Hilbert Space Methods in Signal Processing
Cambridge University Press, Mar 7, 2013 - Mathematics - 420 pages
This lively and accessible book describes the theory and applications of Hilbert spaces and also presents the history of the subject to reveal the ideas behind theorems and the human struggle that led to them. The authors begin by establishing the concept of 'countably infinite', which is central to the proper understanding of separable Hilbert spaces. Fundamental ideas such as convergence, completeness and dense sets are first demonstrated through simple familiar examples and then formalised. Having addressed fundamental topics in Hilbert spaces, the authors then go on to cover the theory of bounded, compact and integral operators at an advanced but accessible level. Finally, the theory is put into action, considering signal processing on the unit sphere, as well as reproducing kernel Hilbert spaces. The text is interspersed with historical comments about central figures in the development of the theory, which helps bring the subject to life.
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2-sphere adjoint operator algebraic Answer to Problem azimuthally symmetric bandlimited bounded linear operator bounded operator cardinality co-latitude commutative compact operator complete orthonormal sequence continuous functions convergent sequence convolution corresponding deﬁnition degree denoted diagonal dimensions domain eigenfunctions eigenvalues eigenvectors equation equivalent Euclidean example expansion f G H Figure ﬁlter ﬁnd ﬁnite number ﬁrst ﬁxed Fourier coefﬁcients Fourier series G L2 given Hilbert space Hilbert-Schmidt integral operator identity inﬁnite-dimensional inner product space input interval isomorphism Legendre polynomials mapping matrix elements non-zero normed space notation operator matrix operator norm orthogonal subspaces orthonormal set output Parseval relation points projection operator Proof real numbers real spherical harmonics Remark representation result RKHS satisﬁes self-adjoint operator separable Hilbert space signal f signiﬁcant SLSHT space H spatial speciﬁc spectral spherical harmonic coefﬁcients spherical harmonics strongly convergent subset sufﬁcient Theorem truncation vector space weakly convergent Wigner d-matrix y-axis z-axis zero