Fourier Series and Orthogonal Polynomials
This text for undergraduate and graduate students illustrates the fundamental simplicity of the properties of orthogonal functions and their developments in related series. Starting with a definition and explanation of the elements of Fourier series, the text follows with examinations of Legendre polynomials and Bessel functions. Boundary value problems consider Fourier series in conjunction with Laplace's equation in an infinite strip and in a rectangle, with a vibrating string, in three dimensions, in a sphere, and in other circumstances. An overview of Pearson frequency functions is followed by chapters on orthogonal, Jacobi, Hermite, and Laguerre polynomials, and the text concludes with a chapter on convergence. 1941 edition.
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application approaches zero arbitrary function becomes inﬁnite Bessel boundary value problem Chapter VII coefﬁcients coeﬁicients constant multiple continuous function convergence coordinates corresponding cos2 cosine sum Courant-Hilbert deﬁned denoted derivative differential equation diﬂerential discussion equal expression factor ﬁnite jumps ﬁnite number ﬁrst ﬁxed Fourier series func function f(x function of period given harmonic polynomial Hermite polynomials hypothesis identity independent integral values integrand Jacobi polynomials kx dx Laguerre Laplace series Lebesgue left-hand member Legendre polynomials Legendre series Let f(x linear combination mathematical nomials non-negative normalized notation nth degree nth order obtained orthogonal polynomials paragraph partial sum particular period 21r poly positive roots power series property of orthogonality recurrence formula relation replaced representation respect right-hand member satisﬁes series for f(x signiﬁcance sin2 sinh solu solution speciﬁed spherical harmonics substitution theory tion trigonometric sum uniformly bounded vanishes weight function