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Consider a string of length 2L plucked at the right end and fixed at the left. The functional form of this configuration is f(x)=x/(2L). (1) The components of the Fourier ...

The G-transform of a function f(x) is defined by the integral (Gf)(x)=(G_(pq)^(mn)|(a_p); (b_q)|f(t))(x) (1) =1/(2pii)int_sigmaGamma[(b_m)+s, 1-(a_n)-s; (a_p^(n+1))+s, ...

Consider a square wave f(x) of length 2L. Over the range [0,2L], this can be written as f(x)=2[H(x/L)-H(x/L-1)]-1, (1) where H(x) is the Heaviside step function. Since ...

The matrix product of a square set of data d and a matrix of basis vectors consisting of Walsh functions. By taking advantage of the nested structure of the natural ordering ...

The Legendre transform of a sequence {c_k} is the sequence {a_k} with terms given by a_n = sum_(k=0)^(n)(n; k)(n+k; k)c_k (1) = sum_(k=0)^(n)(2k; k)(n+k; n-k)c_k, (2) where ...

The integral transform defined by (Kphi)(x)=int_0^infty(x^2-t^2)_+^(lambda/2)P_nu^lambda(t/x)phi(t)dt, where y_+^alpha is the truncated power function and P_nu^lambda(x) is ...

The integral transform (Kf)(x)=int_0^infty((x-t)_+^(c-1))/(Gamma(c))_2F_1(a,b;c;1-t/x)f(t)dt, where Gamma(x) is the gamma function, _2F_1(a,b;c;z) is a hypergeometric ...

Consider a symmetric triangle wave T(x) of period 2L. Since the function is odd, a_0 = 0 (1) a_n = 0, (2) and b_n = (3) = (32)/(pi^2n^2)cos(1/4npi)sin^3(1/4npi) (4) = ...

The Radon inverse transform is an integral transform that has found widespread application in the reconstruction of images from medical CT scans. The Radon and inverse Radon ...

The discrete Fourier transform of length N (where N is even) can be rewritten as the sum of two discrete Fourier transforms, each of length N/2. One is formed from the ...