This paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal (interpolating). They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar functi...

This paper considers the classical Shannon sampling theorem in multiresolution spaceswith scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which...

The kth Markoff-Duffin-Schaeffer inequality provides a bound for the maximum, over the interval -1 </= x </= 1, of the kth derivative of a normalized polynomial of degree n. The bound is the corresponding maximum of the Chebyshev polynomial of degree n, T = cos(n cos(-1)x). The requisite normalization is over the values of the polynomial at the n + 1 points where T achieves its extremal values....

Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the co...

We study the kernels Kn,s(z) in the remainder terms Rn,s(f) of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at ±1, when the weight ω is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel |Kn,s(z)| attains its maximum on the real axis (positive real semi...

Interpolation polynomial pn at the Chebyshev nodes cosπj/n (0 ≤ j ≤ n) for smooth functions is known to converge fast as n → ∞. The sequence {pn} is constructed recursively and efficiently in O(n log2 n) flops for each pn by using the FFT, where n is increased geometrically, n = 2i (i = 2, 3, . . . ), until an estimated error is within a given tolerance of ε. This sequence {2j}, however, grows ...

In this paper, approximate lower and upper Hermite–Hadamard type inequalities are obtained for functions that are approximately convex with respect to a given Chebyshev system. Mathematics subject classification (2010): Primary 39B22, 39B12.

The purpose of this paper is to investigate the use of rational Chebyshev (RC) collocation method for solving high-order linear ordinary differential equations with variable coefficients. Using the rational Chebyshev collocation points, this method transforms the high-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coeffi...