Based on the work of Mauldin and Williams [‘On the Hausdorff dimension of some graphs’, Trans. Amer. Math. Soc. 298(2) (1986), 793–803] on convex Lipschitz functions, we prove that fractal interpolation functions belong to the space of convex Lipschitz functions under certain conditions. Using this, we obtain some dimension results for fractal functions. We also give some bounds on the fractal dimension of fractal functions with the help of oscillation spaces.

The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by $0$ outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.

In this paper we consider polynomials orthogonal with respect to the linear functional defined on the space of all algebraic polynomials by

where α,β> -1/2 are real numbers such that ℓ = |β - α| is a positive integer, and ζ∈ R{0}. We prove the existence of such orthogonal polynomials for some pairs of α and ζ and for all nonnegative integers ℓ. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered. Also, some numerical examples are included.

We introduce a family of orthogonal functions, termed as generalized Slepian functions (GSFs), closely related to the time-frequency concentration problem on a unit disk in D. Slepian [19]. These functions form a complete orthogonal system in with , and can be viewed as a generalization of the Jacobi polynomials with parameter (α, 0). We present various analytic and asymptotic properties of GSFs, and study spectral approximations by such functions.

We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.

In this paper we prove the sharp inequality

$$\left| P_{n}^{\left( s \right)}\left( x \right) \right|\le P_{n}^{\left( s \right)}\left( 1 \right)\left( {{\left| x \right|}^{n}}+\frac{n-1}{2s+1}\left( 1-{{\left| x \right|}^{n}} \right) \right)$$

where $P_{n}^{\left( s \right)}\left( x \right)$ is the classical ultraspherical polynomial of degree $n$ and order $s\ge n\frac{1+\sqrt{5}}{4}$. This inequality can be refined in $\left[ 0,z_{n}^{s} \right]$ and $\left[ z_{n}^{s},1 \right]$, where $z_{n}^{s}$ denotes the largest zero of $P_{n}^{\left( s \right)}\left( x \right)$.

A $q$-discrete analog of the Toda molecule equation and its $N$-soliton solution are constructed by using the bilinear method. The solution is expressed in the Casorati determinant form whose elements are given in terms of the $q$-orthogonal polynomials.