Fractal Jacobi Systems and Convergence of Fourier–Jacobi Expansions of Fractal Interpolation Functions

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of \({\mathbb{R}}\) interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in \({\mathcal{L}^{2}(-1,1)}\) with respect to the weight function \({\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}\), \({r > -1}\) and \({s > -1}\). In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of \({\alpha}\)-fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions \({\mathcal{L}_{\rho}^{2}(-1,1)}\) is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.     

On Fourier Series in Generalized Eigenfunctions of a Discrete Sturm–Liouville Operator

For semicontinuous summation methods generated by Λ = {λ_n(h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class B, some results on the uniform a.e. behavior of Λ-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means λ_n(h) = exp(?u^α(n)h) are obtained. An application to a generalized heat equation is given.     

Upper Bounds of the Lebesgue Constants for Fourier–Jacobi Series Summation Methods Defined by a Multiplier Function

Let be the Jacobi polynomials and let C[a,b] be the space of continuous functions on [a,b] with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators generated by a multiplier matrix defined by the following relations:

(C,α) Summability of Walsh—Fourier Series

The one-dimensional dyadic martingale Hardy spaces H _p are introduced and it is proved that the maximal operator of the (C,) means of a Walsh—Fourier series is bounded from H _p to L _p (1/( + 1) < p < ) and is of weak type (L ₁,L ₁). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,) means of the Walsh—Fourier series of a function f L ₁ converge a.e. to the function in question. Moreover, we prove that the (C,) means are uniformly bounded on H _p whenever 1/( + 1) < p < . We define the two-dimensional dyadic hybrid Hardy space H ₁ and verify that the maximal operator of the (C,,) means of a two-dimensional function is of weak type H ₁ ,L ₁). Consequence, the Walsh—Fourier series of every function f H ₁ is (C,,) summable to the function f.     

On Changes of Variable that Preserve the Absolute Convergence of Fourier–Haar Series of Continuous Functions

Mathematical Notes - In this paper, we discuss the problem of the number of representations of positive integers as sums of triangular numbers. The method we use is similar to Rankin’s way in...     

On Fourier–Jacobi expansions of real analytic Eisenstein series of degree 2

We discuss the Fourier–Jacobi expansion of certain vector valued Eisenstein series of degree $2$ , which is also real analytic. We show that its coefficients of index $\pm 1$ can be described by using a generating series of real analytic Jacobi forms. We also describe all the coefficients of general indices in suitable manners. Our method can be applied to study another Fourier series of Saito-Kurokawa type that is associated with a cusp form of one variable and half-integral weight. Then, following the arguments in the holomorphic case, we find that the Fourier series indeed defines a real analytic Siegel modular form of degree 2.     

Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series

Suppose that d ≥ 1 is an integer, ${\alpha \in (0,d)}$ is a fixed parameter and let I _α be the fractional integral operator associated with d-dimensional Walsh–Fourier series on [0, 1) ^d . The paper contains the proof of the sharp weak-type estimate $$||I_\alpha(f)||_{L^{d/(d-\alpha),\infty}([0,1)^d)}\leq\frac{2^d-1}{(2^{d-\alpha}-1)(2^\alpha-1)}||f||_{L^1([0,1)^d)}.$$ The proof rests on Bellman-function-type method: the above estimate is deduced from the existence of a certain family of special functions.     

Characterization of the Points of ϕ-Strong Summability of Fourier–Laplace Series for Functions of the Class L p(S m), p > 1

We consider the behavior of the -strong means of Fourier–Laplace series for functions that belong to L _p(S ^m), p > 1, on a set of points of full measure on an m-dimensional sphere S ^m.     

On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

Mathematical Notes - Approximations on the closed interval $$[-1,1]$$ of functions that are combinations of classical Markov functions by partial sums of Fourier series on a system of...     

One Inequality of the Landau–Kolmogorov Type for Periodic Functions of Two Variables

Ukrainian Mathematical Journal - We obtain a new sharp inequality of the Landau–Kolmogorov type for a periodic function of two variables estimating the convolution of the best uniform...     

Construction of the Gelfand–Tsetlin Basis for Unitary Principal Series Representations of the Algebra sln(ℂ)

Theoretical and Mathematical Physics - We consider infinite-dimensional unitary principal series representations of the algebra sln(?), implemented on the space of functions of n(n?1)/2...     

Toeplitz–Schur Multipliers of the Class Sp(L2G) for p< 1

We study Toeplitz–Schur multipliers of the Schatten–von Neumann class S _p for 0 < p < 1. We describe all functions F on an arbitrary commutative locally compact group G satisfying the following condition: for any integral operator in S _p with kernel function k(x,y), the kernel function F(x-y)k(x)k(y) defines also an integral operator in S _p. Bibliography: 4 titles.     

Pointwise Inequalities of Landau–Kolmogorov Type for Functions Defined on a Finite Segment

For arbitrary t [0, 1], s [1, ], and A 2, we determine the unimprovable constant B for the inequality

Nonparametric Duality Models for Semi-Infinite Discrete Minmax Fractional Programming Problems Involving Generalized (η, ρ)-Invex Functions

In this paper, we discuss a fairly large number of nonparametric duality results under various generalized (η, ρ)-invexity assumptions for a semi-infinite minmax fractional programming problem.     

Representations of (α, β, γ) triple system

We introduce a notion of (α, β, γ) triple system which generalizes the familiar generalized Jordan triple system. We then discuss its realization by some bilinear algebras and vice versa. We also give a characterization of the structurable algebra of Allison in terms of (?1, 1) Freudenthal–Kantor triple system by imposing some additional triple product constraints.