Fourier transforms and bent functions on finite groups

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis \({\widehat{G}}\), consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on \({\widehat{G}}\), and obtain characterizations of a bent function by its Fourier transforms (as functions on \({\widehat{G}}\)). For a function f from G to another finite group, we define a dual function \({\widetilde{f}}\) on \({\widehat{G}}\), and characterize the nonlinearity of f by its dual function \({\widetilde{f}}\). Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.     

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ^2?γ + h²) and O(τ² + h²), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ² + h²) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.     

An overpartition analogue of the <Emphasis Type="Italic">q</Emphasis>-binomial coefficients

We define an overpartition analogue of Gaussian polynomials (also known as q-binomial coefficients) as a generating function for the number of overpartitions fitting inside the \(M \times N\) rectangle. We call these new polynomials over Gaussian polynomials or over q-binomial coefficients. We investigate basic properties and applications of over q-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers–Ramanujan type partition theorem.     

Permutations of finite fields for check digit systems

Let q be a prime power. For a divisor n of q ? 1 we prove an asymptotic formula for the number of polynomials of the form

$f(X)=\frac{a-b}{n}\left(\sum_{j=1}^{n-1}X^{j(q-1)/n}\right)X+\frac{a+b(n-1)}{n}X\in\mathbb{F}_q[X]$

such that the five (not necessarily different) polynomials f(X), f(X)±X and f(f(X))±X are all permutation polynomials over \({\mathbb{F}_q}\) . Such polynomials can be used to define check digit systems that detect the most frequent errors: single errors, adjacent transpositions, jump transpositions, twin errors and jump twin errors.

     

A probabilistic approach to polynomial inequalities

We define a probability measure on the space of polynomials over ? ⁿ in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities.Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of ? ₁ ⁿ tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.     

Best approximation of reproducing kernels of spaces of analytic functions

We obtain exact values for the best approximation of a reproducing kernel of a system of p-Faber polynomials by functions of the Hardy space H_q, p^-1 + q^-1 = 1 and a Szegö reproducing kernel of the space E²(Ω) in a simply connected domain Ωwith rectifiable boundary.     

An elementary proof of the <Emphasis Type="Italic">A</Emphasis><Subscript>2</Subscript> bound

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp A _p bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L ¹ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A ² bounds in that setting.     

Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

For the self-adjoint Schrödinger operator ? defined on ? by the differential operation ?(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W ₂ ^?1, we describe classes of functions whose spectral expansions corresponding to the operator ? absolutely and uniformly converge on the entire line ?. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

     

Asymptotic equalities for best approximations for classes of infinitely differentiable functions defined by the modulus of continuity

We obtain asymptotic estimates for best approximations by trigonometric polynomials in the metric of the space C(L_p) for classes of periodic functions expressible as convolutions of kernels Ψ_β with Fourier coefficients decreasing to zero faster than any power sequence, and with functions ? ∈ C (? ∈ L_p) whose moduli of continuity do not exceed the given majorant of ω(t). It is proved that, in the spaces C and L₁, for convex moduli of continuity ω(t), the obtained estimates are asymptotically sharp.     

Are there <Emphasis Type="Italic">p</Emphasis>-adic knot invariants?

We suggest using the Hall–Littlewood version of the Rosso–Jones formula to define the germs of p-adic HOMFLY-PT polynomials for torus knots [m, n] as coefficients of superpolynomials in a q-expansion. In this form, they have at least the [m, n] ? [n, m] topological invariance. This opens a new possibility to interpret superpolynomials as p-adic deformations of HOMFLY polynomials and poses a question of generalizing to other knot families, which is a substantial problem for several branches of modern theory.     

Semigroups of Partial Isometries and Symmetric Operators

Let \({{\{ V(t) | \ t \in [0 , \infty) \}}}\) be a one-parameter strongly continuous semigroup of contractions on a separable Hilbert space and let V(?t) : = V*(t) for \({t \in [0, \infty)}\). It is shown that if V(t) is a partial isometry for all \({t \in [-t_0 , t_0], t_0 > 0}\), then the pointwise two-sided derivative of V(t) exists on a dense domain of vectors. This derivative B is necessarily a densely defined symmetric operator. This result can be viewed as a generalization of Stone’s theorem for one-parameter strongly continuous unitary groups, and is used to establish sufficient conditions for a self-adjoint operator on a Hilbert space \({\mathcal{K}}\) to have a symmetric restriction to a dense linear manifold of a closed subspace \({\mathcal H \subset \mathcal K}\). A large class of examples of such semigroups consisting of the compression of the unitary group generated by the operator of multiplication by the independent variable in \({\mathcal {K} := \oplus _{i=1} ^n L^2 (\mathbb {R})}\) to certain model subspaces of the Hardy space of n?compenent vector valued functions which are analytic in the upper half plane is presented.     

General compactly supported solution of an integral equation of the convolution type

We find the general form of solutions of the integral equation ∫k(t ? s)u₁(s) ds = u₂(t) of the convolution type for the pair of unknown functions u₁ and u₂ in the class of compactly supported continuously differentiable functions under the condition that the kernel k(t) has the Fourier transform \(\widetilde {{P_2}}\), where \(\widetilde {{P_1}}\) and \(\widetilde {{P_2}}\) are polynomials in the exponential e^iτx, τ > 0, with coefficients polynomial in x. If the functions \({P_l}\left( x \right) = \widetilde {{P_l}}\left( {{e^{i\tau x}}} \right)\), l = 1, 2, have no common zeros, then the general solution in Fourier transforms has the form U_l(x) = P_l(x)R(x), l = 1, 2, where R(x) is the Fourier transform of an arbitrary compactly supported continuously differentiable function r(t).     

Abstract Coherent State Transforms Over Homogeneous Spaces of Compact Groups

This paper presents theoretical aspects of a unified generalization for the abstract theory of coherent state/voice transforms over homogeneous spaces of compact groups using operator theory. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and \(\mu \) be the normalized G-invariant measure on G/H associated to the Weil’s formula with respect to the probability measures of G, H. Let \((\pi ,\mathcal {H}_\pi )\) be a continuous unitary representation of G with non-zero mean over H. In this article, we introduce the generalized notion of coherent state/voice transform associated to \(\pi \) on the Hilbert function \(L^2(G/H,\mu )\). We then study basic analytic properties of these transforms.     

Very Hyperbolic Polynomials

A real polynomial in one variable is hyperbolic if it has only real roots. A function f is a primitive of order k of a function g if f ^(k) = g. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. In the paper, we prove a property of the domain of very hyperbolic polynomials and describe this domain in the case of degree 4.     

The classification of finite groups by using iteration digraphs

A digraph is associated with a finite group by utilizing the power map f: G → G defined by f(x) = x^kfor all x ∈ G, where k is a fixed natural number. It is denoted by γG(n, k). In this paper, the generalized quaternion and 2-groups are stud- ied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2-group are determined for a 2-group to be a generalized quaternion group. Further, the classification of two generated 2-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed.     

Quaternion <Emphasis Type="Italic">H</Emphasis>-type group and differential operator Δ<Subscript><Emphasis Type="Italic">λ</Emphasis></Subscript>

We study the relations between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q-holomorphic functions is considered. The precise form of Cauchy-Szegö kernel and the orthogonal projection operator is obtained. The fundamental solution for the operator Δ_λ is found.     

Polynomials defining many units

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order n, define a unit in the integral group ring for infinitely many positive integers n. We show that this happens if and only if the polynomial defines generic units in the sense of Marciniak and Sehgal. We also classify the polynomials with integral coefficients which provides units when evaluated on n-roots of a fixed integer a for infinitely many positive integers n.     

Fractal Perturbation of Shaped Functions: Convergence Independent of Scaling

In this paper, we introduce a new class of fractal approximants as a fixed points of the Read–Bajraktarevi? operator defined on a suitable function space. In the development of our fractal approximants, we used the suitable bounded linear operators defined on the space \({\mathcal {C}}(I)\) of continuous functions and \(\alpha \)-fractal functions. The convergence of the proposed fractal approximants towards the continuous function f does not need any condition on the scaling vector. Owing to this reason, the proposed fractal approximants approximate the function f without losing their fractality. We establish constrained approximation by a new class of fractal polynomials. In particular, our constrained fractal polynomials preserve positivity and fractality of the original function simultaneously whenever the original function is positive and irregular. Calculus of the proposed fractal approximants is studied using suitable bounded linear operators defined on the space \({\mathcal {C}}^r(I)\) of all real-valued functions on the compact interval I that are r-times differentiable with continuous r-th derivative. We identify the IFS parameters so that our \(\alpha \)-fractal functions preserve fundamental shape properties such as monotonicity and convexity in addition to the smoothness of f in the given compact interval.     

Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \({\mathbb R^{2}}\) to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.