The Smoothnes and Dimension of Fractal Interpolation Functions

ＴｈｅＳｍｏｏｔｈｎｅｓａｎｄＤｉｍｅｎｓｉｏｎｏｆＦｒａｃｔａｌＩｎｔｅｒｐｏｌａｔｉｏｎＦｕｎｃｔｉｏｎｓ＊ＣｈｅｎＧａｎｇａｂｓｔｒａｃｔ．Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｉｎｖｅｓｔｉｇａｔｅｔｈｅｓｍｏｏｔｈｎｅｓｓｏｆｎｏｎ－ｅｑｕｉｄｉｓｔ...     

On a Class of Fractal Interpolation Functions

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On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)?B(t)y(t) = Δ(t) f(t) on an interval ${\mathcal{I}=[a,b) }$ with the regular endpoint a. It is assumed that the deficiency indices n _±(T _min) of the minimal relation T _min associated with this system in ${L^2_\Delta(\mathcal{I})}$ satisfy ${n_-(T_{\rm min})\leq n_+(T_{\rm min})}$ . We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size ${N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}$ . Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform ${V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}$ where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension ${\tilde{T}}$ of T _min induced by the boundary problem is unitarily equivalent to the multiplication operator in L ²(Σ). Hence the multiplicity of spectrum of ${\tilde{T}}$ does not exceed N _Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.     

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...     

Fourier–Jacobi periods and the central value of Rankin–Selberg L-functions

In this paper, we propose a conjectural formula, relating the Fourier–Jacobi periods of automorphic forms on U(n)×U(n) and the central value of some Rankin–Selberg L-function. This can be viewed as a refinement of the Gan–Gross–Prasad conjecture for unitary groups. We then use the relative trace formula technique to prove this conjectural formula in some cases. We also have give applications to the conjecture of Ichino–Ikeda and N. Harris on the Bessel period of automorphic forms on unitary groups.     

Fourier–Jacobi periods of classical Saito–Kurokawa lifts

Kohnen–Skoruppa (Invent Math 95(3): 541–558, 1989) proved a formula for the ratio of the Petersson inner products of the half integral weight modular form and its Saito–Kurokawa lifting. We give an interpretation of this formula in the framework of the refined Gan–Gross–Prasad conjecture. This provides us with an example of the refined Gan–Gross–Prasad conjecture for the nontempered representations.     

Jacobi Polynomial Estimates and Fourier-Laplace Convergence

ＪａｃｏｂｉＰｏｌｙｎｏｍｉａｌＥｓｔｉｍａｔｅｓａｎｄＦｏｕｒｉｅｒ－ＬａｐｌａｃｅＣｏｎｖｅｒｇｅｎｃｅＧａｖｉｎＢｒｏｗｎ（ＴｈｅＵｎｉｖｅｒｓｉｔｙｏｆＡｄｅｌａｉｄｅ，Ａｄｅｌａｉｄｅ）ＳｏｕｔｈＡｕｓｔｒａｌｉａ，５００５ＷａｎｇＫｕｎｙ...     

On the mean convergence of Fourier–Jacobi series

The convergence of Fourier–Jacobi series in the spaces L _p,A,B is studied in the case where the Lebesgue constants are unbounded.     

Fourier–Jacobi Type Spherical Functions for Discrete Series Representations of Sp(2, R)

In this paper we define a kind of generalized spherical functions on Sp(2, R). We call it Fourier–Jacobi type, since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, R).     

Construction and Dimension Analysis for a Class of Fractal Functions

In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.     

New Exceptional Sets and Convergence of the Square Partial Sums of Walsh–Fourier Series

For double Walsh–Fourier series and with f ∈ L~2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator S_(N(x,y))f(x, y).Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio ■≌2~(n_0) for all x and y, where n_0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n_0, are then extended to L~r, 1 r 2.We give an application to the family N(x, y) = λxy on [0, 1) × [0, 1), any λ 10.     

Potential operators associated with Jacobi and Fourier–Bessel expansions

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier–Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those 1≤p,q≤∞$1\le p,q\le \infty$, for which the potential operators are of strong type (p,q)$(p,q)$, of weak type (p,q)$(p,q)$ and of restricted weak type (p,q)$(p,q)$. These results may be thought of as analogues of the celebrated Hardy–Littlewood–Sobolev fractional integration theorem in the Jacobi and Fourier–Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier–Bessel expansions.     

Uniqueness of Fourier–Jacobi models: The Archimedean case

We prove uniqueness of Fourier–Jacobi models for general linear groups, unitary groups, symplectic groups and metaplectic groups, over an Archimedean local field.     

Fourier–Jacobi expansions in Morrey spaces

In this paper we obtain a characterization of the convergence of the partial sum operator related to Fourier–Jacobi expansions in Morrey spaces.     

Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

We introduce a class of matrix-valued functions W called “L²- regular”. In case W is J-inner, this class coincides with the class of “strongly regular J-inner” matrix functions in the sense of Arov–Dym. We show that the class of L²-regular matrix functions is exactly the class of transfer functions for a discrete-time dichotomous (possibly infinite-dimensional) input-state-output linear system having some additional stability properties. When applied to J-inner matrix functions, we obtain a state-space realization formula for the resolvent matrix associated with a generalized Schur–Nevanlinna–Pick interpolation problem. Communicated by Daniel Alpay Submitted: August 20, 2006; Accepted: September 13, 2006     

Interpolation of Functions from Besov-type Spaces on Gauß–Chebyshev Grids

In this paper, we give a unified approach to error estimates for interpolation on Gauß–Chebyshev grids for functions from certain Besov-type spaces with dominating mixed smoothness properties.