Antonio G. García

Orthogonal Sampling Formulas: A Unified Approach

This paper intends to serve as an educational introduction to sampling theory. Basically, sampling theory deals with the reconstruction of functions (signals) through their values (samples) on an appropriate sequence of points by means of sampling expansions involving these values. In order to obtain such sampling expansions in a unified way, we propose an inductive procedure leading to various orthogonal formulas. This procedure, which we illustrate with a number of examples, closely parallels the theory of orthonormal bases in a Hilbert space. All intermediate steps will be described in detail, so that the presentation is self-contained. The required mathematical background is a basic knowledge of Hilbert space theory. Finally, despite the introductory level, some hints are given on more advanced problems in sampling theory, which we motivate through the examples.

New sampling formulae for the fractional Fourier transform

In this note we obtain two new sampling formulae for reconstructing signals that are band limited or time limited in the fractional Fourier transform sense. In both cases, we use samples from both the signal and its Hilbert transform, but each taken at half the Nyquist rate.

A distributional study of discrete classical orthogonal polynomials

Abstract For the sequences of discrete classical orthogonal polynomials (Charlier, Meixner, Hahn) we can find a functional u, which satisfies the difference distributional equation Δ(φu) = ψu where φ and ψ are polynomials of degrees ⩽2 and 1 respectively. From this it follows that these polynomials are solutions of a second-order difference equation; also, they can be represented by a Rodrigues-type formula. The sequence of difference polynomials derived from them constitutes an orthogonal polynomial sequence. Their weight functions satisfy a Pearson-type difference equation. A structure relation (φΔPn+1 = anPn+2 + bnPn+1 + cnPn) also holds.

Improving treatment adherence in your patients with schizophrenia: the STAY initiative.

Partial and non-adherence to medication is a common problem in schizophrenia, leading to an increased risk of relapse, increased likelihood of hospitalization and poorer long-term outcomes. In contrast, continuous medication in the treatment of schizophrenia is associated with positive outcomes, including improved clinical status, improved quality of life and functioning, and reduced risk of relapse and rehospitalization. Strategies aimed at improving medication adherence are therefore key for patients to achieve their treatment goals. In an attempt to address the issues of partial/non-adherence to antipsychotic medication in schizophrenia, a group of psychiatrists convened to discuss and develop a set of principles aimed at helping patients adhere to their medication. These principles were then refined and developed into the STAY (the Six principles to improve Treatment Adherence in Your patients) initiative following presentation to a wider group of psychiatrists from across Europe. This manuscript summarizes these principles and explains the rationale for their selection. These principles are: (1) recognizing that most patients with schizophrenia are at risk of partial/non-adherence at some time during the course of their illness; (2) the benefits of a good therapeutic alliance for identifying potential adherence issues; (3) tailored treatment plans to meet an individual’s needs, including the most suitable route of delivery of antipsychotic medication; (4) involving family/key persons in care and psychoeducation of the patient, assuming the patient agrees to this; (5) ensuring optimal effectiveness of care; and (6) ensuring continuity in the care of patients with schizophrenia. The application of these six principles should help to raise awareness of and address poor patient adherence, as well as generally improving care of patients with schizophrenia. In turn, this should lead to improved overall clinical outcomes for patients receiving long-term treatment for schizophrenia.

The discrete Kramer sampling theorem and indeterminate moment problems

In this paper we propose candidates to be the kernel appearing in the discrete Kramer sampling theorem. These kernels arise either from orthonormal polynomials associated with indeterminate Hamburger or Stieltjes moment problems, or from the second kind orthogonal polynomials associated with the former ones. The sampling points are given by the zeros of the denominator in the Nevanlinna parametrization of the N-extremal measures. Explicit formulae are given associated with some cases where the Nevanlinna parametrization is known explicitly.

On the properties for modifications of classical orthogonal polynomials of discrete variables

Abstract We consider a modification of moment functionals for some classical polynomials of a discrete variable by adding a mass point at x = 0. We obtain the resulting orthogonal polynomials, identify them as hypergeometric functions and derive the second-order difference equation which these polynomials satisfy. The corresponding tridiagonal matrices and associated polynomials were also studied.

Oversampling in Shift-Invariant Spaces With a Rational Sampling Period

It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang-Fix conditions of order r, this formula also provides an approximation scheme of order r valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support.

Kramer’s Sampling Theorem With Discontinuous Kernels

AbstractKramer’s sampling theorem provides an algorithm for reconstructing a function ƒ, in the form % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

GENERALIZED IRREGULAR SAMPLING IN SHIFT-INVARIANT SPACES

f(t)=\int_{a}^{b}\ F(x)K(x,t)dx,\qquad {\rm for\ some}\ F\ \in\ L^{2}(a,b),