An application of the sampling theorem

Let B_σ,p,1 <-p<-∞, be the set of all functions from L₈(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈B_σ,p, 1 $$f(x) = \mathop \Sigma \limits_{j \in z} f(j\pi /\sigma )\tfrac{{sin\sigma (x - j\pi /\sigma )}}{{\sigma (x - j\pi /\sigma )}}, \sigma > 0$$ It is well known that the Shannon sampling theorem plays an important role in many applied fields [1,2,7]. In this paper we give an application of sampling theorem to approximation theory.     

An application of a general sampling theorem

An extension of the sampling theorem to multi-band signals is discussed and an application to the compression of speech outlined.     

An application of the Riemann-Roch theorem

Applying the Riemann-Roch theorem,we calculate the dimension of a kind of mero- morphicλ-differentials' space on compact Riemann surfaces.And we also construct a basis of theλ-differentials' space.As the main result,the Cauchy type of integral formula on compact Riemann surfaces is established.     

Frames and sampling theorem

The sampling problem on a closed subspace V_o of L²(ℝ) is studied. For the regular sampling, a necessary and sufficient condition on the sampling theorem is presented and the sampling problem proposed by Walter is completely solved. Moreover, a representation formula for the derived function is given. The irregular sampling is studied. The results improve the ones of Walter and Liu’s. Project supported by the National Natural Science Foundation of China (Grant No. 16971047).     

An application of a theorem of Ziegler

The aim of this note is to prove that an artin algebra (resp. a finite dimensional algebra over an algebraically closed field) all of whose algebraically compact (resp. -algebraically compact) indecomposable modules are finitely generated must be of finite-representation type. The result is derived with the aid of a theorem of Ziegler.     

Multivariate irregular sampling theorem

In this paper,we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result,we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.     

On the sampling theorem for wavelet subspaces

In [13], Walter extended the classical Shannon sampling theorem to some wavelet subspaces. For any closed subspace V₀/L² (R), we present a necessary and sufficient condition under which there is a sampling expansion for everyf V₀-Several examples are given.     

An application of Kronecker's theorem to rational functions

Dedicated to the memory of S.K. Pichorides     

An application of the Krasnoselskii theorem to systems of algebraic equations

Based on the Krasnoselskii theorem, we study the existence, multiplicity and nonexistence of positive solutions of general systems of nonlinear algebraic equations under superlinearity and sublinearity conditions. Systems of nonlinear algebraic equations often arise from studies of differential and difference equations. Our results significantly extend and improve those in the literature. A?number of examples and open questions are given to illustrate these results.     

An application of least cost acceptance sampling schemes

Summary A technique is described for determining most economical single sample inspection schemes in the situation where costs associated with inspection and failure to detect defective items can be estimated. A computational procedure is outlined, and a comparison made between the schemes determined by this method, and those previously in use. A least-cost decision rule is also derived for sequential sampling schemes.

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Zusammenfassung Ein Verfahren zum Aufstellen wirtschaftlichster einfacher Stichprobenverfahren wird für den Fall beschrieben, in dem die Kosten für das Überprüfen und das Nichtentdecken fehlerhafter Stücke geschätzt werden können. Es wird ein numerisches Verfahren aufgezeigt und ein Vergleich zwischen dem hier beschriebenen Verfahren und den früher verwendeten durchgeführt. Eine Entscheidungsregel für geringste Kosten wird ebenso für sequentielle Stichprobenpläne hergeleitet.

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Now with the Operations Research Section of B.O.A.C., Comet House, London Airport, Hounslow, Middx., England.

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Vorgel. v.:M. Sasieni     

Nonperiodic sampling and the local three squares theorem

This paper presents an elementary proof of the following theorem: Given {r _j } _j ^m =1 with m=d+1, fix and let Q=[−R, R]^d. Then any f∈ L²(Q) is completely determined by its averages over cubes of side r_j that are completely contained in Q and have edges parallel to the coordinate axes if and only if r_j/r_k is irrational for j≠k. Whend=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials. The third author was supported by NSF Grant DMS9500909 and gratefully acknowledges the support of Bill Moran and the Mathematics Department of Flinders University of South Australia where certain portions of this work were completed. The authors thank the referee for valuable comments and suggestions.     

Optimal investment policy: An application of Stokes' theorem

An application of the Stokes' theorem is illustrated by solving the two-state problem, with inequality constraints, of Dobell and Ho concerning the optimal investment of resources. Whenever applicable, the Stokes' theorem approach seems to be elegant and parsimonious.     

Multiwavelet sampling theorem in Sobolev spaces

This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (H s (R) , H-s (R)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ H s (R) with s 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L 2 (R), whose Sobolev exponents are greater than 1 /2.     

A sampling theorem related to the Wilson transform

In this paper, we establish a sampling theorem related to the nonterminating version of the Wilson polynomials and we give a generalization of the de Branges–Wilson integral.     

General sampling theorem using contour integral

We present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions

     

New trigonometric sums by sampling theorem

We use a sampling theorem associated with second-order discrete eigenvalue problems to derive some trigonometric identities extending the results of Byrne and Smith [G.J. Byrne, S.J. Smith, Some integer-valued trigonometric sums, Proc. Edinburg Math. Soc. 40 (1997) 393-401]. We derive both integral and non-integral valued trigonometric sums. We give illustrative examples involving representations of the trigonometric sums and in an integral-valued polynomial in (2n+1) of degree 2m, .     

A sampling theorem on homogeneous manifolds

We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a manifold is uniquely determined by its values on some discrete sets of points.

The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.