DISCRETE QUADRATIC SPLINES

In the present paper, we define a new class of discrete splines and study the existence, uniqueness, and convergence properties of discrete quadratic splines satisfying the Mean Averaging Condition.

considered error bounds for spllnes satisfying conditions involving functlonals using a different approach.
The existence, uniqueness and convergence properties of discrete cubic spllnes satisfying MAC have been studied by Dikshlt and Power [5].It may be observed that the approach used by Lyche [6] for defining discrete cubic splines is not capable of providing the corresponding definition for discrete splines of degree less than three.
In the present paper, we consider certain interpolatory discrete splines of degree two and study the existence, uniqueness and convergence properties of such spllnes satisfying MAC.
Let a mesh on [a,b] be defined by P: a Xo < Xl <...< Xn b with x i xi-I Pi for i 1,2, n and p mx Pi" The class D(2,P,h) of discrete quadratic splines for h > 0 is the set of all continuous functions s(x,h) such that for i 1,2,...,n the restriction s i of s(x,h) on [Xi_l, x i] is a polynomial of degree 2 or less and S.S. RANA Dh{l} s i(xi-h) Dh{l} si+ l(xi+ h); i 1,2,...,n-I (2.1) where the central difference operator Dh{l}f(x) (f(x+h)-f(x-h))/2h (see Lyche [7]).
For convenience, we set for each i, In the present section, we shall study the following: PROBLEM 2.1.Let f be a b-a periodic locally integrable function with respect to nonnegatlve measure dg.
We investigate restrictions on g such that there exists a unique b-a periodic discrete quadratic spline s(x,h) in D(2,P,h) satisfying the MAC: x i f (f(x) s(x,h))dg 0, i 1,2 n. (2.2)

xi-
We shall answer the Problem 2.1 in the following, THEOREM 2.1.Suppose that the support of the restriction of g over [xi_I, x i] is included in Ji and its total variation over Ji is positive.Then there exists a unique discrete periodic quadratic spllne s(x,h) in D(2,P,h) satisfying MAC (2.2).

REMARK. 2.1.
In the case in which h O, Theorem 2.1 gives the corresponding result for continuous quadratic spline interpolation under MAC studied in [3].It is also interesting to note that condition (2.2) reduces to some other interpolating conditions by suitable choice of g(x).
PROOF OF THEOREM 2.1.
In order to prove Theorem 2.1, we shall show that the system of equations (2.7) has a unique set of solutions.Since the support of the restriction of g over [Xi_l,Xl] is included in Ji' we observe that the coefficients of Mi_I, M i and Mi+ are all nonnegative by virtue of the hypothesis that the total variation of g over Ji is positive.Further, the excess of the positive value of the coefficient of M i over the sum of the positive values of the coefficients of Mi_ and Mi+ is tl(h 2[(Pl-2h)Rl+ + (Pi+l-2h) Ri] (2.8) where R i di(l,l) with h 0. Clearly ti(h) > 0 under the conditions of Theorem 2.1.Thus, the coefficient matrix of the system of equations (2.7) is diagonally dominant and hence invertible.This completes the proof of Theorem 2.1.

ANOTHER INTERPOLATION PROBLEM.
In this section of the paper, we shall answer problem 2.1 with MAC replaced by the interpolatory condition: It may be mentioned that it is possible to deduce the Interpolatory condition (3.1) from the MAC (2.2) by a suitable choice of dg.Further, it is easy to see that the conditions of Theorem 2.1 are not satisfied for the interpolatory condition (3.1).To answer Problem 2.1 for the interpolatory condition (3.1), we use (3.1) where fi f(xi) for all i.Eliminating we have the following system of equations: from the equations (3.2) and (2.5), PiMi_l + (Pi+Pi+1)Mi + Pi+lMi+l 2(fi+l-fi_l). (3.3) Clearly the system of equations (3.3) is not diagonally dominant.However, we may rewrite (3.3) in the following form, pi(Mi_l+Mi) 2(fl-fi_l)+ Pi+l(Mi+Mi+l 1,2, n.

2(fi+l-fi) 0 (3.4)
Assuming that n is odd and s,f are b-a periodic, we get from (3.4) (of.Sharma and Tzimbalario [3]) Thus, we have proved the following: THEOREM 3.1.
For every b-a periodic function f, there exists a unique discrete periodic quadratic spline s(x,h) in D(2,P,h) satisfying the interpolatory condition (3.1) if and only if n is odd.

ERROR BOUNDS.
In this section, we shall estimate the error function e(x) s(x,h) f(x) for the spllne interpolant of Theorem 2.1.over the discrete interval [a,b] h which is the intersection of [a,b] with Rha {a + Jh: J is an integer a is real }.
For a function f defined over [a,b]h, w(f,p) denotes the discrete modulus of continuity.
Throughout this section for convenience, we assume that P is a uniform mesh so that x i xi_ p and g(x + p)-g(x) K (a constant) We first begin with the following result, (see Lyche [7]) {i i for some h > 0 and LEMMA 4.
{al}i=0, {bl}i= 0 are given sequences of nonnegative real numbers such that n m iai iD bi' then for any real valued function f defined on the discrete interval [a,b]h, we have We now proceed to estimate the error bounds at the points xl-h for all i.Observing that, A i(j,k) A(j,k) and H i H for uniform mesh, we shall prove the following, LEMMA 4.2.Let s(x,h) be the interpolating spllne of Theorem 2.1.If g is a step function with Jumps of only at the points of [a,b]h, then where t(h) is some positive function of h and Jl(A,h) is some positive function of A(J,k) and h.
PROOF OF LEMMA 4.2.It may be observed that the system of equations (2.7) may be written as , where A(h) is the coefficient matrix and M(h) and F(h) are single column matrices (Mi(h)) and (Fi(h)) respectively.Further, it may be seen that the row max norm Using the fact that g is a step function with Jumps of at the points of [a,b]h, we wish to apply Lemma 4.1 to estimate the right hand side of (4.5).For this, we notice that Dl}f(xl-h) [xl-2h,xl]f and that the sum of the coefficients of the Dl}f(xi-h)'son the right hand side of (4.x i Thus, rearranging the terms suitably and then "applylng Lemma 4.1, we see that the row max norm of the matrix on the right hand side of (4.5) does not exceed P Jl(A'h) W(Dh{l}f, p) where Jl(A,h)is some positive function of A(J,k) and h. (4.2) now follows when we appeal to (4.4).We are now set to prove the following:

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

First
Round of Reviews March 1, 2009