Approximation of smooth functions by weighted means of n

A new estimate for the degree of approximation of a function class by means of its Fourier series has been determined. Here, we extend the results of Singh and Mahajan which in turn generalize the result of Lal and Yadav Some corollaries have also been deduced from our main theorem.

The degree of approximation of a function belonging to various classes using different summability method has been determined by several investigators like Khan [ 12 ], V. Mishra and L. Mishra [ 3 ], Mishra et al. Summability of Fourier series is useful for engineering analysis, for example, [ 11 ]. Recently, Mursaleen and Mohiuddine [ 12 ] discussed convergence methods for double sequences and their applications in various fields.

Mishra et al. Analysis of signals or time functions is of great importance, because it conveys information or attributes of some phenomenon. The engineers and scientists use properties of Fourier approximation for designing digital filters. Especially, Psarakis and Moustakides [ 21 ] presented a new based method for designing the finite impulse response FIR digital filters and got corresponding optimum approximations having improved performance.

We also discuss an example when the Fourier series of the signal has Gibbs phenomenon. For a -periodic signalperiodic integrable in the sense of Lebesgue. Then the Fourier series of is given by with th partial sum called trigonometric polynomial of degree or order and given by The conjugate series of Fourier series 1 is given by with th partial sum. Let and denote two given moduli of continuity such that Let denote the Banach space of all -periodic continuous functions defined on under the sup-norm.

The space where includes the space. For some positive constantthe function space is defined by with norm defined by where and are increasing functions ofwith the understanding that. Clearly is a Banach space which decreases as increases; that is, Let be a given infinite series with the sequence of th partial sums.

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The series is said to be summable to the sum if exists and is equal to. An infinite series is said to be summable to if The transform is defined as the th partial sum of summability and we denote it by. If then the infinite series is said to be summable to.

The transform of the transform defines the transform of the partial sums of the series ; that is, the product summability is obtained by superimposing summability on summability. Thus, if where denotes the transform ofthen the series with the partial sums is said to be summable to the definite number and we can write The transform of the transform defines product transform and denote it by.

We note thatand are also trigonometric polynomials of degree or order. Remark 1.Metrics details. The smooth approximation and weighted energy estimates for delta 6-convex functions are derived in this research. In the recent decade, the study of convex functions and convex sets has developed rapidly because of its use in applied mathematics, specially in non-linear programming and optimization theory.

Furthermore, the elegance shape and properties of a convex function develop interest in studying this branch of mathematics.

But the classical definitions of convex function and convex set are not enough to overcome advanced applied problems. In the last few years, many efforts have been made on generalization of the notion of convexity to meet the hurdles in advanced optimization theory.

Among many generalizations, some are quasi convex [ 1 ], pseudo convex [ 2 ], logarithmically convex [ 3 ], n-convex [ 4 ], delta convex [ 5 ], s-convex [ 6 ], h-convex [ 7 ], mid convex [ 8 ] and [ 9 — 14 ]. The weighted energy estimates for the convex function and 4-convex function are derived in [ 15 ] and [ 16 ].

These estimates are important in hedging strategies in finance [ 17 ]. In the present paper, we deal with a delta 6-convex function. We derive some basic properties of the delta 6-convex function under certain conditions. Moreover, we approximate an arbitrary delta 6-convex function by smooth ones and derive weighted energy estimates for the derivative of delta 6-convex function.

The mollification of an arbitrary function is very well explained in the book by Evans [ 19 ]. Let f be a delta 2-convex function, 4-convex function, and 6-convex function.

Weighted approximations by means of entire functions of exponential type

Now we give the statement of our main theorem. Let f x be an arbitrary delta 6- convex function over the interval I.

Alsolet f x be delta 4- convex and delta 2- convexthen the following holds :. We come to the following result of Hussain, Pecaric, and Shashiashvili [ 15 ]. The results of 4-convex functions are established in [ 16 ].

Also f x is delta 4- convex as well as delta 2- convex. Then the following energy estimate is valid :. To prove Theorem 3. Using the integration by parts formula and making use of condition 1. Now take the first integral of 3.

Using the integration by parts formula and condition 1. Now take the first and the second integrals on the right-hand side of the latter expression. Proceeding in the similar way and using condition 1. Using Theorem 2.

Now, using 2.

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Substituting 3. Then we have. Now, writing inequality 3. From the result 2. Definetti, B.

Approximation of functions

Pures Appl. Mangasarian, O. SIAM J. Control 3— Mathematics in Science and Engineering, vol.Malczewskiego 20a, Radom, Poland. Modelling of the contact between moving rough surfaces allows a better understanding of friction and wear mechanisms, which can be used in engineering solutions. This issue has been examined using a number of approaches. The statistical type of a contact model is still the most popular model used in rough surfaces contact. This statement does not mean that it is the best solution of contact rough surfaces.

Instead of using the complete roughness data, only probability density function is used. This function means the probability of the asperity with the height between and.

The first well-known statistical model was introduced by Greenwood and Williamson [ 1 ] GW. They joined a statistical process with a classical Hertzian contact to deal with the rough surfaces contact. They adopted the following assumptions: the asperity height distribution is Gaussian, asperity contact is modelled by the Hertzian spherical contact theory, the asperity tip radius is assumed constant, and adhesion contact between asperities is ignored. A rough surface was described by three parameters: standard deviation of asperity height distribution, average asperity summit radius of curvature, and areal asperity density.

This model has been widely accepted and developed by numerous researchers. Some interesting review articles in the frame of dry friction are written by Bhushan [ 23 ], Buczkowski and Kleiber [ 4 ], and Jedynak and Sulek [ 5 ]. They showed that the contact between two rough surfaces can be modelled by a contact between an equivalent single rough surface and a flat one.

The equivalent rough surface is characterized by an asperity curvature and the peak-height distribution of the equivalent surface. They used the simple formula for a standard deviation of the statistical distribution as the square root of the sum of squares of the standard deviations of asperity height distributions on the two surfaces.

The GT model gained large popularity in the field of elastohydrodynamic analysis, and all the following papers refer to [ 6 ]. The most frequently cited equations given by the GT model are for the following asperity contact area: and load carried by the following asperities: where —roughness parameter, —nominal contact area,—Stribeck oil film parameter, first defined by Stribeck [ 36 ] as—effective elastic modulus, and—statistical functions introduced to match the assumed Gaussian distribution of asperities.

A parameter, ratio of film thickness to the composite roughness of the contiguous surfacesis used to ascertain any boundary contributions, which occur because of asperity interactions. It is a very important parameter for EHD analysis because interruptions in a coherent film may occur when [ 7 ].

At this point, it is worth noting that GT equations are only valid for surfaces which have a Gaussian distribution of asperities. A lot of real engineering surfaces do not meet these requirements. For example, a typical analysis of the compression ring of a piston against the cylinder bore is not correct if we use a Gaussian distribution.

Spencer et al. This geometry provides a smooth surface to allow for hydrodynamic film buildup between the piston rings and the cylinder liner surface. On the basis of these facts we can definitely say that the mentioned surfaces do not have a Gaussian distribution of asperities ringcylinder. On the other hand, cam-tappet contact is a good example for using the mentioned distribution.

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Both surfaces are well known to follow Gaussian topography. This kind of conjunction is the most loaded contact in the valve train mechanism. Jedynak and Sulek [ 5 ] showed that some kinds of galvanic coatings have also a Gaussian distribution of asperities.In practice, the necessity to approximate functions arises in various situations when it is needed to replace a function by a smoother function or by one which is simpler and more convenient in computations, or when it is required to establish a functional dependence on the basis of experimental data, etc.

Their widespread application as approximating sets is due, in particular, to the fact that it is in principle possible to approximate any continuous function by means of algebraic of trigonometric polynomials with any degree of accuracy specified in advance. The accuracy of the approximation can be increased at the expense of increasing the degree of the polynomial; however, this complicates the approximating process and increases the difficulties of applying it.

In practice one takes subspaces of algebraic or trigonometric polynomials of a fixed degree as approximating sets and one aims at obtaining the accuracy required while keeping the degrees of the polynomials used as low as possible.

A more general, and at the same time more flexible, approximation tool is obtained by considering generalized polynomials.

In many problems splines cf. Spline have been found to be more natural and more convenient from the viewpoint of computation than classical polynomials. Splines in two or more variables are similarly defined. Non-periodic functions defined on the whole real axis can also be approximated by entire functions of exponential type. Interpolation with splines has its peculiarities connected with the choice of the interpolation nodes and with the boundary conditions ensuring the existence and uniqueness of the interpolating spline.

In the approximation of functions an important role is also played by linear methods constructed on the basis of an expansion of the function in a Fourier series with respect to an orthogonal system.

In particular, in the periodic case a widely used approximation tool consists of Fourier sums, and their various means the case of the trigonometric orthogonal system see Approximation of functions, linear methods. The investigation and estimation of the error of approximation is important from the practical point of view and at the same time is the branch of the approximation of functions that is richest in ideas.

More precisely, the development of methods dealing with estimating the error, the investigation of this dependence on the smoothness of the approximated function, as well as the study and comparison of the approximation properties of various approximation methods, have led to the creation of the approximation theory of functions, one of the most rapidly developing branches of mathematical analysis.

The foundation of the approximation theory of functions was laid in the papers of P. Chebyshev in — see [1] on best uniform approximation of continuous functions by polynomials and rational fractions and in the work of K. Similar assertions hold in the case when the measure of the error of approximation is defined by an integral metric, as well as in the case of functions of several variables. An important stage in the development of the approximation theory of functions is connected with the names of Ch.

Jackson, and S. Bernstein [S. This yields a new, constructive characterization of continuous and differentiable functions. In the first third of the 20th century this kind of problem was dominating approximation theory; for this reason this field was also thought of as the constructive theory of functions. In the s and s there appeared papers by A.

Kolmogorov, J. Favard and S. Nikol'skii which initiated a new direction of research connected with the approximation of classes of functions by finite-dimensional subspaces and with obtaining accurate estimates of the error, using difference-differential properties defining the class.

The objective is to obtain the quantities. Research in this direction, based both on the properties of specific approximation methods and on advanced results of functional analysis, turned out to be also very fertile in ideas, because, by proceeding in this way, new facts about the mutual connections between many extremal problems of various types were discovered, making it possible to obtain deep and delicate relations in the theory of functions.In mathematicsleast squares function approximation applies the principle of least squares to function approximationby means of a weighted sum of other functions.

The best approximation can be defined as that which minimises the difference between the original function and the approximation; for a least-squares approach the quality of the approximation is measured in terms of the squared differences between the two. A generalization to approximation of a data set is the approximation of a function by a sum of other functions, usually an orthogonal set : [1].

For example, the magnitude, or norm, of a function g x over the interval [a, b] can be defined by: [2]. Substituting function f n into these equations then leads to the n -dimensional Pythagorean theorem : [4]. Due to the frequent difficulty of evaluating integrands involving absolute value, one can instead define. Furthermore, this can be applied with a theorem:. From Wikipedia, the free encyclopedia.

See also: Fourier series and Generalized Fourier series. Applied analysis Reprint of Prentice—Hall ed. Dover Publications. American Mathematical Society Bookstore. Fourier Analysis and Its Applications.

American Mathematical Society. Saville, Graham R. Wood Statistical methods: the geometric approach 3rd ed. Categories : Least squares Approximation theory.

Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.The second NPA is computed with the reduced amount of information by removing the last coefficient from the expansion of f at x 1. We assume that f is sufficiently smooth, e. Whether this is the case for a given function f is not necessarily known a priori, however, as illustrated by examples below it holds for many functions of practical interest.

In this case, further steps become relatively simple. We select a known function s having the two-sided estimates property with values s x i as close as possible to the values f x i.

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