UDC 519.652 517.518.85 About some generalizations trigonometric splines

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UDC 519.652 + 517.518.85

About some generalizations

trigonometric splines

Denysiuk V.P.

Dr. of Ps.-M.. Sciences, Professor, Kiev, Ukraine

National Aviation University

kvomden@nau.edu.ua

Annotation

Methods of constructing trigonometric fundamental splines with constant sign and sign-changing

convergence factors are given. An example and graphics illustrating the concepts of convergence and inter-

polation grids are given. Some methods of constructing constant-sign and sign-changing coefficients of

convergence of trigonometric splines are considered.

Keywords: trigonometric splines, sign-constant and sign-changing convergence factors, equivalent

infinitely small.

Introduction

Approximation, respectively representation, of an arbitrary known or unknown function through a

set of some special functions can be considered as a central topic of analysis. We will use the term "special

functions" to refer to classes of algebraic and trigonometric polynomials and their modifications; at the same

time, we believe that the classes of trigonometric polynomials also include trigonometric series. As a rule,

such special functions are easy to calculate and have interesting analytical properties [1].

One of the most successful modifications of algebraic polynomials are polynomial splines that are

stitched together from segments of these polynomials according to a certain scheme. The theory of polyno-

mial splines appeared relatively recently and is well developed (see, for example, [2], [3], [4] [5], etc.). The

advantages of polynomial splines include the fact that they can be given certain smoothness properties, as

well as their approximate properties [6]. The main disadvantage of polynomial splines, in our opinion, is

their piecemeal structure, which greatly complicates their use in analytical transformations.

Later it turned out [7], [8] that there are also modifications of trigonometric series whose sums de-

pend on several parameters and have the same properties as polynomial splines [9]; moreover, the class of

such modified series is quite broad and includes the class of polynomial periodic splines. This gave reason to

call the class of such series trigonometric interpolation splines.

Convergence of trigonometric series that determine trigonometric interpolation splines, provided by

convergence factors [7], [8], which have the order of decreasing

( )

(1 )r

−+

, (

0r>

) ; since these series co-

incide uniformly, they are trigonometric Fourier series with special coefficients. In this work, we will limit

ourselves to consideration of integer values of the parameter

(i.e. case

1,2,...r=

); note that trigonometric

splines of fractional powers (that is, of non-integer parameter values

) was considered in [10].

Trigonometric series that provide trigonometric interpolation splines can naturally be divided into two

components - even and odd. In turn, each of these components can be broken down into three more compo-

nents: low-frequency, medium-frequency, and high-frequency. These components can be considered together

with the coefficients

{ }

123

γγγ

Γ=

and

{ }

123

ηηη

Η=

, where the components of the vectors

and

real

numbers and at least one of the components

2323

,,,

γ γηη

not equal to 0. It is clear that with vectors

{ }

,0,0

Γ=

and

{ }

,0,0

Η=

we have a trigonometric interpolation polynomial. Note that with the help of

vectors

and

it is easy to model low-, medium-, and high-pass filters, which is important in the problems

of the theory of modeling and signal processing.

Trigonometric splines with vectors

{ }

1,1,1Γ=

and

{ }

1,1,1Η=

we will call simple. In the notation of

simple trigonometric splines, we will omit their dependence on vectors

and

Trigonometric interpolation splines assume a number of generalizations; below, we consider only

generalizations that relate to types of convergence factors.

The goal of the work.

1. Consideration of methods for constructing one- and two-parameter convergence factors of trig-

onometric fundamental simple splines.

Main part.

General methods of constructing trigonometric fundamental splines with constant coefficients of

convergence.

Let

- class

-periodic functions having a completely continuous derivative of order

1r−

(

1,2, ...r=

), and the derivative is of order

is a function of bounded variation; it is clear that

is a class of

continuous functions. A symbol

denote the class of piecewise constant functions with a finite number of

discontinuity points.

At a stretch

[

)

0,2π

we will consider grids

() ()

{}

I IN

N jj

∆=

, (

0,1I=

.)1(),1(

)1()0(

−=−= j

ππ

Let

also on

[

)

0,2π

given a periodic function

() W

ft∈

, and its values are known

() ()

{( }

I IN

j jj

f ft

in grid nodes

()I

∆

Let's introduce the concepts of stitching grids and interpolation of trigonometric splines. .

We will call the mesh a stitching mesh

()I

∆

, on which the polynomial analog of the trigonometric

spline of the 0th degree has discontinuities of the 1st kind of the jump type or the polynomial analogs of the

trigonometric splines of the 1st degree are stitched together; indicator

we will mark the stitching grids

through

, (

0,1I=

). Note that trigonometric splines of degree 0.1 were chosen for the reasons that in these

cases the points of discontinuity or splicing appear in the most relief.

We will call the grid an interpolation grid or an interpolation grid

()I

∆

, in the nodes of which interpola-

tion of the approximated function is carried out by trigonometric splines; indicator

interpolation grids will

be denoted by

, (

0,1I=

As we said before [11], the system

(,) W

st r t

−

∈

1,2, ... ,rN=

trigonometric fundamental splines on

an interpolation grid

()I

∆

meets the conditions:

( )

() ()

1, ;

,0, .

st r t kj



=≠



(

, 1, ...,kj N=

). (1)

Using a system of trigonometric fundamental splines

( )

() ,

st r t

1, 2, , ,kN=

trigonometric interpola-

tion spline can be written as:

() () ()

(,) (,)

I II

n kk

St rt f st rt

∑

. (2)

Note that in the definition of trigonometric fundamental splines we are talking about interpolation

grids. Next, stitch and interpolation grid indices, as well as vectors

and

we will enter splines in notation.

Parameter

, (

1,2, ...r=

), determines the order of decreasing convergence factors

(, )rj

. The most

natural as convergence multipliers are the constant multipliers :

0( , )

rj j



=



, (

1,2, ...j=

). (3)

It is clear that the parameter

, (

1, 2, ...r=

) determines the number of continuous derivatives of trigo-

nometric fundamental splines.

Note that otherwise, we considered trigonometric splines with sign-changing convergence factors of

the Riemann type [9]; in this work, we consider constant coefficients of convergence (3), the only property

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UDC519.652+517.518.85AboutsomegeneralizationstrigonometricsplinesDenysiukV.P.Dr.ofPs.-M..Sciences,Professor,Kiev,UkraineNationalAviationUniversitykvomden@nau.edu.uaAnnotationMethodsofconstructingtrigonometricfundamentalsplineswithconstantsignandsign-changingconvergencefactorsaregiven.Anexampleandgra...

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UDC 519.652 517.518.85 About some generalizations trigonometric splines

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