Intelligent Mathematics: Computational Analysis

Mathematics provides a complete language for describing systems and methods with rigor. So we are able to represent and manipulate knowledge in an abstract way and make computations that lead to useful conclusions for the real world.

Continuous functions are approximated by wavelet like operators. These preserve convexity and

r

-convexity and transform continuous probability distribution functions into probability distribution functions at the same time preserving certain convexity conditions. The degree of this approximation is estimated by presented Jackson type inequalities.

Shape-preserving properties of some naturally arising bivariate wavelet operators

B

n

are presented. Namely, let

$f\in C^{k}(\mathbb{R}\mathrm{ ^{2}}),\ k>0,\ r,s\geq 0$

all integers such that

r

 + 

s

 = 

k

. If

$$ \frac{\partial ^{r+s}f}{\partial x^{r}\partial y^{r}}(x,y)\geq 0, $$

then it is established, under mild conditions on

B

n

, that

$$ \frac{\partial ^{r+s}}{\partial x^{r}\partial y^{r}}B_{n}f(x,y)\geq 0, $$

also pointwise convergence of

B

n

(

f

) to

f

is given with rates through a Jackson type inequality. Related simultaneous shape-preserving results are also given for special type of wavelet operators

B

n

. This chapter relies on [89].

Multivariate probabilistic distribution functions are approximated by some naturally arising wavelet type operators involving a scale function. These transform multivariate distribution functions to multivariate distribution functions. The degree of this approximation is given by establishing some sharp Jackson type inequalities. This chapter relies on [90].

Let

$$ \varphi _{0}\left( x,y\right) :=\left\{ \begin{array}{l} 1,\text{ \ \ \ }x,y\geq 0 \\ 0,\text{ \ \ \ otherwise} \end{array} \right. $$

and

$F\left( x,y\right) $

be a continuous probability distribution function on ℝ

2

.

Then there exist linear wavelet type operators

$L_{n}\left( F,x,y\right) $

which are also distribution functions and where the defining them wavelet function is

$\varphi _{0}\left( x,y\right) $

. These approximate

$F\left( x,y\right) $

in the supnorm. The degree of this approximation is estimated by establishing a Jackson type inequality. Furthermore we give generalizations for the case of a wavelet function ≠ 

ϕ

0

, which is just any distribution function on ℝ

2

, also we extend these results in ℝ

r

,

r

 > 2. This chapter relies on [87].

Let

F

be a normed space and let

B

be a subspace of

F

. Assume that

$ L\colon F\rightarrow L_{\infty }(\Omega )$

,

$\Omega \subset \mbox{I\kern-.3468em R}^{m}$

, is a linear bounded operator and

M

(

L

) = {

f

 ∈ 

F

:

Lf

 ≥ 0 a.e. on Ω}. We establish some inequalities for best approximation of

f

 ∈ 

M

(

L

) by elements from

B

 ∩ 

M

(

L

). In the case when

L

is a differential operator and

F

is the Sobolev space

$ W_{p}^{\ell }(\Omega )$

we obtain Jackson type estimates for simultaneous approximation of

f

 ∈ 

M

(

L

) by multivariate polynomials and entire functions of exponential type from

M

(

L

). This chapter relies on [75].

Results concerning shape preserving weighted uniform approximation on the real line are presented. This chapter is based on [74].

In this chapter the integral of a function over a finite interval, is approximated by Jackson-type approximations that are non-positive linear functionals. Several important cases are treated, in which approximations are given with rates by using higher order moduli of smoothness. Real applications of these results might be, e.g., in Communications and Medical Imaging. This chapter relies on [70].

A discrete theory is presented for the best approximation in the ”gauges” sense. This chapter relies on [8].

In [249], A.Pinkus and O. Shisha introduced novel measures of size (“gauges”) of real functions of a real variable, continuous on [0, 1]. In their simplest form, these measures can be described roughly as follows.

In this chapter we study the smooth Picard singular integral operators on the line of very general kind. We establish their convergence to the unit operator with rates. The estimates are mostly sharp and they are pointwise and uniform. The presented inequalities involve the higher order modulus of smoothness. To prove optimality we apply mainly the geometric moment theory method. This chapter relies on [34].

In this chapter we study the smooth Picard singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the

L

p

norm, 1 ≤ 

p

 ≤ ∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the first modulus of continuity with respect to the uniform norm. The established Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the involved function. This chapter is based on [33].

In this chapter we continue with the study of smooth Picard singular integral operators on the line regarding their convergence to the unit operator with rates in the

L

p

norm,

p

 ≥ 1. The related established inequalities involve the higher order

L

p

modulus of smoothness of the engaged function or its higher order derivative. This chapter relies on [36].

In this chapter we study the very general fractional smooth Picard singular integral operators on the real line, regarding their convergence to the unit operator with fractional rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the involved function. Furthermore we present a fractional Voronovskaya type of result giving the fractional asymptotic expansion of the basic error of our approximation.

In this chapter, we study the type of Picard singular integral operators on ℝ

n

constructed by means of the nonisotropic

β

-distance and the

q

-exponential functions. The central role here is played by the concept of nonisotropic

β

-distance, which allows us to improve and generalize the results given for classical Picard and

q

-Picard singular integral operators. In order to obtain the rate of convergence we introduce a modulus of continuity depending on the nonisotropic

β

-distance with respect to the uniform norm. Then we give the definition of

β

-Lebesgue points depending on nonisotropic

β

-distance and a pointwise approximation result shown at these points. Futhermore, we present the global smoothness preservation property of these type of Picard singular integral operators and prove a sharp inequality. This chapter relies on [61].

In this chapter, we present a generalization of Gauss-Weierstrass operators based on

q

-integers using the

q

-integral and we call them

q

-Gauss- Weierstrass integral operators. For these operators, we obtain a convergence property in a weighted function space using Korovkin theory. Then we estimate the rate of convergence of these operators in terms of a weighted modulus of continuity. We also give optimal global smoothness preservation property of these operators. This chapter is based on [62].

High order differentiable functions of one real variable are approximated by univariate shift-invariant integral operators wavelet-like, and their generalizations. The high order of this approximation is estimated by establishing some Jackson type inequalities, involving the modulus of continuity of the

N

th order derivative of the function under approximation. At the end we give applications to Probability. This chapter is based on [28].

High order differentiated functions of several variables are approximated by multivariate shift-invariant convolution type operators and their generalizations. The high order of this approximation is determined by giving some multivariate Jackson-type inequalities, involving the first multivariate usual modulus of continuity of the

N

th order partial derivatives of the multivariate function to be approximated. This chapter follows [30].

The aim of this chapter is that by using the so-called max-product method, to associate to Cardaliaguet-Euvrard linear operator, a nonlinear neural network operator, for which a Jackson-type approximation order is obtained. In some classes of functions, the order of approximation is essentially better than the order of approximation of the corresponding linear operator. This chapter relies on [65].

We present here a generalized Shisha-Mond type inequality which implies a generalized Korovkin theorem. These are regarding the convergence with rates of a sequence of positive linear operators to the unit. This chapter is based on [39].

This is a quantitative study for the rate of pointwise convergence of a sequence of bounded linear operators to an arbitrary operator in a very general setting involving the modulus of continuity. This is accomplished via the Riesz representation theorem and the weak convergence of the corresponding signed measures to zero, studied quantitatively in various important cases. This chapter relies on [25].

Here we study very general stochastic positive linear operators induced by general positive linear operators that are acting on continuous functions. These are acting on the space of real differentiable stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related stochastic Shisha–Mond type inequalities of

L

q

-type 1 ≤ 

q

 < ∞ and corresponding stochastic Korovkin type theorems. These are regarding the stochastic

q

-mean convergence of a sequence of stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are produced with rates and are given via the stochastic inequalities involving the stochastic modulus of continuity of the

n

 − 

th

derivative of the engaged stochastic process,

n

 ≥ 0. The impressive fact is that the basic real Korovkin test functions assumptions are enough for the conclusions of our stochastic Korovkin theory. We give an application. This chapter is based on [38].

Here we study very general multivariate stochastic positive linear operators induced by general multivariate positive linear operators that are acting on multivariate continuous functions. These are acting on the space of real differentiable multivariate time stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we present related multidimensional stochastic Shisha–Mond type inequalities of

L

q

-type 1 ≤ 

q

 < ∞ and corresponding multidimensional stochastic Korovkin type theorems. These are regarding the stochastic

q

-mean convergence of a sequence of multivariate stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are given with rates and are shown via the stochastic inequalities involving the maximum of the multivariate stochastic moduli of continuity of the

n

th order partial derivatives of the engaged stochastic process,

n

 ≥ 0. The astonishing fact here is that basic real Korovkin test functions assumptions are enough for the conclusions of the multidimensional stochastic Korovkin theory. We give an application. This chapter relies on [40].

Here we present fractional Taylor type formulae with fractional integral remainder and fractional differential formulae, regarding the right Caputo fractional derivative, the right generalized fractional derivative of Canavati type ([126]) and their corresponding right fractional integrals.

In this chapter we study quantitatively with rates the weak convergence of a sequence of finite positive measures to the unit measure. Equivalently we study quantitatively the pointwise convergence of sequence of positive linear operators to the unit operator, all acting on continuous functions. From there we obtain with rates the corresponding uniform convergence of the latter. The inequalities for all of the above in their right hand sides contain the moduli of continuity of the right and left Caputo fractional derivatives of the involved function. From the uniform Shisha-Mond type inequality we derive the fractional Korovkin type theorem regarding the uniform convergence of positive linear operators to the unit.We give applications, especially to Bernstein polynomials for which we establish fractional quantitative results.

In this chapter we study quantitatively with rates the trigonometric weak convergence of a sequence of finite positive measures to the unit measure. Equivalently we study quantitatively the trigonometric pointwise convergence of sequence of positive linear operators to the unit operator, all acting on continuous functions on [ − 

π

,

π

]. From there we obtain with rates the corresponding trigonometric uniform convergence of the latter. The inequalities for all of the above in their right hand sides contain the moduli of continuity of the right and left Caputo fractional derivatives of the involved function. From these uniform trigonometric Shisha-Mond type inequality we derive the trigonometric fractional Korovkin type theorem regarding the trigonometric uniform convergence of positive linear operators to the unit. We give applications, especially to Bernstein polynomials over [ − 

π

,

π

] for which we establish fractional trigonometric quantitative results. This chapter relies on [46].

Here we present very general Taylor formulae, and then a representation formula. Based on the latter we give general integral inequalities of Opial type, Ostrowski type, Comparison of integral means, Information Theory Csiszar

f

- divergence type, and Grüss type. This chapter is based on [45].

Here we study

L

p

,

p

 > 1, fractional Opial integral inequalities subject to high order boundary conditions. They engage the right and left Caputo, Riemann-Liouville fractional derivatives. These derivatives are mixed together into the balanced Caputo, Riemann-Liouville, respectively, fractional derivative.

In this chapter we develop some integral identities and inequalities for the fractional integral. We obtain Montgomery identities for fractional integrals and a generalization for double fractional integrals. We also give Ostrowski and Grüss inequalities for fractional integrals. This chapter is based on [80].

In this chapter some general representation formulae for (

C

0

)

m

-parameter operator semigroups with rates of convergence are given by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (

C

0

) one-and

m

-parameter operator semigroups as special cases. When we consider special semigroups well-known convergence theorems for multivariate approximation operators are regained. This chapter is based on [92].

In this chapter a quantitative estimate for the simultaneous approximation of a function and its derivatives by the Feller probabilistic operator is given using probabilistic approach. This covers the cases of some classical approximation operators such as the Bernstein, Szász, Baskakov and Gamma operator. This chapter relies on [91].

In this chapter, we study the fuzzy global smoothness and fuzzy uniform convergence of fuzzy Picard, Gauss- Weierstrass and Poisson- Cauchy singular fuzzy integral operators to the fuzzy unit operator. These are given with rates involving the fuzzy modulus of continuity of a fuzzy derivative of the involved function. The established fuzzy Jackson type inequalities are tight, containing elegant constants, and they reflect the order of the fuzzy differentiability of the involved fuzzy function. This chapter is based on [55].

Here we transfer basic real approximations to corresponding vectorial and fuzzy setting of: Bernstein polynomials, Bernstein-Durrmeyer operators, genuine Bernstein-Durrmeyer operators, Stancu type operators and special Stancu operators. These are convergences to the unit operator with rates. We also give the convergence with rates to zero of the difference of genuine Bernstein-Durrmeyer and special Stancu operators. All approximations involve Jackson type inequalities and moduli of smoothness of various orders. In order to transfer we develop basic and important general results at the vectorial and fuzzy level. Our technique goes from real to vectorial and then to fuzzy setting. This chapter is based on [58].

Here are studied in terms of multivariate fuzzy high approximation to the multivariate unit several basic sequences of multivariate fuzzy wavelet type operators and multivariate fuzzy neural network operators. These operators are multivariate fuzzy analogs of earlier studied multivariate real ones. The produced results generalize earlier real ones into the fuzzy setting. Here the high order multivariate fuzzy pointwise convergence with rates to the multivariate fuzzy unit operator is established through multivariate fuzzy inequalities involving the multivariate fuzzy moduli of continuity of the

N

th order (

N

 ≥ 1) H-fuzzy partial derivatives, of the engaged multivariate fuzzy number valued function. The purpose of embedding fuzziness into multivariate classical analysis is to better understand, explain and describe the imprecise, uncertain and chaotic phenomena of the real world and then derive useful conclusions.

Here we introduce and study the right and left fuzzy fractional Riemann- Liouville integrals and the right and left fuzzy fractional Caputo derivatives. Then we present the right and left fuzzy fractional Taylor formulae. Based on these we establish a fuzzy fractional Ostrowski type inequality with applications. The last inequality provides an estimate for the deviation of a fuzzy real number valued function from its fuzzy average, and the related upper bounds are given in terms of the right and left fuzzy fractional derivatives of the involved function. The purpose of embedding fuzziness into fractional calculus and have them act together, is to better understand, explain and describe the imprecise, uncertain and chaotic phenomena of the real world and then derive useful conclusions. This chapter is based on [54].

Here we define a Caputo like discrete fractional difference and we compare it to the earlier defined Riemann-Liouville fractional discrete analog. Then we present discrete fractional Taylor formulae and we estimate their remainders. Finally we give related discrete fractional Ostrowski, Poincare and Sobolev type inequalities. This chapter is based on [48].

Here we define a Caputo like discrete nabla fractional difference and we give discrete nabla fractional Taylor formulae. We estimate their remainders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincaré and Sobolev type inequalities. This chapter relies on [51].

We give here forward and reverse

q

 −Hölder inequalities,

q

 −Poincaré inequality,

q

 −Sobolev inequality,

q

 −reverse Poincaré inequality,

q

 −reverse Sobolev inequality,

q

 −Ostrowski inequality,

q

 − Opial inequality and

q

 −Hilbert-Pachpatte inequality. Some interesting background is mentioned and built in the introduction. This chapter relies on [47].

Here we present

q

 −fractional Poincaré type, Sobolev type and Hilbert-Pachpatte type integral inequalities, involving

q

 −fractional derivatives of functions. We give also their generalized versions. This chapter relies on [50].

Here first we collect and develop necessary background on time scales required for this chapter. Then we give time scales integral inequalities of types: Poincaré, Sobolev, Opial, Ostrowski and Hilbert-Pachpatte. We present also the generalized analogs of all these inequalities involving high order delta derivatives of functions on time scales. We finish with many applications: all these inequalities on the specific time scales ℝ, ℤ and

$q^{\overline{\mathbb{Z}}}$

,

q

 > 1. This chapter relies on [57].

Here first we collect and develop necessary background on nabla time scales required for this chapter. Then we give nabla time scales integral inequalities of types: Poincaré, Sobolev, Opial, Ostrowski and Hilbert-Pachpatte. We present also the generalized analogs of all these nabla inequalities involving high order nabla derivatives of functions on time scales. We finish with many applications: all these nabla inequalities on the specific time scales ℝ, ℤ and

$q^{\overline{ \mathbb{Z}}}$

,

q

 > 1. In most of these nabla inequalities the nabla differentiability order is any

n

 ∈ ℕ, as opposed to delta time scales approach where

n

is always odd. This chapter relies on [59].

Here we present and extend the principle of duality in time scales. Using this principle and based on a variety of important delta inequalities we produce the corresponding nabla ones. We give several applications. This chapter relies on [52].

Here we present the Delta Fractional Calculus on Time Scales. Then we prove related integral inequalities of types: Poincaré, Sobolev, Opial, Ostrowski and Hilbert-Pachpatte. At the end we give inequalities applications on the time scale ℝ. This chapter is based on [56].

Here we present the Nabla Fractional Calculus on Time Scales. Then we prove related integral inequalities of types: Poincaré, Sobolev, Opial, Ostrowski and Hilbert-Pachpatte. At the end we give inequalities applications on the time scales ℝ, ℤ. This chapter relies on [53].

For the multivariate Dirichlet problem of the Poisson equation on an arbitrary compact domain, this chapter examines convergence properties with rates of approximate solutions, obtained by a standard difference scheme over inscribed uniform grids. Sharp quantitative estimates are proved by the use of second moduli of continuity of the second single partial derivatives of the exact solution. This is achieved by engaging the probabilistic method of simple random walk. This chapter is based on [63].

For the multidimensional Dirichlet problem of the heat equation on a cylinder, this chapter examines convergence properties with rates of approximate solutions, obtained by a naturally arising difference scheme over inscribed uniform grids. Sharp quantitative estimates are presented by the use of first and second moduli of continuity of some first and second order partial derivatives of the exact solution. This is achieved by using the probabilistic method of an appropriate random walk. This chapter is based on [64].

The classical time dependent partial differential equations of mathematical physics involve evolution in one dimensional time. Space can be multidimensional, but time stays one dimensional. There are various mathematical situations (such as multiparameter Brownian motion) which suggest that there should be a mathematical theory of evolution in multidimensional time. We formulate a rather general class of equations that involve two “time dimensions” and we prove a related uniqueness theorem.