\r\n\tA quark exhibits confinement, which means that the quarks are not observed independently but always in combination with other quarks. This makes determining the properties (mass, spin, and parity) impossible to measure directly; these traits must be inferred from the particles composed of them. There are six flavors of quarks: up, down, strange, charm, bottom, and top. The flavor of the quark determines its properties.
\r\n\tThere are three generations of quarks, based on pairs of weak positive/negative, weak isospin. The first generation quarks are up and down quarks, the second-generation quarks are strange and charm quarks, the third generation quarks are top and bottom quarks. The up and down quarks make up protons and neutrons, seen in the nucleus of ordinary matter. They are the lightest and most stable. The heavier quarks are produced in high-energy collisions and rapidly decay into up and down quarks.
\r\n\tThe baryons and mesons known at the time fell into symmetric families of multiplets (octuplets, decuplets) sharing two identical quantum numbers (spin and parity), but differing in an ordered way in others (mass, charge, baryon number and strangeness). The mathematical group to fit this complex situation-SU3, the symmetric, unitary group of dimension 3-was proposed independently by Gell-Mann and Ne'eman. The validity of SU3 was demonstrated by the experiment. A major prediction was that a particle (the omega-minus), an isotopic singlet with spin = 3/2, positive parity, mass of roughly 1,680 MeV, negative charge, baryon number +1, strangeness = -3, and stable to strong decay, should exist to complete the 3/2+ baryon decuplet. It was therefore a major triumph for the scheme when the omega-minus, a baryon with the precise mass, charge, and strangeness predicted, was discovered in 1964. All these facts introduced a quark idea fully into modern physics.
\r\n\r\n\tThis book will be a self-contained collection of scholarly papers targeting an audience of practicing researchers, academics, PhD students and other scientists. The contents of the book will be written by multiple authors and edited by experts in the field.
",isbn:"978-1-83968-313-8",printIsbn:"978-1-83968-312-1",pdfIsbn:"978-1-83968-314-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"0d9403b5c874f6e63b0686cd7c432e00",bookSignature:"Prof. Zbigniew Piotr Szadkowski",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10205.jpg",keywords:"Chiral Symmetries, Weak Interactions, Neutrinoless Double Beta Decay, Deep Inelastic Scattering, Quantum Chromodynamics (QCD), Color Confinement, Quarks Mixing, Cabibbo Angle, Kobayashi-Maskawa Matrix, Quarks Multiplets, CP-Nonconservation, Neutrino Oscillation",numberOfDownloads:66,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 6th 2020",dateEndSecondStepPublish:"October 8th 2020",dateEndThirdStepPublish:"December 7th 2020",dateEndFourthStepPublish:"February 25th 2021",dateEndFifthStepPublish:"April 26th 2021",remainingDaysToSecondStep:"5 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"The designer of the 2nd level trigger for the fluorescence detector and the designer of the 1st level trigger and the Front-End Boards for the surface detector of the Pierre Auger Observatory.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"67836",title:"Prof.",name:"Zbigniew Piotr",middleName:null,surname:"Szadkowski",slug:"zbigniew-piotr-szadkowski",fullName:"Zbigniew Piotr Szadkowski",profilePictureURL:"https://mts.intechopen.com/storage/users/67836/images/system/67836.jpeg",biography:"Dr. Szadkowski completed his Ph.D. with a thesis 'Quarks mixing in chiral symmetries SU4 x SU4 and SU6 x SU6”. Habilitation: „Triggers in the Pierre Auger Observatory: Designs, Implementation and the Impact on the Experimental Results”.\r\nDevelopment of the FPGA-based 2nd level trigger for 24 fluorescence detectors and 1st level trigger for 1660 surface detectors of the Pierre Auger Observatory, FPGA based filters suppressing radio-frequency interferences (RFI) in radio detector of Auger Engineering Radio Array, FPGA based triggers for the Auger surface detectors dedicated for a recognition of very inclined EAS induced by 'old” proton showers or 'young” neutrino showers.\r\nDr. Szadkowski has worked as a research scientist in Michigan Technological University, Associate Professor in College de France, Senior Wissenschaftler, Bergische Universität Wuppertal, and currently is working as the head of the Department of High-Energy Astrophysics and as an Associate Professor at the University of Łódź.",institutionString:"University of Łódź",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"4",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"University of Łódź",institutionURL:null,country:{name:"Poland"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"20",title:"Physics",slug:"physics"}],chapters:[{id:"73971",title:"The Inter-Nucleon Up-to-Down Quark Bond and its Implications for Nuclear Binding",slug:"the-inter-nucleon-up-to-down-quark-bond-and-its-implications-for-nuclear-binding",totalDownloads:67,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"247041",firstName:"Dolores",lastName:"Kuzelj",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/247041/images/7108_n.jpg",email:"dolores@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"38462",title:"Lipid Peroxidation in Hepatic Fibrosis",doi:"10.5772/46180",slug:"lipid-peroxidation-in-hepatic-fibrosis",body:'Hepatic fibrosis is a complex dynamic process which is mediated by death of hepatocytes and activation of hepatic stellate cells (HSCs). Lipid peroxidation including the generation of reactive oxygen species (ROS), transforming growth factor-β, and tumor necrosis factor-α can be implicated as a cause of hepatic fibrosis.
Damage of any etiology, such as infection with hepatitis C virus (HCV) or hepatitis B virus (HBV), heavy alcohol intake, and iron overload, to hepatocytes can produce oxygen-derived free radicals and other ROS derived from lipid peroxidative processes. Persistent production of ROS constitutes a general feature of a sustained inflammatory response and liver injury, once antioxidant mechanisms have been depleted. The major source of ROS production in hepatocytes is NADH and NADPH oxidases localized in mitochondria (Figure 1). NADH and NADPH oxidases leak ROS as part of its operation. Kupffer cells (hepatic resident macrophages), infiltrating inflammatory cells such as macrophages and neutrophils, and HSCs also produce ROS in the injured liver.
ROS include the free radicals superoxide (O2-) and hydroxyl radical (HO-) and non-radicals such as hydrogen peroxide (H2O2). A number of reactive nitrogen species including nitric oxide (NO) and peroxynitrite (ONOO-) are also ROS. Superoxide production is mediated mainly by NADH oxidase. Hydrogen peroxide is more stable and membrane permeable in comparison to other ROS. Thus, hydrogen peroxide plays an important role in the intracellular signaling under physiological conditions. With respect to pathological actions, ROS participate in the development of liver disease. In this regard, hydrogen peroxide is converted into the hydroxyl radical, a harmful and highly reactive ROS, in the presence of transition metals such as iron (Figure 1). The hydroxyl radical is able to induce not only lipid peroxidation in the structure of membrane phospholipids, which results in irreversible modifications of cell membrane structure and function (membrane injury), but DNA cleavage (DNA injury) as well. Such a chain of events due to increased ROS production exceeding cellular antioxidant defense systems are called oxidative stress, inducing cell death.
Malondialdehyde (MDA) and 4-hydroxynonenal (HNE) (Figure 1), end products of lipid peroxidation, are discharged from destroyed hepatocytes into the space of Disse (Figure 2). Cells are well equipped to neutralize the effects of ROS by virtue of a series of the antioxidant protective systems, including superoxide dismutase (SOD), glutathione peroxidase, glutathione (GSH), and thioredoxin. Upon oxidation, GSH forms glutathione disulfide (GSSG).
Oxidative stress and hepatocyte damage (Shimizu et al., 2012). A primary source of reactive oxygen species (ROS) production is mitochondrial NADPH/NADH oxidase. Hydrogen peroxide (H2O2) is converted to a highly reactive ROS, the hydroxyl radical, in the presence of transition metals such as iron (+Fe) and copper. The hydroxyl radical induces DNA cleavage and lipid peroxidation in the structure of membrane phospholipids, leading to cell death and discharge of products of lipid peroxidation, malondialdehyde (MDA) and 4-hydroxynonenal (HNE) into the space of Disse. Cells have comprehensive antioxidant protective systems, including SOD, glutathione peroxidase and glutathione (GSH). Upon oxidation, GSH forms glutathione disulfide (GSSG).
A single liver injury eventually results in an almost complete resolution, but the persistence of the original insult causes a prolonged activation of tissue repair mechanisms, thereby leading to hepatic fibrosis rather than to effective tissue repair. Hepatic fibrosis, or the excessive collagen deposition in the liver (see next section), is associated with oxidative stimuli and cell death. Cell death is a consequence of severe liver damage that occurs in many patients with chronic liver disease, regardless of the etiology such as HCV/HBV infection, heavy alcohol intake, and iron overload.
At the cellular levels, origin of hepatic fibrosis is initiated by the damage of hepatocytes, followed by the accumulation of neutrophils and macrophages including Kupffer cells on the sites of injury and inflammation in the liver. When hepatocytes are continuously damaged, leading to cell death, production of extracellular matrix proteins such as collagens predominates over hepatocellular regeneration. Overproduced collagens are deposited in injured areas instead of destroyed hepatocytes. In other words, hepatic fibrosis is fibrous scarring of the liver in which excessive collagens build up along with the duration and extent of persistence of liver injury. Hepatic fibrosis itself causes no symptoms but can lead to the end-stage cirrhosis. In cirrhosis the failure to properly replace destroyed hepatocytes and the excessive collagen deposition to distort blood flow through the liver (portal hypertension) result in severe liver dysfunction. Cirrhosis is an important host-related risk factor for the development of hepatocellular carcinoma (HCC) in chronic hepatitis C and B, as well as a major factor predicting a poor response to interferon-based antiviral therapy in chronic hepatitis C. Staging of chronic liver disease by assessment of hepatic fibrosis always is a major function of prognostic interpretation of individual data including liver biopsy. Of the commonly used staging systems, the METAVIR fibrosis score has been widely used (Huwart et al., 2008). The stages are determined by both the quantity and location of the fibrosis. With this score, F0 represents no fibrosis; F1 (mild fibrosis), portal fibrosis without septa; F2 (moderate fibrosis), portal fibrous and few septa; F3 (severe fibrosis), numerous septa without cirrhosis; and F4, cirrhosis (Figure 3), the tissue is eventually composed of nodules surrounded completely by fibrosis.
Schema of the sinusoidal wall of the liver (Shimizu et al., 2012). Schematic representation of hepatic stellate cells (HSCs) was based on the studies by Wake (Wake, 1999). Kupffer cells (hepatic resident macrophages) rest on fenestrated endothelial cells. HSCs are located in the space of Disse in close contact with endothelial cells and hepatocytes, functioning as the primary retinoid storage area. Collagen fibrils course through the space of Disse between endothelial cells and the cords of hepatocytes.
Normal liver has a connective tissue matrix which includes collagen type IV (non-fibrillary), glycoproteins such as fibronectin and laminin, and proteoglycans such as heparan sulphate. These comprise the low density basement membrane in the space of Disse. Following liver injury there is a 3- to 8-fold increase in the extracellular matrix which is of a high density interstitial type, containing fibril-forming collagens (types 1 and III) as well as fibronectin, hyaluronic acid and proteoglycans. Collagen types 1 and III are major components of the extracellular matrix, which is principally produced by cells known as HSCs. HSCs are located in the space of Disse in close contact with hepatocytes and sinusoidal endothelial cells (Figure 2). Their three-dimensional structure consists of the cell body and several long and branching cytoplasmic processes (Wake, 1999). In the resting liver, HSCs have intracellular droplets containing retinoids. Retinoids refer to a group of chemical compound associated with vitamin A. HSCs contain approximately 50-80% of the whole body stores of retinoids (Blomhoff et al., 1990). In contrast, in the injured liver, HSCs are regarded as the primary target cells for inflammatory and oxidative stimuli, and they are proliferated, enlarged and transformed into myofibroblast-like cells. These HSCs are referred to as activated cells and are responsible for the overproduction of collagens during hepatic fibrosis to cirrhosis. This activation is accompanied by a loss of cellular retinoids, and the synthesis of α-smooth muscle actin (α-SMA), and large quantities of the major components of the extracellular matrix including collagen types I, III, and IV, fibronectin, laminin and proteoglycans. α-SMA is produced by activated HSCs (myofibroblast-like cells) but not by resting (quiescent) HSCs, thereby a marker of HSC activation. Moreover, activated HSCs produce ROS and transforming growth factor-β (TGF-β) (Figure 4). TGF-β is a major fibrogenic cytokine, regulating the production, degradation and accumulation of the extracellular matrix in hepatic fibrosis. TGF-β expression correlates with the extent of hepatic fibrosis (Castilla et al., 1991). This cytokine induces its own expression in activated HSCs, thereby creating a self-perpetuating cycle of events, referred to as an autocrine loop. TGF-β is also released in a paracrine manner from Kupffer cells, endothelial cells, and infiltrating inflammatory cells following liver injury. Similarly, ROS are produced by activated HSCs in response to ROS released from adjacent cells such as destroyed hepatocytes and activated Kupffer cells.
Stages of hepatic fibrosis in chronic hepatitis according to the five stages (0-4) of the METAVIR scoring system (1994). With this score, F0 represents no fibrosis; F1 (mild fibrosis), fibrous expansion of portal areas without septa; F2 (moderate fibrosis), fibrous septa extend to form bridges between adjacent vascular structures, both portal to portal and portal to central, occasional bridges; F3 (severe fibrosis), numerous bridges or septa without cirrhosis; and F4 (cirrhosis), the tissue is eventually composed of nodules surrounded completely by fibrosis.
During liver injury, HSCs are proliferated, enlarged and transformed into myofibroblast-like cells (Shimizu, 2001). These activated HSCs produce large quantities of collagens, α-smooth muscle actin (α-SMA), ROS, and transforming growth factor-β (TGF-β), and lose cellular retinoids.
HSCs are activated mainly by ROS, products of lipid peroxidation (MDA and HNE) (Lee et al., 1995; Parola et al., 1993), and TGF-β, which are released from destroyed hepatocytes, activated Kupffer cells and infiltrating macrophages and neutrophils in the injured liver (Figure 5). In addition to ROS, exogenous TGF-β increases the production of ROS, particularly hydrogen peroxide, by HSCs, whereas the addition of hydrogen peroxide induces ROS and TGF-β production and secretion by HSCs (De Bleser et al., 1999). This so-called autocrine loop of ROS by HSCs is regarded as mechanism corresponding to the autocrine loop of TGF-β which HSCs produce in response to this cytokine with an increased collagen expression in the injured liver (Itagaki et al., 2005).
Activation of HSCs. HSCs are activated by such factors as ROS, lipid peroxidation products (MDA and HNE), and TGF-β released when adjacent cells including hepatocyte, Kupffer cells, and endothelial cells are injured. ROS and TGF-β are also produced by HSCs in response to exogenous ROS and TGF-β in an autocrine manner.
Other important factors for HSC activation are platelet-derived growth factor (PDGF) released from platelets, and endothelin-1 from endothelial cells. PDGF is the most potent mitogen. HSCs congregate in the area of injury, through proliferation and migration from elsewhere, in response to the release of PDGF and monocyte chemotactic peptide-1 (MCP-1). MCP-1 is produced by activated Kupffer cells and infiltrating macrophages and neutrophils. The number of activated HSCs also increases after liver injury (Enzan et al., 1994).
At the molecular levels, HSCs express the genes which encode for enzymes such as matrix metalloproteinase (MMP)-1 (interstitial collagenase) (Casini et al., 1994), which digests native fibrillar collagen types I and III, and MMP-2 (Milani et al., 1994), which digests denatured collagen types I and III and native collagen type IV, as well as tissue inhibitors of MMPs (TIMP)-1 and TIMP-2 (Iredale et al., 1992). Imbalance between matrix synthesis and degradation plays a major role in hepatic fibrosis (Shimizu, 2001). Matrix degradation depends upon the balance between MMPs, TIMPs and converting enzymes (MT1-MMP and stromelysin) (Li and Friedman, 1999). Collagen types I and III constitute the main framework of the so-called “fibrillar matrix”. The space of Disse is a virtual space constituted by an extracellular matrix network composed of collagen type IV and non-collagenous components such as laminin. The large majority of collagen types III and IV, and laminin are synthesized by HSCs and endothelial cells, whereas all cell types synthesize small amounts of collagen type I. During hepatic fibrosis, however, HSCs become the major extracellular matrix producing cell type, with a predominant production of collagen type I (Maher and McGuire, 1990). In the resting liver, a balance between matrix synthesis and degradation exists, whereas, in the injured liver, the balance is disrupted. The net result of the changes during hepatocyte damage is increased degradation of the normal basement membrane collagen, and reduced degradation of interstitial-type collagens. The latter can be explained by increased TIMP-1 and TIMP-2 expressions relative to MMP-1. The degradative portion of the remodeling process is coordinated by MMPs and TIMPs.
Origin of hepatic fibrosis is initiated by the damage of hepatocytes, resulting in the recruitment of inflammatory cells and platelets, and activation of kupffer cells, with subsequent release of cytokines and growth factors. HSCs are the primary target cells for these inflammatory and oxidative stimuli, because during hepatic fibrosis, HSCs undergo an activation process to a myofibroblast-like cell, which represents the major matrix-producing cell. In the injured liver, hydrogen peroxide seems to act as a second messenger to regulate signaling events including mitogen activated protein kinase (MAPK) activation. The MAPK family includes three major subgroups, extracellular signal regulated kinase (ERK), p38 MAPK (p38), and c-Jun N-terminal kinase/stress activated protein kinase (JNK). MAPK participates in the intracellular signaling to: (1) induce the gene expression of redox sensitive transcription factors, such as activator protein-1 (AP-1) and nuclear factor κB (NF-κB) (Pinkus et al., 1996), (2) stimulate apoptosis (Clement and Pervaiz, 1999), and (3) modulate cell proliferation (Lundberg et al., 2000). ERK and JNK lie upstream of AP-1. JNK and p38 activation are more important in stress responses such as inflammation, which can also activate NF-κB. AP-1 and NF-κB induce the expression of multiple genes involved in inflammation and oxidative stress response, cell death and fibrosis, including proinflammatory cytokines such as tumor necrosis factor-α (TNF-α), interleukin-1 and interleukin-6 and growth factors such as PDGF and TGF-β. TGF-β is a major fibrogenic cytokine, acting as a paracrine and autocrine (from HSCs) mediator as already noted. TGF-β triggers and activates the proliferation, enlargement and transformation of HSCs, but it exerts its inhibitory effect on hepatocyte proliferation (Nakamura et al., 1985).
Since many cytokines exert growth factor like activity, in addition to their specific proinflammatory effects, the distinction between cytokines and growth factors is somewhat artificial. No growth factor or cytokine acts independently. The injured liver, predominantly Kupffer cells and infiltrating macrophages and neutrophils, produces TNF-α, interleukin-1 and interleukin-6. These proinflammatory cytokines may also also inhibit hepatic regeneration. In particular, TNF-α plays a dichotomous role in the liver, where it not only induces hepatocyte proliferation and liver regeneration but also acts as a mediator of cell death (Schwabe and Brenner, 2006). During TNF-α-induced apoptosis in hepatocytes, hydrogen peroxide is an important mediator of cell death (Bohler et al., 2000).
In liver injury of hepatitis virus infection, transgenic mice expressing HBsAg exhibit the generation of oxidative stress and DNA damage, leading to the progression of hepatic fibrosis and carcinogenesis (Hagen et al., 1994; Nakamoto et al., 2004). In addition, HBV X protein changes the mitochondrial transmembrane potential and increases ROS production in the liver (Waris et al., 2001). Moreover, structural and non-structural (NS) proteins of HCV are involved in the production of ROS in an infected liver. HCV core protein is associated with increased ROS, decreased intracellular and/or mitochondrial glutathione content, and increased levels of lipid peroxidation products (Moriya et al., 2001). Glutathione is an antioxidant. NS3 protein of HCV activates NADPH oxidase in Kupffer cells to increase production of ROS, which can exert oxidative stress on nearby cells (Thoren et al., 2004).
Hepatic fibrosis is a complex dynamic process which is mediated by death of hepatocytes and activation of HSCs. Lipid peroxidation including the generation of ROS, TGF-β, and TNF-α can be implicated as a cause of hepatic fibrosis. HSCs are regarded as the primary target cells for inflammatory stimuli, and produce extracellular matrix components. HSCs are activated by such factors as ROS, lipid peroxidation products (MDA and HNE), and TGF-β released when adjacent cells including hepatocyte, Kupffer cells, and endothelial cells are injured. ROS and TGF-β are also produced by HSCs in response to exogenous ROS and TGF-β in an autocrine manner. During TNF-α-induced death in hepatocytes, ROS is an important mediator of cell death. The most common cause of hepatic fibrosis is currently chronic HCV/HBV infection.
Understanding the basic mechanisms underlying the ROS-mediated fibrogenesis provides valuable information on the search for effective antifibrogenic therapies.
One of the classical tools of Mathematics is the apparatus of Fourier series, based on the orthogonal system
Since the beginning of the last century, a huge amount of research has been carried out, devoted to the methods of effective summation of the Fourier series. Let us briefly dwell on the works to overcome this situation (for detiles see, for example, articles [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and their links).
The pioneer of “overcoming the Gibbs phenomenon” is A.N.Krylov, who at the dawn of the 20th century (see [2]) proposed methods that he later developed in the monograph [3]. In particular, he proposed the following approach. Let a piecewise smooth function
In the neighborhoods of other points we assume that
where
Therefore, taking into account only first
However, the computing of the jumps
The “spectral” method proposed by Kurt Eckhoff in [4] (1993) turned out to be more practical, as it is based only on the use of coefficients
As a function
Denoting
According to Krylov’s scheme, the sequence
where the quantities
K. Eckhoff suggests to find approximate values of jumps
Further, the problem is solved by the Krylov method. It is natural to call this acceleration scheme the Krylov-Eckhoff method (KE-method). Over the past two to three decades, in the KE-method scheme, not only polynomials have been used as functions of
In [14], for a smooth function
The classical definition of the partial sums of the Fourier series is based on a gradual increase in the frequencies of the Fourier system. Below we use a more general notations from the work [17].
Definition 1. Let us call the truncated Fourier series any sum of the form
where
are Fourier coefficients of
Definition 2. Let
It is easy to see that
Definition 3. We call the system
Definition 4. Let parameters
Remark 1. The set of quasi-polynomials
For a fixed integer
Remark 2. The last product in(3)is invariant with respect to the numbering of the parameters
In the general case, the parameters
Further we denote (see [17] for details)
The system
Here we confine ourselves to the one-dimensional case. A similar universal algorithm for multivariate truncated Fourier series was proposed and numerically implemented in [1].
The following obvious formula (
where
To include the case of a quasi-polynomial
Lemma 1. If
where
Since the Bernoulli polynomial
Remark 3. Like the Bernoulli polynomials in the KE- method, quasi-polynomials
If in the sequence (9) parameters
where
Consider now the possibility of including in the consideration of some integer parameters in (10). First of all, note that if for some
If
Finally, we note that in the latter case, the sequence
Lemma 2. Let in (
If there is an
If
With a
If
The following result is a generalization of Theorem 1 in [17] to the case that in formula (15) there are integer values among the parameters
Theorem 1. Suppose the sequence(15)is given and the possible integer parameters in
where
In the main Theorem of paper [17], the phenomenon of over-convergence was theoretically substantiated. The basis of this result was the following Algorithm
Let
regarding parameters
Here we show how Algorithm
Step 1. Using the representation (9), we formally bring the left side of (18) to a common denominator, and fix the conditions
Step 2. Using the Vieta’s formula decompose the products in (18) containing the parameters
It is not difficult to see that, as a result, Eq.(18) will take the form of a linear system of equations with respect to the variables
Step 3. Solve resulting equation by the least square method.
Step 4. Again, according to the Vieta’s formulae find all roots
Step 5. Realize the approximation
Remark 4. Least square method provides only one solution in Step 3.
The following theorem is the main result of the paper [17].
Theorem 2. {The phenomenon of the over-convergence}. Let
It is now natural to formulate the following definition of the acceleration of the convergence of a truncated Fourier series.
Definition 5. Let
we define by formula (17).
Remark 5. There are no Fourier coefficients
Consider the Fourier transform
The method of the item 2 can be applied by linear change of variable to function
On the other hand, our method allows one to approximate function
Our idea is as follows. First, based on the required 1 error, you can leave a fnite number of intervals. Second, on the remaining intervals, the integrals are calculated explicitly. Really, each function
On the complex plane
Remark 6. Depending on function
Let us turn to the main details of the algorithm for such a numerical implementation of the Fourier transform for a piecewise smooth function
To make our approach understandable to a wide range of specialists, when developing an applied algorithm, we will assume that (see formulas (7–19)) for a fixed
Consider now the following when writing code to implement the one-dimensional Fourier transform.
The computer used must be installed with programs that can perform symbolic mathematical operations as well as numerical integration (for example, as system Wolfram Mathematica, see [19]). It is desirable to use the highest possible accuracy of calculations.
The singularities of piecewise-smooth function
The computer errs or freezes most often due to type
For example, let in the process of computer operation at the output of Algorithm
During the operation of the proposed algorithm for Fourier transform, \t
Let us explain the details on the following example of an approximate finding of the Fourier transform (see notation (21)).
Our goal is to provide a final value of a
and when deleting these semi-infinite intervals, further actions should lead to an error no greater than
On the other hand, we see that function
Monitoring showed (see item 4) that
The total
These calculations were performed using system Wolfram Mathematica 9.
Those who have mastered the technique of using Algorithm
There are good prerequisites for other classical (in a broad sense) Fourier tools for efficient applications based on the method of item 2 (see articles [14, 15]).
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