Discrete Taylor Transform and Inverse Transform

• Record Nr.: UNINA9911018812903321
• Autore: Baghai-Wadji Alireza
• Pubbl/distr/stampa: Newark : , : John Wiley & Sons, Incorporated, , 2024; ©2025
• ISBN: 9781394240081; 1394240082; 9781394240098; 1394240090; 9781394240104; 1394240104
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (0 pages)
• Disciplina: 515/.723
• Soggetti: Integral transforms
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Cover -- Title Page -- Copyright -- Contents -- About the Author -- Preface -- Introduction -- I.1 Notation and Elementary Notions -- I.2 Orthonormal Bases and Their Corresponding Dual Bases -- I.3 Fourier Transform and Inverse Transform and the AssociatedResolution of Identity -- Chapter 1 Toy Model I‐1: {−Δ,0,Δ} -- 1.1 Introduction -- 1.1.1 Symmetric Equidistant Sampling -- 1.1.2 Difference Operators -- 1.2 Frames and Dual Frames Induced by the Monomials 1, x, and x2 -- 1.2.1 Brief Summary of the Essentials -- 1.2.2 Frame Vectors -- 1.2.3 Frame Operator -- 1.2.4 Inverse Frame Operator -- 1.2.5 Dual‐Frame Vectors -- 1.2.6 Dual‐Frame Operator -- 1.2.7 The Resolution of the Identity -- 1.2.8 D‐TTIT in 3D -- Chapter 2 Toy Model I‐2:{0,Δ,2Δ} -- 2.1 Introduction -- 2.1.1 Difference Operators -- 2.2 Frames and Dual Frames Induced by Monomials 1, x, and x2 -- 2.2.1 Frame Vectors -- 2.2.2 Frame Operator -- 2.2.3 Inverse Frame Operator -- 2.2.4 Dual‐Frame Vectors -- 2.2.5 Dual‐Frame Operator -- 2.2.6 The Resolution of the Identity -- 2.2.7 D‐TTIT in 3‐D -- Chapter 3 Toy Model I‐3: {−2Δ,−Δ,0} -- 3.1 Introduction -- 3.1.1 Difference Operators -- 3.2 Frames and Dual Frames -- 3.2.1 Frame Vectors -- 3.2.2 Frame Operator -- ; 3.2.3 Inverse Frame Operator -- 3.2.4 Dual‐Frame Vectors -- 3.2.5 The Resolution of the Identity -- 3.2.6 D‐TTIT in 3‐D -- Chapter 4 Toy Model I‐4: {−Δ,0,Δ} -- 4.1 Overcompleteness -- 4.1.1 Difference Operators -- 4.2 Frames and Dual Frames -- 4.2.1 Frame Vectors -- 4.2.2 Frame Operator -- 4.2.3 Inverse Frame Operator -- 4.2.4 Dual Frame Vectors -- 4.2.5 The Resolution of the Identity -- 4.2.6 Establishing Relationships Between the Dual Frame Vectors |1˜&gt -- , |x˜&gt -- , |x˜2&gt -- , |x˜3&gt -- , and |x˜4&gt -- , and the Difference Operators |D(0)&gt -- , |D(1)&gt -- , and |D(2)&gt.; 4.2.7 Establishing Relationships Between the Dual Frame Vectors |1˜&gt -- , |x˜&gt -- , |x˜2&gt -- , |x˜3&gt -- , |x˜4&gt -- , |x˜5&gt -- , and |x˜6&gt -- , and the Difference Operators |D(0)&gt -- , |D(1)&gt -- , and |D(2)&gt -- -- Chapter 5 Toy Model I‐5: {−2Δ, −Δ, 0, Δ, 2Δ} -- 5.1 Introduction -- 5.2 Difference Operators -- 5.3 Frames and Dual Frames -- 5.3.1 Dual‐Frame Vectors -- 5.3.1.1 On the Construction of |1˜&gt -- -- 5.3.1.2 On the Construction of |x˜&gt -- -- 5.3.1.3 On the Construction of |x˜2&gt -- -- 5.3.1.4 On the Construction of |x˜3&gt -- -- 5.3.1.5 On the Construction of |x˜4&gt -- -- 5.3.2 Dual‐Frame Operator -- Chapter 6 Toy Model I‐7: {−3Δ,−2Δ,−Δ,0,Δ,2Δ,3Δ} -- 6.1 Introduction -- 6.2 Difference Operators -- 6.3 Frame Vectors -- 6.4 Frame Operator -- 6.5 Inverse Frame Operator -- 6.6 Constructing Skeleton Matrices for S7×7−1 -- 6.7 Practical Implementation -- 6.8 Dual Vectors -- 6.8.1 Summarizing the Results Obtained -- 6.9 Dual‐Frame Operator -- 6.10 Conclusions -- Chapter 7 Self‐consistent Expressions for |D(n)&gt -- -- 7.1 The Interval [−Δ,Δ] -- 7.2 The Interval [−2Δ,2Δ] -- 7.3 The Interval [−3Δ,3Δ] -- Chapter 8 Toy Model I‐3: {Δ−1,Δ0,Δ1} -- 8.1 A Guide Through the Chapter -- 8.2 Univariate Functions on Three Nonuniformly Distributed Lattice Points: Derivatives at an Inner Cluster Point -- 8.3 Setting Up the System of Equations for the Determination of Df(n) (n&amp -- equals -- 0,1,2) -- 8.4 Matrix Multiplication Expressed in Terms of Exterior Products -- 8.4.1 General Considerations -- 8.4.2 The Resolution of Identity -- 8.4.3 The Frame Operator -- 8.4.4 Preliminary Summary -- 8.5 Solving the System of Equations in (8.7) by Successive Elimination (Method 1) -- 8.5.1 Obtaining the Expressions of the Universal Derivative Kets |D(n)&gt -- Defined by Df(n)&amp -- equals -- &lt -- D(n)|F&gt -- (n&amp -- equals -- 0,1,2).; 8.6 Exterior Products |xn&gt -- &lt -- D(n)| (n&amp -- equals -- 0,1,2) and the Resolution of Identity (Property 1) -- 8.7 Inner Products &lt -- xn|D(n)&gt -- &amp -- equals -- δmn (m,n&amp -- equals -- 0,1,2) (Property 2) -- 8.8 Calculation of the Derivative Operators Based on the Inverse of the Δ‐Matrix (Method 2) -- 8.9 Calculating the Derivative Operators Based on the Frame Operator (Method 3) -- 8.9.1 The Exterior Product of the Kets |xn&gt -- with Their Bra Counterpart &lt -- xn| -- 8.9.2 The Exterior Product of the Ket |1&gt -- with Its Bra Counterpart -- 8.9.3 The Exterior Product of the Ket |x&gt -- with Its Bra Counterpart -- 8.9.4 The Exterior Product of the Ket |x2&gt -- with Its Bra Counterpart -- 8.9.5 The S‐Matrix and Its Properties -- 8.9.6 Calculation of &lt -- D(0)| Utilizing S−1 and the Position Bra &lt -- x(0)| -- 8.9.7 Calculation of &lt -- D(1)| Utilizing S−1 and the Position Bra &lt -- x(1)| -- 8.9.8 Calculation of &lt -- D(2)| Utilizing S−1 and the Position Bra &lt -- x(2)| -- 8.10 Construction of the Derivative Operators in Terms of Rational Polynomials (Method 4) -- 8.11 Construction of the Derivative Operators Simply‐by‐Inspection of Indices (Method 5) -- 8.12 Uniform Lattices -- 8.12.1 Properties of the Derivative Operators on Uniform Lattices -- 8.12.2 Relating &lt -- D(n)| to f(n)(0) (n&amp -- equals -- 0,1,2) -- 8.13 Conclusions -- Chapter 9 Toy Model I‐5: {Δ−2,Δ−1,Δ0,Δ1,Δ2} -- 9.1 The Resolution of Identity ; -- 9.2 Setting Up the System of Equations -- 9.3 Solving the System of Equation in (9.18) by Successive Elimination -- 9.4 Obtaining the Expressions of the Universal Difference Operators |D(n)&gt -- Defined by Df(n)&amp -- equals -- &lt -- D(n)|F&gt -- -- 9.5 Simplifying the Expressions of the Difference Operators -- 9.6 Exterior Products of the Position Kets and their Dual Difference Kets -- 9.7 Uniform Lattices.; 9.7.1 Derivative Operators -- 9.7.2 Properties of the Derivative Operators on Uniform Lattices -- 9.7.3 Position Kets on the Five Point Uniform Lattice -- 9.7.4 Biorthogonality -- 9.8 The Frame Operator S -- 9.8.1 The Exterior Product of the Ket |1&gt -- with its Dual Bra Counterpart -- 9.8.2 The Exterior Product of the Ket |x&gt -- with its Dual Bra Counterpart -- 9.8.3 The Exterior Product of the Ket |x2&gt -- with its Dual Bra Counterpart -- 9.8.4 The Exterior Product of the Ket |x3&gt -- with its Dual Bra Counterpart -- 9.8.5 The Exterior Product of the Ket |x4&gt -- with its Dual Bra Counterpart -- 9.8.6 Properties of the S‐Matrices -- 9.9 The Relationship Between the Resolution of Identity and Biorthogonality -- 9.9.1 Biorthogonality Implies the Resolution of Identity -- 9.9.2 The Resolution of Identity Implies Biorthogonality -- 9.10 The Construction of the Derivative Operators by Calculating Residues -- Chapter 10 Toy Model I‐6: {Δ−3,Δ−2,Δ−1,Δ0,Δ1,Δ2,Δ3} -- 10.1 Generating Formulas for the Difference Operators by Residue Method -- 10.2 Summary of the Relevant Formulas for the Calculation of Df(k) -- Chapter 11 Toy Model I‐7: {Δ−3,Δ−2,Δ−1,Δ0,Δ1,Δ2,Δ3} -- 11.1 A Guide Through the Chapter -- 11.2 Univariate Functions on 7 Nonuniformly Distributed Lattice Points -- 11.3 Setting Up the System of Equations -- 11.4 Generating Formulas for the Derivative Operators Simply‐by‐Inspection -- 11.5 Differential and Position Coordinate Bras -- 11.6 Differential Bras -- 11.7 Position Coordinate Bras -- 11.8 Differential and Position Kets: Uniformly Distributed Lattice Points -- 11.8.1 The Seven Common Denominators -- 11.8.2 The Expression of |D(6)&gt -- -- 11.8.3 The Expression of |D(5)&gt -- -- 11.8.4 The Expression of |D(4)&gt -- -- 11.8.5 The Expression of |D(3)&gt -- -- 11.8.6 The Expression of |D(2)&gt -- -- 11.8.7 The Expression of |D(1)&gt.; 11.9 The Biorthogonality and the Resolution of Identity Conditions -- 11.10 Conclusions: A Brief Philosophical Detour -- Chapter 12 Toy Model II: {{−Δ1,0,Δ1},{−Δ2,0,Δ2}} -- 12.1 Introduction -- 12.2 Determination of the Expansion Coefficients F(m,n) (m,n&amp -- equals -- 0,1,2) -- 12.2.1 On the Construction of |1&gt -- -- 12.2.2 On the Construction of |x&gt -- -- 12.2.3 On the Construction of |y&gt -- -- 12.2.4 On the Construction of |x2&gt -- -- 12.2.5 On the Construction of |xy&gt -- -- 12.2.6 On the Construction of |y2&gt -- -- 12.2.7 On the Construction of |x2y&gt -- -- 12.2.8 On the Construction of |xy2&gt -- -- 12.2.9 On the Construction of |x2y2&gt -- -- 12.3 The Biorthonormality Property -- 12.4 The Resolution of Identity -- Chapter 13 Toy Model III: {−Δ1,Δ1} × {−Δ2,Δ2} × {−Δ3,Δ3} -- 13.1 Discrete Taylor Transform and Inverse Transform of Trivariate Functions -- 13.2 Determination of the Expansion Coefficients F(m,n,p) (m,n,p&amp -- equals -- 0,1,2) -- 13.2.1 Sample f(x,y,z) at the Node 1, Defined by the Coordinates (−Δ1,−Δ2,−Δ3) -- 13.2.2 Sample f(x,y,z) at the Node 2 Defined by the Coordinates (0,−Δ2,−Δ3) -- 13.2.3 Sample f(x,y,z) at the Node 3 Defined by the Coordinates (Δ1,−Δ2,−Δ3) -- 13.2.4 Sample f(x,y,z) at the Node 4 Defined by the Coordinates (−Δ1,0,−Δ3) -- 13.2.5 Sample f(x,y,z) at the Node 5 Defined by the Coordinates (0,0,−Δ3) -- 13.2.6 Sample f(x,y,z) at the Node 6 Defined by the Coordinates (Δ1,0,−Δ3) -- 13.2.7 Sample f(x,y,z) at the Node 7 Defined by the Coordinates (−Δ1,Δ2,−Δ3) -- 13.2.8 Sample f(x,y,z) at the Node 8 Defined by the Coordinates (0,Δ2,−Δ3) -- 13.2.9 Sample f; (x,y,z) at the Node 9 Defined by the Coordinates (Δ1,Δ2,−Δ3) -- 13.2.10 Sample f(x,y,z) at the Node 10 Defined by the Coordinates (−Δ1,−Δ2,0) -- 13.2.11 Sample f(x,y,z) at the Node 11 Defined by the Coordinates (0,−Δ2,0).; 13.2.12 Sample f(x,y,z) at the Node 12 Defined by the Coordinates (Δ1,−Δ2,0).
• Sommario/riassunto: "The use of different types of transforms are very common in engineering. Typically, responses of linear systems are described and analyzed using various transforms and inverse transforms. Discrete transforms and inverse transforms are useful when such responses are analyzed using computer algorithms. Transform techniques allow researchers and designers to operate in transformed domains which ordinarily facilitate the work substantially. After the work has been completed the results must be transformed back into the original real domain. Ordinarily carrying out the inverse transform presents a challenge."--