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Discrete Taylor Transform and Inverse Transform
Discrete Taylor Transform and Inverse Transform
Autore Baghai-Wadji Alireza
Edizione [1st ed.]
Pubbl/distr/stampa Newark : , : John Wiley & Sons, Incorporated, , 2024
Descrizione fisica 1 online resource (0 pages)
Disciplina 515/.723
Soggetto topico Integral transforms
ISBN 9781394240081
1394240082
9781394240098
1394240090
9781394240104
1394240104
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
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˜> -- , |x˜> -- , |x˜2> -- , |x˜3> -- , and |x˜4> -- , and the Difference Operators |D(0)> -- , |D(1)> -- , and |D(2)>.
4.2.7 Establishing Relationships Between the Dual Frame Vectors |1˜> -- , |x˜> -- , |x˜2> -- , |x˜3> -- , |x˜4> -- , |x˜5> -- , and |x˜6> -- , and the Difference Operators |D(0)> -- , |D(1)> -- , and |D(2)> -- -- 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˜> -- -- 5.3.1.2 On the Construction of |x˜> -- -- 5.3.1.3 On the Construction of |x˜2> -- -- 5.3.1.4 On the Construction of |x˜3> -- -- 5.3.1.5 On the Construction of |x˜4> -- -- 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)> -- -- 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& -- 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)> -- Defined by Df(n)& -- equals -- < -- D(n)|F> -- (n& -- equals -- 0,1,2).
8.6 Exterior Products |xn> -- < -- D(n)| (n& -- equals -- 0,1,2) and the Resolution of Identity (Property 1) -- 8.7 Inner Products < -- xn|D(n)> -- & -- equals -- δmn (m,n& -- 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> -- with Their Bra Counterpart < -- xn| -- 8.9.2 The Exterior Product of the Ket |1> -- with Its Bra Counterpart -- 8.9.3 The Exterior Product of the Ket |x> -- with Its Bra Counterpart -- 8.9.4 The Exterior Product of the Ket |x2> -- with Its Bra Counterpart -- 8.9.5 The S‐Matrix and Its Properties -- 8.9.6 Calculation of < -- D(0)| Utilizing S−1 and the Position Bra < -- x(0)| -- 8.9.7 Calculation of < -- D(1)| Utilizing S−1 and the Position Bra < -- x(1)| -- 8.9.8 Calculation of < -- D(2)| Utilizing S−1 and the Position Bra < -- 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 < -- D(n)| to f(n)(0) (n& -- 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)> -- Defined by Df(n)& -- equals -- < -- D(n)|F> -- -- 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> -- with its Dual Bra Counterpart -- 9.8.2 The Exterior Product of the Ket |x> -- with its Dual Bra Counterpart -- 9.8.3 The Exterior Product of the Ket |x2> -- with its Dual Bra Counterpart -- 9.8.4 The Exterior Product of the Ket |x3> -- with its Dual Bra Counterpart -- 9.8.5 The Exterior Product of the Ket |x4> -- 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)> -- -- 11.8.3 The Expression of |D(5)> -- -- 11.8.4 The Expression of |D(4)> -- -- 11.8.5 The Expression of |D(3)> -- -- 11.8.6 The Expression of |D(2)> -- -- 11.8.7 The Expression of |D(1)>.
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& -- equals -- 0,1,2) -- 12.2.1 On the Construction of |1> -- -- 12.2.2 On the Construction of |x> -- -- 12.2.3 On the Construction of |y> -- -- 12.2.4 On the Construction of |x2> -- -- 12.2.5 On the Construction of |xy> -- -- 12.2.6 On the Construction of |y2> -- -- 12.2.7 On the Construction of |x2y> -- -- 12.2.8 On the Construction of |xy2> -- -- 12.2.9 On the Construction of |x2y2> -- -- 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& -- 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).
Record Nr. UNINA-9911018812903321
Baghai-Wadji Alireza  
Newark : , : John Wiley & Sons, Incorporated, , 2024
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Mathematical Quantum Physics for Engineers and Technologists : Governing Equations
Mathematical Quantum Physics for Engineers and Technologists : Governing Equations
Autore Baghai-Wadji Alireza
Edizione [1st ed.]
Pubbl/distr/stampa Stevenage : , : Institution of Engineering & Technology, , 2023
Descrizione fisica 1 online resource (516 pages)
Collana Electromagnetic Waves Series
Soggetto topico Quantum theory
Engineering mathematics
ISBN 1-83724-453-7
1-5231-6325-9
1-83953-869-4
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Contents -- About the author -- Preface -- 1. Quantum harmonic oscillator and allied topics -- 2. QHO revisited—the Method of Frobenius -- 3. The hydrogen atom -- 4. Governing equations -- 5. Feynman path integral -- 6. Quantum electrodynamics -- Index
Record Nr. UNINA-9911007173303321
Baghai-Wadji Alireza  
Stevenage : , : Institution of Engineering & Technology, , 2023
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui