Approximation on a rectangular grid

I: The Primitive Functions.- §1. The Variational-Difference Method.- §2. An Example.- §3. The Basic Properties of Variational-Difference Matrices.- §4. Primitive Functions and Coordinate Functions.- §5. Interpolatory Properties of Primitive Systems of Functions.- II: Completeness and Fundamental Completeness Conditions.- §1. Approximation of Smooth Functions.- §2. Extensions of Functions.- §3. Completeness in Sobolev Spaces.- §4. On the Minimum Number of Primitive Functions.- §5. The Necessity of the Fundamental Completeness Conditions.- §6. One-Dimensional Primitive Systems.- §7. Primitive Systems of Higher Dimensions with Zero Degree.- §8. Primitive Systems with m = s = 2.- §9. Product Primitive Systems.- III: Order of Approximation.- §1. Order of Approximation using the Uniform Norm.- §2. On the Averaging of Functions.- §3. The Order of Approximation for Sobolev Spaces.- §4. Estimation of the Constants for the Simplest Case.- §5. Approximation Using Product Primitive Functions.- §6. Strengthened Fundamental Completeness Conditions.- §7. Some General Considerations.- §8. A More General Class of Primitive Systems.- IV: Primitive Functions with Wide Support.- §1. Definitions.- §2. Fundamental Completeness Conditions for One-Dimensional Systems.- §3. Example: The Parabolic Approximation.- §4. Fundamental Completeness Conditions for Systems of Arbitrary Dimension.- V: Approximation in One-Dimensional Degenerate Norms.- §1. The Formulation of the Problem.- §2. On the Completeness of Coordinate Systems Which are Complete with Respect to Non-Degenerate Norms.- §3. Equations of Second Order with Weak Degeneracy.- §4. The Case 1 ? ? ? 2.- §5. Properties of the Solution.- §6. Improved Estimates.- §7. The Case ? ? 2.- §8. More GeneralEquations.- §9. Approximation in L2.- §10. Other Boundary Conditions.- VI: Some Degenerate Two-Dimensional Norms.- §1. Approximations for Radially-Symmetric Grids.- §2. Estimation of the First Integral.- §3. Estimation of the Second Integral.- §4. The Class C(2,?).- §5. Approximation on Lp and C.- §6. Degenerate Second Order Elliptic Equations.- VII: Approximation of Eigenvalues.- §1. On the Order of the Largest Approximate Eigenvalue. Formulation of the Problem.- §2. The Rayleigh-Ritz Process.- §3. One-Dimensional Variational-Difference Processes.- §4. The Case of Several Variables.- §5. Error Estimate for Fixed Eigenvalues.- VIII: Construction of Variational Difference Equations.- §1. First Boundary Value Problems: Equations with Constant Coefficients on a Cube.- §2. First Boundary Value Problems: Equations with Variable Coefficients on a Cube.- §3. First Boundary Value Problems: Natural Boundary Conditions.- §4. First Boundary Value Problems: Approximation of the Boundary Conditions.- §5. Variational-Difference Methods on an Axial-Symmetric Grid.- §6. Variational-Difference Schemes Containing a Boundary Layer: The One-Dimensional Situation.- §7. Variational-Difference Schemes Containing a Boundary Layer: The Multidimensional Situation.- §8. Non-Self Adjoint Problems.- IX: Error Estimates for the Variational-Difference Method.- §1. On the Stability of Numerical Processes.- §2. The Stability of Variational-Difference Processes - The One-Dimensional Problem.- §3. The Stability of Variational-Difference Processes - Multi-Dimensional Problems.- §4. The Stability of Variational-Difference Processes - Eigenvalue Problems.- §5. On the Condition Number of the Variational-Difference Matrix.- §6. The Case of Arbitrary Domains and ArbitraryBoundary Conditions.- §7. Numerical Example - A Degenerate Second Order Ordinary Differential Equation.- X: The Euler-Maclaurin Sum Formula.- §1. A New Derivation of the Euler-Maclaurin Sum Formula.- §2. A Related Euler-Maclaurin Sum Formula.- §3. An Euler-Maclaurin Sum Formula for the Multidimensional Cube.- §4. Integration Over a Ball.- XI: On Integral Equations.- §1. Approximation of the Kernel and Resolvent.- §2. The Accuracy of the Approximation.- §3. Rounding Error Accumulation - Absolute Estimates.- §4. Rounding Error Accumulation - Probabilistic Estimates.- §5. Integral Equations Which Can be Solved by Iteration.- §6. Some Additional Notes.- §7. Equations with Weak Singularities.- §8. Integral Equations of Heat Conduction.- References.