**Mathematical Surveys and Monographs**

Volume: 9;
1963;
544 pp;
Softcover

MSC: Primary 41;

Print ISBN: 978-0-8218-1509-0

Product Code: SURV/9

List Price: $86.00

Individual Member Price: $68.80

**Electronic ISBN: 978-1-4704-1237-1
Product Code: SURV/9.E**

List Price: $86.00

Individual Member Price: $68.80

# Linear Approximation

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*Arthur Sard*

Many approximations are linear, that is, conform to the principle of super-position, and may profitably be studied by means of the theory of linear spaces. “Linear approximation” sets forth the pertinent parts of that theory, with particular attention to the key spaces \(C_n, B, K\), and Hilbert space, spaces which are powerful tools in the analysis, appraisal, and design of approximations, ranging from formulas of mechanical quadrature to approximations of operators by operators.

Because it affords a detailed treatment of a timely and important subject, “Linear approximation” is of interest to scientists and engineers as well as to mathematicians. The book includes worked illustrative examples and discussions of the rationale of its formulation of problems.

#### Table of Contents

# Table of Contents

## Linear Approximation

Table of Contents pages: 1 2

- TABLE OF CONTENTS vii8 free
- INTRODUCTION 114 free
- CHAPTER 1. FUNCTIONALS IN TERMS OF DERIVATIVES 1124
- The spaces C[sub(n)], C[sub(n)], V of functions 1124
- The space l*[sub(n)] of functionals 1326
- A standard form for elements of l*[sub(n-1)] 1427
- Figure 1. Step functions and their integrals 1629
- Inequalities 1932
- Symmetry and skew symmetry 2336
- Functionals that vanish for degree n – 1 2538
- An approximation of ∫[sup(1)][sub(-1)] x(s) ds 2639
- An approximation of ∫[sup(1)][sub(0)] x{s) ds|√s 3144
- An approximation of the derivative x[sub(1)](s) at s = 1/4 3144
- Linear interpolation 3245
- A theorem on convex families of functions 3346

- CHAPTER 2. APPLICATIONS 3649
- Part 1. Integrals 3649
- Introduction. Best formulas 3649
- The approximation of ∫[sup(m)][sub(0)] x by c[sub(0)]x(0) +ƒ+ c[sub(m)]x(m) 5164
- Best and nearly best integration formulas 5366
- Derivation of the formulas 6174
- m even 6578
- m odd 7184
- A formula of Gaussian type involving two ordinates 7487
- Approximations of Gaussian type 8093
- Approximations that involve derivatives 8194
- Stepwise solution of differential equations 8295

- Part 2. Values of functions 8497
- Part 3. Derivatives 111124
- Part 4. Sums 116129

- CHAPTER 3. LINEAR CONTINUOUS FUNCTIONALS ON C[sub(n)] 118131
- CHAPTER 4. FUNCTIONALS IN TERMS OF PARTIAL DERIVATIVES 160173
- CHAPTER 5. APPLICATIONS 214227
- Approximation of an integral in terms of its integrand at the center of mass 214227
- Circular domain of integration 221234
- Use of several values of the integrand 224237
- Use of derivatives of the integrand 226239
- A functional not in K* unless n is large 229242
- Circular domain and partial derivatives 230243
- An interpolation 232245
- Double linear interpolation 232245
- An approximate differentiation 233246

- CHAPTER 6. LINEAR CONTINUOUS FUNCTIONALS ON B, Z, K 240253
- CHAPTER 7. FUNCTIONS OF m VARIABLES 271284
- The space B 271284
- The full core φ 274287
- The norm in B 278291
- Functions of bounded variation 279292
- The space B* 279292
- The space B* 280293
- The spaces Z and Z* 281294
- The covered core 281294
- The retracted core ρ and the space K 284297
- The norm in K 286299
- The space K* 287300
- The space K* 288301
- K* as a subspace of B* 289302
- Illustration 290303
- Figures and tables 295308

- CHAPTER 8. FACTORS OF OPERATORS 300313
- Banach spaces 301314
- Baire's Theorem 303316
- The inverse of a linear continuous map 305318
- The factor space X/X[sub(0)] 308321
- The quotient theorem 310323
- Import thereof 313326
- An instance in which U = D[sup(n)][sub(s)] 314327
- Related instances 315328
- U a linear homogeneous differential operator 316329
- Approximation of a function by a solution of a linear homogeneous differential equation 316329
- A trigonometric approximation 322335
- An instance in which U involves difference operators 324337
- Functions of several variables 325338
- Use and design of machines 326339

- CHAPTER 9. EFFICIENT AND STRONGLY EFFICIENT APPROXIMATION 328341
- Norms based on integrals 328341
- Inner product spaces. Hilbert spaces 329342
- Orthogonality 331344
- L[sup(2)]-spaces and other function spaces 333346
- The Pythagorean theorem and approximation 337350
- Bases. Fourier coefficients 340353
- Projections 344357
- Orthogonalization 345358
- Elementary harmonic analysis of a derivative 349362
- The direct sum of two Hilbert spaces 353366
- The direct product of two Hilbert spaces 354367
- Direct products and function spaces 358371
- Matrices 369382
- Probability spaces 370383
- Extension of operators to direct products 372385
- The general problem of approximation 377390
- Efficient and strongly efficient approximation 382395
- Digression on unbiased approximation 385398
- Characterization of efficient operators 385398
- Calculation of operators near efficiency 389402
- Conditions that L[sup(ο)] = L 390403
- The subspaces M[sub(t)] 392405
- Characterization of strongly efficient operators 393406
- A sufficient condition for strong efficiency 400413
- Applications 404417
- Smoothing of one observation 410423
- Weak efficiency 414427
- Approximation based on a table of contaminated values 415428
- Use of the nearest tabular entry 419432
- Use of the two nearest tabular entries 421434
- Estimation of the pertinent stochastic processes 423436
- Stationary data 430443
- The roles of the correlations in Ψ 432445
- The spaces M[sub(t)] 434447
- The normal equation 438451
- A calculation 441454

- CHAPTER 10. MINIMAL RESPONSE TO ERROR 443456
- Minimal response among unbiased approximations 443456
- Minimal operators and projections 446459
- Linear continuous functionals on a Hilbert space. Adjoint operators 451464
- Operators of finite Schmidt norm 454467
- Trace 458471
- Nonnegative operators. Square roots 458471
- The variance of δx 460473
- Variance and inner product 468481
- Minimality in terms of variance 469482
- A digression on statistical estimation 473486
- Partitioned form 474487
- Minimizing sequences 477490
- The approximation of x by A(x + δx) 480493
- Illustration 481494
- Least square approximation 491504
- Existence thereof 493506
- The choice of a suitable weight 494507

- CHAPTER 11. THE STEP FUNCTIONS θ AND ψ 496509
- CHAPTER 12. STIELTJES INTEGRALS, INTEGRATION BY PARTS, FUNCTIONS OF BOUNDED VARIATION 500513
- The integral ∫[sup(a)][sub(a)] x(s) df(s) 500513
- Increasing functions 503516
- Functions of bounded variation 503516
- Integration relative to a function of bounded variation 506519
- |f|-null sets 509522
- The Lebesgue integral 510523
- Absolutely continuous masses 514527
- m-fold integrals 515528
- Intervals and increments 519532
- The extension of a function 521534
- Entirely increasing functions 521534
- Functions of bounded variation 524537
- The variations of f 526539
- Integration relative to a function of bounded variation 529542
- The Lebesgue double integral 533546
- Absolutely continuous masses 534547

- BIBLIOGRAPHY 535548
- INDEX AND LIST OF SYMBOLS 539554

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