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INTRODUCTION1

Functionals1

General linear formulas5

The effect of error in input7

The use of probability8

Efficient approximation9

Minimal response to error．Variance10

CHAPTER 1．FUNCTIONALS IN TERMS OF DERIVATIVES11

The spaces Cn，Cn，V of functions11

The space ？﹡n of functionals13

A standard form for elements of ？﹡n-114

Figure 1．Step functions and their integrals16

Inequalities19

Symmetry and skew symmetry23

Functionals that vanish for degree n-125

An approximation of ∫1 -1 x(s) ds26

An approximation of ∫1 0 x(s) ds/√s31

An approximation of the derivative x1(s) at s=1/431

Linear interpolation32

A theorem on convex families of functions33

CHAPTER 2．APPLICATIONS36

Part 1．Integrals36

Introduction．Best formulas36

The approximation of ∫m 0 x by c0x(0) +．．．+cmx(m)51

Best and nearly best integration formulas53

Derivation of the formulas61

m even65

m odd71

A formula of Gaussian type involving two ordinates74

Approximations of Gaussian type80

Approximations that involve derivatives81

Stepwise solution of differential equations82

Part 2．Values of functions84

Conventional interpolation with distinct arguments84

Conventional interpolation with at most q coincident arguments88

Interpolation90

Special formulas and best narrow formulas93

Broad interpolation104

Interpolations that use derivatives107

Part 3．Derivatives111

Conventional approximate differentiation111

Best formulas111

Part 4．Sums116

An instance116

CHAPTER 3．LINEAR CONTINUOUS FUNCTIONALS ON Cn118

Normed linear spaces118

Additive operators122

The adjoint space125

Riesz＇s Theorem127

The space C﹡n138

An illustration144

The spaces Z and Z﹡145

Taylor operators151

The operator δ154

The spaces Bn and Kn155

CHAPTER 4．FUNCTIONALS IN TERMS OF PARTIAL DERIVATIVES160

The space Bp，q160

Taylor＇s formula162

The spaces ？﹡p，q and Bp，q170

The spaces B181

The spaces ？﹡ and B194

Symmetry202

Appraisals203

Figures and tables206

CHAPTER 5．APPLICATIONS214

Approximation of an integral in terms of its integrand at the center of mass214

Circular domain of integration221

Use of several values of the integrand224

Use of derivatives of the integrand226

A functional not in ？﹡ unless n is large229

Circular domain and partial derivatives230

An interpolation232

Double linear interpolation232

An approximate differentiation233

CHAPTER 6．LINEAR CONTINUOUS FUNCTIONALS ON B，Z，K240

The norm in B240

Riesz＇s Theorem242

The space B﹡246

The space ？﹡249

Illustration252

The operators δs and δt256

The spaces Z and Z﹡256

The space K262

The norm in K265

The space K﹡266

K﹡ as a subspace of B﹡269

CHAPTER 7．FUNCTIONS OF m VARIABLES271

The space B271

The full core ф274

The norm in B278

Functions of bounded variation279

The space B﹡279

The space ？﹡280

The spaces Z and Z﹡281

The covered core281

The retracted core ρ and the space K284

The norm in K286

The space K﹡287

The space ？﹡288

K﹡ as a subspace of B﹡289

Illustration290

Figures and tables295

CHAPTER 8．FACTORS OF OPERATORS300

Banach spaces301

Baire＇s Theorem303

The inverse of a linear continuous map305

The factor space X/X0308

The quotient theorem310

Import thereof313

An instance in which U=Dns314

Related instances315

U a linear homogeneous differential operator316

Approximation of a function by a solution of a linear homogeneous differential equation316

A trigonometric approximation322

An instance in which U involves difference operators324

Functions of several variables325

Use and design of machines326

CHAPTER 9．EFFICIENT AND STRONGLY EFFICIENT APPROXIMATION328

Norms based on integrals328

Inner product spaces．Hilbert spaces329

Orthogonality331

L2-spaces and other function spaces333

The Pythagorean theorem and approximation337

Bases．Fourier coefficients340

Projections344

Orthogonalization345

Elementary harmonic analysis of a derivative349

The direct sum of two Hilbert spaces353

The direct product of two Hilbert spaces354

Direct products and function spaces358

Matrices369

Probability spaces370

Extension of operators to direct products372

The general problem of approximation377

Efficient and strongly efficient approximation382

Digression on unbiased approximation385

Characterization of efficient operators385

Calculation of operators near efficiency389

Conditions that L°=L390

The subspaces Mt392

Characterization of strongly efficient operators393

A sufficient condition for strong efficiency400

Applications404

Smoothing of one observation410

Weak efficiency414

Approximation based on a table of contaminated values415

Use of the nearest tabular entry419

Use of the two nearest tabular entries421

Estimation of the pertinent stochastic processes423

Stationary data430

The roles of the correlations in ψ432

The spaces Mt434

The normal equation438

A calculation441

CHAPTER 10．MINIMAL RESPONSE TO ERROR443

Minimal response among unbiased approximations443

Minimal operators and projections446

Linear continuous functionals on a Hilbert space．Adjoint operators451

Operators of finite Schmidt norm454

Trace458

Nonnegative operators．Square roots458

The variance of δx460

Variance and inner product468

Minimality in terms of variance469

A digression on statistical estimation473

Partitioned form474

Minimizing sequences477

The approximation of x by A(x+δx)480

Illustration481

Least square approximation491

Existence thereof493

The choice of a suitable weight494

CHAPTER 11．THE STEP FUNCTIONS θ AND ψ496

Introduction496

Formulas of integration497

CHAPTER 12．STIELTJES INTEGRALS，INTEGRATION BY PARTS，FUNCTIONS OF BOUNDED VARIATION500

The integral ∫ā ɑx(s) df(s)500

Increasing functions503

Functions of bounded variation503

Integration relative to a function of bounded variation506

∣f∣-null sets509

The Lebesgue integral510

Absolutely continuous masses514

m-fold integrals515

Intervals and increments519

The extension of a function521

Entirely increasing functions521

Functions of bounded variation524

The variations of f526

Integration relative to a function of bounded variation529

The Lebesgue double integral533

Absolutely continuous masses534

BIBLIOGRAPHY535

INDEX AND LIST OF SYMBOLS539