INTRODUCTION 1
Functionals 1
General linear formulas 5
The effect of error in input 7
The use of probability 8
Efficient approximation 9
Minimal response to error．Variance 10
CHAPTER 1．FUNCTIONALS IN TERMS OF DERIVATIVES 11
The spaces Cn，Cn，V of functions 11
The space ？﹡n of functionals 13
A standard form for elements of ？﹡n-1 14
Figure 1．Step functions and their integrals 16
Inequalities 19
Symmetry and skew symmetry 23
Functionals that vanish for degree n-1 25
An approximation of ∫1 -1 x(s) ds 26
An approximation of ∫1 0 x(s) ds/√s 31
An approximation of the derivative x1(s) at s=1/4 31
Linear interpolation 32
A theorem on convex families of functions 33
CHAPTER 2．APPLICATIONS 36
Part 1．Integrals 36
Introduction．Best formulas 36
The approximation of ∫m 0 x by c0x(0) +．．．+cmx(m) 51
Best and nearly best integration formulas 53
Derivation of the formulas 61
m even 65
m odd 71
A formula of Gaussian type involving two ordinates 74
Approximations of Gaussian type 80
Approximations that involve derivatives 81
Stepwise solution of differential equations 82
Part 2．Values of functions 84
Conventional interpolation with distinct arguments 84
Conventional interpolation with at most q coincident arguments 88
Interpolation 90
Special formulas and best narrow formulas 93
Broad interpolation 104
Interpolations that use derivatives 107
Part 3．Derivatives 111
Conventional approximate differentiation 111
Best formulas 111
Part 4．Sums 116
An instance 116
CHAPTER 3．LINEAR CONTINUOUS FUNCTIONALS ON Cn 118
Normed linear spaces 118
Additive operators 122
The adjoint space 125
Riesz＇s Theorem 127
The space C﹡n 138
An illustration 144
The spaces Z and Z﹡ 145
Taylor operators 151
The operator δ 154
The spaces Bn and Kn 155
CHAPTER 4．FUNCTIONALS IN TERMS OF PARTIAL DERIVATIVES 160
The space Bp，q 160
Taylor＇s formula 162
The spaces ？﹡p，q and Bp，q 170
The spaces B 181
The spaces ？﹡ and B 194
Symmetry 202
Appraisals 203
Figures and tables 206
CHAPTER 5．APPLICATIONS 214
Approximation of an integral in terms of its integrand at the center of mass 214
Circular domain of integration 221
Use of several values of the integrand 224
Use of derivatives of the integrand 226
A functional not in ？﹡ unless n is large 229
Circular domain and partial derivatives 230
An interpolation 232
Double linear interpolation 232
An approximate differentiation 233
CHAPTER 6．LINEAR CONTINUOUS FUNCTIONALS ON B，Z，K 240
The norm in B 240
Riesz＇s Theorem 242
The space B﹡ 246
The space ？﹡ 249
Illustration 252
The operators δs and δt 256
The spaces Z and Z﹡ 256
The space K 262
The norm in K 265
The space K﹡ 266
K﹡ as a subspace of B﹡ 269
CHAPTER 7．FUNCTIONS OF m VARIABLES 271
The space B 271
The full core ф 274
The norm in B 278
Functions of bounded variation 279
The space B﹡ 279
The space ？﹡ 280
The spaces Z and Z﹡ 281
The covered core 281
The retracted core ρ and the space K 284
The norm in K 286
The space K﹡ 287
The space ？﹡ 288
K﹡ as a subspace of B﹡ 289
Illustration 290
Figures and tables 295
CHAPTER 8．FACTORS OF OPERATORS 300
Banach spaces 301
Baire＇s Theorem 303
The inverse of a linear continuous map 305
The factor space X/X0 308
The quotient theorem 310
Import thereof 313
An instance in which U=Dns 314
Related instances 315
U a linear homogeneous differential operator 316
Approximation of a function by a solution of a linear homogeneous differential equation 316
A trigonometric approximation 322
An instance in which U involves difference operators 324
Functions of several variables 325
Use and design of machines 326
CHAPTER 9．EFFICIENT AND STRONGLY EFFICIENT APPROXIMATION 328
Norms based on integrals 328
Inner product spaces．Hilbert spaces 329
Orthogonality 331
L2-spaces and other function spaces 333
The Pythagorean theorem and approximation 337
Bases．Fourier coefficients 340
Projections 344
Orthogonalization 345
Elementary harmonic analysis of a derivative 349
The direct sum of two Hilbert spaces 353
The direct product of two Hilbert spaces 354
Direct products and function spaces 358
Matrices 369
Probability spaces 370
Extension of operators to direct products 372
The general problem of approximation 377
Efficient and strongly efficient approximation 382
Digression on unbiased approximation 385
Characterization of efficient operators 385
Calculation of operators near efficiency 389
Conditions that L°=L 390
The subspaces Mt 392
Characterization of strongly efficient operators 393
A sufficient condition for strong efficiency 400
Applications 404
Smoothing of one observation 410
Weak efficiency 414
Approximation based on a table of contaminated values 415
Use of the nearest tabular entry 419
Use of the two nearest tabular entries 421
Estimation of the pertinent stochastic processes 423
Stationary data 430
The roles of the correlations in ψ 432
The spaces Mt 434
The normal equation 438
A calculation 441
CHAPTER 10．MINIMAL RESPONSE TO ERROR 443
Minimal response among unbiased approximations 443
Minimal operators and projections 446
Linear continuous functionals on a Hilbert space．Adjoint operators 451
Operators of finite Schmidt norm 454
Trace 458
Nonnegative operators．Square roots 458
The variance of δx 460
Variance and inner product 468
Minimality in terms of variance 469
A digression on statistical estimation 473
Partitioned form 474
Minimizing sequences 477
The approximation of x by A(x+δx) 480
Illustration 481
Least square approximation 491
Existence thereof 493
The choice of a suitable weight 494
CHAPTER 11．THE STEP FUNCTIONS θ AND ψ 496
Introduction 496
Formulas of integration 497
CHAPTER 12．STIELTJES INTEGRALS，INTEGRATION BY PARTS，FUNCTIONS OF BOUNDED VARIATION 500
The integral ∫ā ɑx(s) df(s) 500
Increasing functions 503
Functions of bounded variation 503
Integration relative to a function of bounded variation 506
∣f∣-null sets 509
The Lebesgue integral 510
Absolutely continuous masses 514
m-fold integrals 515
Intervals and increments 519
The extension of a function 521
Entirely increasing functions 521
Functions of bounded variation 524
The variations of f 526
Integration relative to a function of bounded variation 529
The Lebesgue double integral 533
Absolutely continuous masses 534
BIBLIOGRAPHY 535
INDEX AND LIST OF SYMBOLS 539