Approximation methods obtained by using the umbral calculus

Introduction

1. Unidimensional umbral calculus
   1.1. Translation invariant operators and binomial sequences
   1.2. Sheffer sequences
   l.3. Umbral operators and umbral composition
   l.4. Umbral translation operators and operatorial formulas
   1.5. Other versions of umbral calculus
   1.6. Classical umbral calculus
2. Multidimensional umbral calculus
   2.1. Finite dimensional umbral calculus
   • 2.1.1.Binomial sequences
     2.1.2. Umbral translation operators
     2.1.3. Explicit formulas for basic sequences
     2.1.4. Examples of binomial sequences
     2.1.5. Sheffer sequences
     2.1.6. Bi-indexed Sheffer sequences
     2.2. Invariant umbral calculus
     2.2.1. Generalized polynomials and quasi-polynomials
     2.2.2. Binomial sequences
     2.2.3. Invariant orthogonal quasi-polynomials and invariant orthogonal operators
     2.2.4. Examples of invariant binomial sequences
     2.2.5. Invariant Appell and Sheffer sequences

3. Binomial approximation operators
   3.1. Binomial approximation operators of T. Popoviciu type
   • 3.1.1. Definition and convergence
     3.1.2. Examples of binomial operators
     3.1.3. D. Stancu operator
     3.1.4. Evaluation of approximation orders
     3.1.5. Generalized operators of T. Popoviciu type
     3.2.  Modified binomial operators
     3.3. Bivariate binomial operators
4. Approximation operators constructed using Sheffer sequences
   4.1. Definition and convergence
   4.2. Examples
   4.3. Evaluation of approximation orders
   4.4. A Kantorovich generalization
   4.5. Lipschitz constants for operators5. Compound operators
   5.1. Compound binomial operators of D.D. Stancu type
   5.1.1. Definition and convergence
   5.1.2. Evaluation of remainder
   5.2. Compound operators depending on s parameters
   5.2.1. Definition and convergence
   5.2.2. Examples
   5.2.3. An integral representation of remainder

   6. Another class of approximation operators
   6.1. Definition and convergence
   6.2. Special cases
   6.3. Evaluation of approximation orders using modulus of continuity
   6.4. D. Stancu operator depending on many parameters

   7. Sheffer sequences, probabililty distributions and approximation operators
   7.1. Results for the standard umbral calculus
   7.2. Generalized umbral calculus
   7.2.1 q- umbral calculus
   7.2.2. Hyperbolic umbral calculus
   References

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